List of space groups

There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point groups of the unit cell.

Symbols

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

• P primitive
• I body centered (from the German "Innenzentriert")
• F face centered (from the German "Flächenzentriert")
• A centered on A faces only
• B centered on B faces only
• C centered on C faces only
• R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

• $a$, $b$, or $c$ glide translation along half the lattice vector of this face
• $n$ glide translation along with half a face diagonal
• $d$ glide planes with translation along a quarter of a face diagonal.
• $e$ two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis is noted by a number, n, where the angle of rotation is $\color{Black}\tfrac{360^\circ}{n}$. The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 2₁ is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 3₁ is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector.

The possible screw axis are: 2₁, 3₁, 3₂, 4₁, 4₂, 4₃, 6₁, 6₂, 6₃, 6₄, and 6₅.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group. It is related to the order in which Shoenflies derived space groups.

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups. Symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm ($\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$): the symmorphic space groups are P4/mmm ($P\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$, 36s) and I4/mmm ($I\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$, 37s); hemisymmorphic space groups should contain axial combination 422, these are P4/mcc ($P\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{c}$, 35h), P4/nbm ($P\tfrac{4}{n}\tfrac{2}{b}\tfrac{2}{m}$, 36h), P4/nnc ($P\tfrac{4}{n}\tfrac{2}{n}\tfrac{2}{c}$, 37h), and I4/mcm ($I\tfrac{4}{m}\tfrac{2}{c}\tfrac{2}{m}$, 38h).

List of Triclinic

Triclinic Bravais lattice

Triclinic crystal system

Number	Point group	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
1	1	P1	P 1	$C_1^1$	1s	$(a/b/c)\cdot 1$
2	1	P1	P 1	$C_i^1$	2s	$(a/b/c)\cdot \tilde 2$

List of Monoclinic

Monoclinic Bravais lattice

Simple (P)	Base (C)

Monoclinic crystal system

Number	Point group	Short name	Full name(s)		Schoenflies	Fedorov	Shubnikov	Fibrifold
3	2	P2	P 1 2 1	P 1 1 2	$C_2^1$	3s	$(b:(c/a)):2$
4	2	P21	P 1 21 1	P 1 1 21	$C_2^2$	1a	$(b:(c/a)):2_1$
5	2	C2	C 1 2 1	B 1 1 2	$C_2^3$	4s	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) :2$
6	m	Pm	P 1 m 1	P 1 1 m	$C_s^1$	5s	$(b:(c/a))\cdot m$
7	m	Pc	P 1 c 1	P 1 1 b	$C_s^2$	1h	$(b:(c/a))\cdot \tilde c$
8	m	Cm	C 1 m 1	B 1 1 m	$C_s^3$	6s	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m$
9	m	Cc	C 1 c 1	B 1 1 b	$C_s^4$	2h	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c$
10	2/m	P2/m	P 1 2/m 1	P 1 1 2/m	$C_{2h}^1$	7s	$(b:(c/a))\cdot m:2$
11	2/m	P21/m	P 1 21/m 1	P 1 1 21/m	$C_{2h}^2$	2a	$(b:(c/a))\cdot m:2_1$
12	2/m	C2/m	C 1 2/m 1	B 1 1 2/m	$C_{2h}^3$	8s	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot m:2$
13	2/m	P2/c	P 1 2/c 1	P 1 1 2/b	$C_{2h}^4$	3h	$(b:(c/a))\cdot \tilde c:2$
14	2/m	P21/c	P 1 21/c 1	P 1 1 21/b	$C_{2h}^5$	3a	$(b:(c/a))\cdot \tilde c:2_1$
15	2/m	C2/c	C 1 2/c 1	B 1 1 2/b	$C_{2h}^6$	4h	$\left ( \tfrac{a+b}{2}/b:(c/a)\right ) \cdot \tilde c:2$

List of Orthorhombic

Orthorhombic crystal system

Number	Point group	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
16	222	P222	P 2 2 2	$D_2^1$	9s	$(c:a:b):2:2$
17	222	P2221	P 2 2 21	$D_2^2$	4a	$(c:a:b):2_1:2$
18	222	P21212	P 21 21 2	$D_2^3$	7a	$(c:a:b):2$ $2_1$
19	222	P212121	P 21 21 21	$D_2^4$	8a	$(c:a:b):2_1$ $2_1$
20	222	C2221	C 2 2 21	$D_2^5$	5a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :2_1:2$
21	222	C222	C 2 2 2	$D_2^6$	10s	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :2:2$
22	222	F222	F 2 2 2	$D_2^7$	12s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :2:2$
23	222	I222	I 2 2 2	$D_2^8$	11s	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) :2:2$
24	222	I212121	I 21 21 21	$D_2^9$	6a	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :2:2_1$
25	mm2	Pmm2	P m m 2	$C_{2v}^1$	13s	$(c:a:b):m \cdot 2$
26	mm2	Pmc21	P m c 21	$C_{2v}^2$	9a	$(c:a:b): \tilde c \cdot 2_1$
27	mm2	Pcc2	P c c 2	$C_{2v}^3$	5h	$(c:a:b): \tilde c \cdot 2$
28	mm2	Pma2	P m a 2	$C_{2v}^4$	6h	$(c:a:b): \tilde a \cdot 2$
29	mm2	Pca21	P c a 21	$C_{2v}^5$	11a	$(c:a:b): \tilde a \cdot 2_1$
30	mm2	Pnc2	P n c 2	$C_{2v}^6$	7h	$(c:a:b): \tilde c \odot 2$
31	mm2	Pmn21	P m n 21	$C_{2v}^7$	10a	$(c:a:b): \widetilde{ac} \cdot 2_1$
32	mm2	Pba2	P b a 2	$C_{2v}^8$	9h	$(c:a:b): \tilde a \odot 2$
33	mm2	Pna21	P n a 21	$C_{2v}^9$	12a	$(c:a:b): \tilde a \odot 2_1$
34	mm2	Pnn2	P n n 2	$C_{2v}^{10}$	8h	$(c:a:b): \widetilde{ac} \odot 2$
35	mm2	Cmm2	C m m 2	$C_{2v}^{11}$	14s	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2$
36	mm2	Cmc21	C m c 21	$C_{2v}^{12}$	13a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) :\tilde c \cdot 2_1$
37	mm2	Ccc2	C c c 2	$C_{2v}^{13}$	10h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) : \tilde c \cdot 2$
38	mm2	Amm2	A m m 2	$C_{2v}^{14}$	15s	$\left ( \tfrac{b+c}{2}/c:a:b\right ):m \cdot 2$
39	mm2	Aem2	A b m 2	$C_{2v}^{15}$	11h	$\left ( \tfrac{b+c}{2}/c:a:b\right ) :m \cdot 2_1$
40	mm2	Ama2	A m a 2	$C_{2v}^{16}$	12h	$\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2$
41	mm2	Aea2	A b a 2	$C_{2v}^{17}$	13h	$\left ( \tfrac{b+c}{2}/c:a:b\right ) : \tilde a \cdot 2_1$
42	mm2	Fmm2	F m m 2	$C_{2v}^{18}$	17s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) :m \cdot 2$
43	mm2	Fdd2	F dd2	$C_{2v}^{19}$	16h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b \right ) : \tfrac{1}{2} \widetilde{ac} \odot 2$
44	mm2	Imm2	I m m 2	$C_{2v}^{20}$	16s	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) :m \cdot 2$
45	mm2	Iba2	I b a 2	$C_{2v}^{21}$	15h	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde c \cdot 2$
46	mm2	Ima2	I m a 2	$C_{2v}^{22}$	14h	$\left ( \tfrac{a+b+c}{2}/c:a:b \right ) : \tilde a \cdot 2$
47	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pmmm	P 2/m 2/m 2/m	$D_{2h}^1$	18s	$\left ( c:a:b \right ) \cdot m:2 \cdot m$
48	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pnnn	P 2/n 2/n 2/n	$D_{2h}^2$	19h	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \widetilde{ac}$
49	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pccm	P 2/c 2/c 2/m	$D_{2h}^3$	17h	$\left ( c:a:b \right ) \cdot m:2 \cdot \tilde c$
50	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pban	P 2/b 2/a 2/n	$D_{2h}^4$	18h	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \odot \tilde a$
51	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pmma	P 21/m 2/m 2/a	$D_{2h}^5$	14a	$\left ( c:a:b \right ) \cdot \tilde a :2 \cdot m$
52	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pnna	P 2/n 21/n 2/a	$D_{2h}^6$	17a	$\left ( c:a:b \right ) \cdot \tilde a:2 \odot \widetilde{ac}$
53	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pmna	P 2/m 2/n 21/a	$D_{2h}^7$	15a	$\left ( c:a:b \right ) \cdot \tilde a:2_1 \cdot \widetilde{ac}$
54	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pcca	P 21/c 2/c 2/a	$D_{2h}^8$	16a	$\left ( c:a:b \right ) \cdot \tilde a:2 \cdot \tilde c$
55	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pbam	P 21/b 21/a 2/m	$D_{2h}^9$	22a	$\left ( c:a:b \right ) \cdot m:2 \odot \tilde a$
56	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pccn	P 21/c 21/c 2/n	$D_{2h}^{10}$	27a	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot \tilde c$
57	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pbcm	P 2/b 21/c 21/m	$D_{2h}^{11}$	23a	$\left ( c:a:b \right ) \cdot m:2_1 \odot \tilde c$
58	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pnnm	P 21/n 21/n 2/m	$D_{2h}^{12}$	25a	$\left ( c:a:b \right ) \cdot m:2 \odot \widetilde{ac}$
59	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pmmn	P 21/m 21/m 2/n	$D_{2h}^{13}$	24a	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2 \cdot m$
60	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pbcn	P 21/b 2/c 21/n	$D_{2h}^{14}$	26a	$\left ( c:a:b \right ) \cdot \widetilde{ab}:2_1 \odot \tilde c$
61	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pbca	P 21/b 21/c 21/a	$D_{2h}^{15}$	29a	$\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot \tilde c$
62	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Pnma	P 21/n 21/m 21/a	$D_{2h}^{16}$	28a	$\left ( c:a:b \right ) \cdot \tilde a:2_1 \odot m$
63	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Cmcm	C 2/m 2/c 21/m	$D_{2h}^{17}$	18a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2_1 \cdot \tilde c$
64	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Cmca	C 2/m 2/c 21/a	$D_{2h}^{18}$	19a	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2_1 \cdot \tilde c$
65	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Cmmm	C 2/m 2/m 2/m	$D_{2h}^{19}$	19s	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m$
66	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Cccm	C 2/c 2/c 2/m	$D_{2h}^{20}$	20h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot \tilde c$
67	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Cmme	C 2/m 2/m 2/e	$D_{2h}^{21}$	21h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot m$
68	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Ccce	C 2/c 2/c 2/e	$D_{2h}^{22}$	22h	$\left ( \tfrac{a+b}{2}:c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c$
69	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Fmmm	F 2/m 2/m 2/m	$D_{2h}^{23}$	21s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot m:2 \cdot m$
70	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Fddd	F 2/d 2/d 2/d	$D_{2h}^{24}$	24h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:c:a:b\right ) \cdot \tfrac{1}{2}\widetilde{ab}:2 \odot \tfrac{1}{2}\widetilde{ac}$
71	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Immm	I 2/m 2/m 2/m	$D_{2h}^{25}$	20s	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot m$
72	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Ibam	I 2/b 2/a 2/m	$D_{2h}^{26}$	23h	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot m:2 \cdot \tilde c$
73	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Ibca	I 2/b 2/c 2/a	$D_{2h}^{27}$	21a	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot \tilde c$
74	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	Imma	I 2/m 2/m 2/a	$D_{2h}^{28}$	20a	$\left ( \tfrac{a+b+c}{2}/c:a:b\right ) \cdot \tilde a :2 \cdot m$

List of Tetragonal

Tetragonal crystal system

Number	Point group	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
75	4	P4	P 4	$C_4^1$	22s	$(c:a:a):4$
76	4	P41	P 41	$C_4^2$	30a	$(c:a:a) :4_1$
77	4	P42	P 42	$C_4^3$	33a	$(c:a:a) :4_2$
78	4	P43	P 43	$C_4^4$	31a	$(c:a:a) :4_3$
79	4	I4	I 4	$C_4^5$	23s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4$
80	4	I41	I 41	$C_4^6$	32a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1$
81	4	P4	P 4	$S_4^1$	26s	$(c:a:a):\tilde 4$
82	4	I4	I 4	$S_4^2$	27s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4$
83	4/m	P4/m	P 4/m	$C_{4h}^1$	28s	$(c:a:a)\cdot m:4$
84	4/m	P42/m	P 42/m	$C_{4h}^2$	41a	$(c:a:a)\cdot m:4_2$
85	4/m	P4/n	P 4/n	$C_{4h}^3$	29h	$(c:a:a)\cdot \widetilde{ab}:4$
86	4/m	P42/n	P 42/n	$C_{4h}^4$	42a	$(c:a:a)\cdot \widetilde{ab}:4_2$
87	4/m	I4/m	I 4/m	$C_{4h}^5$	29s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4$
88	4/m	I41/a	I 41/a	$C_{4h}^6$	40a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1$
89	422	P422	P 4 2 2	$D_4^1$	30s	$(c:a:a):4:2$
90	422	P4212	P4212	$D_4^2$	43a	$(c:a:a):4$ $2_1$
91	422	P4122	P 41 2 2	$D_4^3$	44a	$(c:a:a):4_1:2$
92	422	P41212	P 41 21 2	$D_4^4$	48a	$(c:a:a):4_1$ $2_1$
93	422	P4222	P 42 2 2	$D_4^5$	47a	$(c:a:a):4_2:2$
94	422	P42212	P 42 21 2	$D_4^6$	50a	$(c:a:a):4_2$ $2_1$
95	422	P4322	P 43 2 2	$D_4^7$	45a	$(c:a:a):4_3:2$
96	422	P43212	P 43 21 2	$D_4^8$	49a	$(c:a:a):4_3$ $2_1$
97	422	I422	I 4 2 2	$D_4^9$	31s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2$
98	422	I4122	I 41 2 2	$D_4^{10}$	46a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4:2_1$
99	4mm	P4mm	P 4 m m	$C_{4v}^1$	24s	$(c:a:a):4\cdot m$
100	4mm	P4bm	P 4 b m	$C_{4v}^2$	26h	$(c:a:a):4\odot \tilde a$
101	4mm	P42cm	P 42 c m	$C_{4v}^3$	37a	$(c:a:a):4_2\cdot \tilde c$
102	4mm	P42nm	P 42 n m	$C_{4v}^4$	38a	$(c:a:a):4_2\odot \widetilde{ac}$
103	4mm	P4cc	P 4 c c	$C_{4v}^5$	25h	$(c:a:a):4\cdot \tilde c$
104	4mm	P4nc	P 4 n c	$C_{4v}^6$	27h	$(c:a:a):4\odot \widetilde{ac}$
105	4mm	P42mc	P 42 m c	$C_{4v}^7$	36a	$(c:a:a):4_2\cdot m$
106	4mm	P42bc	P 42 b c	$C_{4v}^8$	39a	$(c:a:a):4\odot \tilde a$
107	4mm	I4mm	I 4 m m	$C_{4v}^9$	25s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot m$
108	4mm	I4cm	I 4 c m	$C_{4v}^{10}$	28h	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4\cdot \tilde c$
109	4mm	I41md	I 41 m d	$C_{4v}^{11}$	34a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot m$
110	4mm	I41cd	I 41 c d	$C_{4v}^{12}$	35a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :4_1\odot \tilde c$
111	42m	P42m	P 4 2 m	$D_{2d}^1$	32s	$(c:a:a):\tilde 4 :2$
112	42m	P42c	P 4 2 c	$D_{2d}^2$	30h	$(c:a:a):\tilde 4$ $2$
113	42m	P421m	P 4 21 m	$D_{2d}^3$	52a	$(c:a:a):\tilde 4 \cdot \widetilde{ab}$
114	42m	P421c	P 4 21 c	$D_{2d}^4$	53a	$(c:a:a):\tilde 4 \cdot \widetilde{abc}$
115	42m	P4m2	P 4 m 2	$D_{2d}^5$	33s	$(c:a:a):\tilde 4 \cdot m$
116	42m	P4c2	P 4 c 2	$D_{2d}^6$	31h	$(c:a:a):\tilde 4 \cdot \tilde c$
117	42m	P4b2	P 4 b 2	$D_{2d}^7$	32h	$(c:a:a):\tilde 4 \odot \tilde a$
118	42m	P4n2	P 4 n 2	$D_{2d}^8$	33h	$(c:a:a):\tilde 4 \cdot \widetilde{ac}$
119	42m	I4m2	I 4 m 2	$D_{2d}^9$	35s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot m$
120	42m	I4c2	I 4 c 2	$D_{2d}^{10}$	34h	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \cdot \tilde c$
121	42m	I42m	I 4 2 m	$D_{2d}^{11}$	34s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 :2$
122	42m	I42d	I 4 2 d	$D_{2d}^{12}$	51a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) :\tilde 4 \odot \tfrac{1}{2}\widetilde{abc}$
123	4/m 2/m 2/m	P4/mmm	P 4/m 2/m 2/m	$D_{4h}^1$	36s	$(c:a:a)\cdot m:4\cdot m$
124	4/m 2/m 2/m	P4/mcc	P 4/m 2/c 2/c	$D_{4h}^2$	35h	$(c:a:a)\cdot m:4\cdot \tilde c$
125	4/m 2/m 2/m	P4/nbm	P 4/n 2/b 2/m	$D_{4h}^3$	36h	$(c:a:a)\cdot \widetilde{ab}:4\odot \tilde a$
126	4/m 2/m 2/m	P4/nnc	P 4/n 2/n 2/c	$D_{4h}^4$	37h	$(c:a:a)\cdot \widetilde{ab}:4\odot \widetilde{ac}$
127	4/m 2/m 2/m	P4/mbm	P 4/m 21/b 2/m	$D_{4h}^5$	54a	$(c:a:a)\cdot m:4\odot \tilde a$
128	4/m 2/m 2/m	P4/mnc	P 4/m 21/n 2/c	$D_{4h}^6$	56a	$(c:a:a)\cdot m:4\odot \widetilde{ac}$
129	4/m 2/m 2/m	P4/nmm	P 4/n 21/m 2/m	$D_{4h}^7$	55a	$(c:a:a)\cdot \widetilde{ab}:4\cdot m$
130	4/m 2/m 2/m	P4/ncc	P 4/n 21/c 2/c	$D_{4h}^8$	57a	$(c:a:a)\cdot \widetilde{ab}:4\cdot \tilde c$
131	4/m 2/m 2/m	P42/mmc	P 42/m 2/m 2/c	$D_{4h}^9$	60a	$(c:a:a)\cdot m:4_2\cdot m$
132	4/m 2/m 2/m	P42/mcm	P 42/m 2/c 2/m	$D_{4h}^{10}$	61a	$(c:a:a)\cdot m:4_2\cdot \tilde c$
133	4/m 2/m 2/m	P42/nbc	P 42/n 2/b 2/c	$D_{4h}^{11}$	63a	$(c:a:a)\cdot \widetilde{ab}:4_2\odot \tilde a$
134	4/m 2/m 2/m	P42/nnm	P 42/n 2/n 2/m	$D_{4h}^{12}$	62a	$(c:a:a)\cdot \widetilde{ab}:4_2\odot \widetilde{ac}$
135	4/m 2/m 2/m	P42/mbc	P 42/m 21/b 2/c	$D_{4h}^{13}$	66a	$(c:a:a)\cdot m:4_2\odot \tilde a$
136	4/m 2/m 2/m	P42/mnm	P 42/m 21/n 2/m	$D_{4h}^{14}$	65a	$(c:a:a)\cdot m:4_2\odot \widetilde{ac}$
137	4/m 2/m 2/m	P42/nmc	P 42/n 21/m 2/c	$D_{4h}^{15}$	67a	$(c:a:a)\cdot \widetilde{ab}:4_2\cdot m$
138	4/m 2/m 2/m	P42/ncm	P 42/n 21/c 2/m	$D_{4h}^{16}$	65a	$(c:a:a)\cdot \widetilde{ab}:4_2\cdot \tilde c$
139	4/m 2/m 2/m	I4/mmm	I 4/m 2/m 2/m	$D_{4h}^{17}$	37s	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot m$
140	4/m 2/m 2/m	I4/mcm	I 4/m 2/c 2/m	$D_{4h}^{18}$	38h	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot m:4\cdot \tilde c$
141	4/m 2/m 2/m	I41/amd	I 41/a 2/m 2/d	$D_{4h}^{19}$	59a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot m$
142	4/m 2/m 2/m	I41/acd	I 41/a 2/c 2/d	$D_{4h}^{20}$	58a	$\left ( \tfrac{a+b+c}{2}/c:a:a\right ) \cdot \tilde a :4_1\odot \tilde c$

List of Trigonal

Unit cells for trigonal crystal system

Rhombohedral (R)	Hexagonal (P)

Trigonal crystal system

Number	Point group	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
143	3	P3	P 3	$C_3^1$	38s	$(c:(a/a)):3$
144	3	P31	P 31	$C_3^2$	68a	$(c:(a/a)):3_1$
145	3	P32	P 32	$C_3^3$	69a	$(c:(a/a)):3_2$
146	3	R3	R 3	$C_3^4$	39s	$(a/a/a)/3$
147	3	P3	P 3	$C_{3i}^1$	51s	$(c:(a/a)):\tilde 6$
148	3	R3	R 3	$C_{3i}^2$	52s	$(a/a/a)/\tilde 6$
149	32	P312	P 3 1 2	$D_3^1$	45s	$(c:(a/a)):2:3$
150	32	P321	P 3 2 1	$D_3^2$	44s	$(c:(a/a))\cdot 2:3$
151	32	P3112	P 31 1 2	$D_3^3$	72a	$(c:(a/a)):2:3_1$
152	32	P3121	P 31 2 1	$D_3^4$	70a	$(c:(a/a))\cdot 2:3_1$
153	32	P3212	P 32 1 2	$D_3^5$	73a	$(c:(a/a)):2:3_2$
154	32	P3221	P 32 2 1	$D_3^6$	71a	$(c:(a/a))\cdot 2:3_2$
155	32	R32	R 3 2	$D_3^7$	46s	$(a/a/a)/3:2$
156	3m	P3m1	P 3 m 1	$C_{3v}^1$	40s	$(c:(a/a)):m\cdot 3$
157	3m	P31m	P 3 1 m	$C_{3v}^2$	41s	$(c:(a/a))\cdot m\cdot 3$
158	3m	P3c1	P 3 c 1	$C_{3v}^3$	39h	$(c:(a/a)):\tilde c:3$
159	3m	P31c	P 3 1 c	$C_{3v}^4$	40h	$(c:(a/a))\cdot\tilde c :3$
160	3m	R3m	R 3 m	$C_{3v}^5$	42s	$(a/a/a)/3\cdot m$
161	3m	R3c	R 3 c	$C_{3v}^6$	41h	$(a/a/a)/3\cdot\tilde c$
162	3 2/m	P31m	P 3 1 2/m	$D_{3d}^1$	56s	$(c:(a/a))\cdot m\cdot\tilde 6$
163	3 2/m	P31c	P 3 1 2/c	$D_{3d}^2$	46h	$(c:(a/a))\cdot\tilde c \cdot\tilde 6$
164	3 2/m	P3m1	P 3 2/m 1	$D_{3d}^3$	55s	$(c:(a/a)):m\cdot\tilde 6$
165	3 2/m	P3c1	P 3 2/c 1	$D_{3d}^4$	45h	$(c:(a/a)):\tilde c \cdot\tilde 6$
166	3 2/m	R3m	R 3 2/m	$D_{3d}^5$	57s	$(a/a/a)/\tilde 6 \cdot m$
167	3 2/m	R3c	R 3 2/c	$D_{3d}^6$	47h	$(a/a/a)/\tilde 6 \cdot\tilde c$

List of Hexagonal

Hexagonal lattice cell
(P)

Hexagonal crystal system

Number	Point group	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
168	6	P6	P 6	$C_6^1$	49s	$(c:(a/a)):6$
169	6	P61	P 61	$C_6^2$	74a	$(c:(a/a)):6_1$
170	6	P65	P 65	$C_6^3$	75a	$(c:(a/a)):6_5$
171	6	P62	P 62	$C_6^4$	76a	$(c:(a/a)):6_2$
172	6	P64	P 64	$C_6^5$	77a	$(c:(a/a)):6_4$
173	6	P63	P 63	$C_6^6$	78a	$(c:(a/a)):6_3$
174	6	P6	P 6	$C_{3h}^1$	43s	$(c:(a/a)):3:m$
175	6/m	P6/m	P 6/m	$C_{6h}^1$	53s	$(c:(a/a))\cdot m :6$
176	6/m	P63/m	P 63/m	$C_{6h}^2$	81a	$(c:(a/a))\cdot m :6_3$
177	622	P622	P 6 2 2	$D_6^1$	54s	$(c:(a/a))\cdot 2 :6$
178	622	P6122	P 61 2 2	$D_6^2$	82a	$(c:(a/a))\cdot 2 :6_1$
179	622	P6522	P 65 2 2	$D_6^3$	83a	$(c:(a/a))\cdot 2 :6_5$
180	622	P6222	P 62 2 2	$D_6^4$	84a	$(c:(a/a))\cdot 2 :6_2$
181	622	P6422	P 64 2 2	$D_6^5$	85a	$(c:(a/a))\cdot 2 :6_4$
182	622	P6322	P 63 2 2	$D_6^6$	86a	$(c:(a/a))\cdot 2 :6_3$
183	6mm	P6mm	P 6 m m	$C_{6v}^1$	50s	$(c:(a/a)):m\cdot 6$
184	6mm	P6cc	P 6 c c	$C_{6v}^2$	44h	$(c:(a/a)):\tilde c \cdot 6$
185	6mm	P63cm	P 63 c m	$C_{6v}^3$	80a	$(c:(a/a)):\tilde c \cdot 6_3$
186	6mm	P63mc	P 63 m c	$C_{6v}^4$	79a	$(c:(a/a)):m\cdot 6_3$
187	6m2	P6m2	P 6 m 2	$D_{3h}^1$	48s	$(c:(a/a)):m\cdot 3:m$
188	6m2	P6c2	P 6 c 2	$D_{3h}^2$	43h	$(c:(a/a)):\tilde c \cdot 3:m$
189	6m2	P62m	P 6 2 m	$D_{3h}^3$	47s	$(c:(a/a))\cdot m:3\cdot m$
190	6m2	P62c	P 6 2 c	$D_{3h}^4$	42h	$(c:(a/a))\cdot m:3\cdot \tilde c$
191	6/m 2/m 2/m	P6/mmm	P 6/m 2/m 2/m	$D_{6h}^1$	58s	$(c:(a/a))\cdot m:6\cdot m$
192	6/m 2/m 2/m	P6/mcc	P 6/m 2/c 2/c	$D_{6h}^2$	48h	$(c:(a/a))\cdot m:6\cdot\tilde c$
193	6/m 2/m 2/m	P63/mcm	P 63/m 2/c 2/m	$D_{6h}^3$	87a	$(c:(a/a))\cdot m:6_3\cdot\tilde c$
194	6/m 2/m 2/m	P63/mmc	P 63/m 2/m 2/c	$D_{6h}^4$	88a	$(c:(a/a))\cdot m:6_3\cdot m$

List of Cubic

| | | |}

(221) Caesium chloride. Different colors for the two atom types.

(216) Sphalerite

(223) Weaire–Phelan structure

Cubic crystal system

Number	Point group	Short name	Full name	Schoenflies	Fedorov	Shubnikov	Fibrifold
195	23	P23	P 2 3	$T^1$	59s	$\left ( a:a:a\right ) :2/3$	2^o
196	23	F23	F 2 3	$T^2$	61s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :2/3$	1^o
197	23	I23	I 2 3	$T^3$	60s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2/3$	4^oo
198	23	P213	P 21 3	$T^4$	89a	$\left ( a:a:a\right ) :2_1/3$	1^o/4
199	23	I213	I 21 3	$T^5$	90a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :2_1/3$	2^o/4
200	2/m 3	Pm3	P 2/m 3	$T_h^1$	62s	$\left ( a:a:a\right ) \cdot m/ \tilde 6$	4^−
201	2/m 3	Pn3	P 2/n 3	$T_h^2$	49h	$\left ( a:a:a\right ) \cdot \widetilde{ab} / \tilde 6$	4^+o
202	2/m 3	Fm3	F 2/m 3	$T_h^3$	64s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot m/ \tilde 6$	2^−
203	2/m 3	Fd3	F 2/d 3	$T_h^4$	50h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) \cdot \tfrac{1}{2}\widetilde{ab} / \tilde 6$	2^+o
204	2/m 3	Im3	I 2/m 3	$T_h^5$	63s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot m/\tilde 6$	8^−o
205	2/m 3	Pa3	P 21/a 3	$T_h^6$	91a	$\left ( a:a:a\right ) \cdot \tilde a /\tilde 6$	2^−/4
206	2/m 3	Ia3	I 21/a 3	$T_h^7$	92a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) \cdot \tilde a /\tilde 6$	4^−/4
207	432	P432	P 4 3 2	$O^1$	68s	$\left ( a:a:a\right ) :4/3$	4^−o
208	432	P4232	P 42 3 2	$O^2$	98a	$\left ( a:a:a\right ) :4_2//3$	4^+
209	432	F432	F 4 3 2	$O^3$	70s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/3$	2^−o
210	432	F4132	F 41 3 2	$O^4$	97a	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//3$	2^+
211	432	I432	I 4 3 2	$O^5$	69s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/3$	8^+o
212	432	P4332	P 43 3 2	$O^6$	94a	$\left ( a:a:a\right ) :4_3//3$	2^+/4
213	432	P4132	P 41 3 2	$O^7$	95a	$\left ( a:a:a\right ) :4_1//3$	2^+/4
214	432	I4132	I 41 3 2	$O^8$	96a	$\left ( \tfrac{a+b+c}{2}/:a:a:a\right ) :4_1//3$	4^+/4
215	43m	P43m	P 4 3 m	$T_d^1$	65s	$\left ( a:a:a\right ) :\tilde 4 /3$	2^o:2
216	43m	F43m	F 4 3 m	$T_d^2$	67s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 /3$	1^o:2
217	43m	I43m	I 4 3 m	$T_d^3$	66s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 /3$	4^o:2
218	43m	P43n	P 4 3 n	$T_d^4$	51h	$\left ( a:a:a\right ) :\tilde 4 //3$	4^o
219	43m	F43c	F 4 3 c	$T_d^5$	52h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :\tilde 4 //3$	2^oo
220	43m	I43d	I 4 3 d	$T_d^6$	93a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :\tilde 4 //3$	4^o/4
221	4/m 3 2/m	Pm3m	P 4/m 3 2/m	$O_h^1$	71s	$\left ( a:a:a\right ) :4/\tilde 6 \cdot m$	4^−:2
222	4/m 3 2/m	Pn3n	P 4/n 3 2/n	$O_h^2$	53h	$\left ( a:a:a\right ) :4/\tilde 6 \cdot \widetilde{abc}$	8^oo
223	4/m 3 2/m	Pm3n	P 42/m 3 2/n	$O_h^3$	102a	$\left ( a:a:a\right ) :4_2//\tilde 6 \cdot \widetilde{abc}$	8^o
224	4/m 3 2/m	Pn3m	P 42/n 3 2/m	$O_h^4$	103a	$\left ( a:a:a\right ) :4_2//\tilde 6 \cdot m$	4^+:2
225	4/m 3 2/m	Fm3m	F 4/m 3 2/m	$O_h^5$	73s	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot m$	2^−:2
226	4/m 3 2/m	Fm3c	F 4/m 3 2/c	$O_h^6$	54h	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4/\tilde 6 \cdot \tilde c$	4^−−
227	4/m 3 2/m	Fd3m	F 41/d 3 2/m	$O_h^7$	100a	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot m$	2^+:2
228	4/m 3 2/m	Fd3c	F 41/d 3 2/c	$O_h^8$	101a	$\left ( \tfrac{a+c}{2}/\tfrac{b+c}{2}/\tfrac{a+b}{2}:a:a:a\right ) :4_1//\tilde 6 \cdot \tilde c$	4^++
229	4/m 3 2/m	Im3m	I 4/m 3 2/m	$O_h^9$	72s	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4/\tilde 6 \cdot m$	8^o:2
230	4/m 3 2/m	Ia3d	I 41/a 3 2/d	$O_h^{10}$	99a	$\left ( \tfrac{a+b+c}{2}/a:a:a\right ) :4_1//\tilde 6 \cdot \tfrac{1}{2}\widetilde{abc}$	8^o/4

External links

• International Union of Crystallography
• Point Groups and Bravais Lattices
• Full list of 230 crystallographic space groups
• Interactive 3D visualization of all 230 crystallographic space groups