Space Group → Bravais Lattice, Point Group

The space group of a crystal uniquely determines the Bravais lattice and the point group. The point group does not uniquely determine the Bravais lattice.

Space Group	Bravais lattice	Point group International / Schoenflies
1 P1	Triclinic	1	C₁
2 P1	Triclinic	1	S₂ = C_i
3 P2	Simple Monoclinic	2	C₂
4 P2₁	Simple Monoclinic	2	C₂
5 C2	Base-Centered Monoclinic	2	C₂
6 Pm	Simple Monoclinic	m	C_1h = C_s
7 Pc	Simple Monoclinic	m	C_1h = C_s
8 Cm	Base-Centered Monoclinic	m	C_1h = C_s
9 Cc	Base-Centered Monoclinic	m	C_1h = C_s
10 P2/m	Simple Monoclinic	$\frac{2}{m}$	C_2h
11 P2₁/m	Simple Monoclinic	$\frac{2}{m}$	C_2h
12 C2/m	Base-Centered Monoclinic	$\frac{2}{m}$	C_2h
13 P2/c	Simple Monoclinic	$\frac{2}{m}$	C_2h
14 P2₁/c	Simple Monoclinic	$\frac{2}{m}$	C_2h
15 C2/c	Base-Centered Monoclinic	$\frac{2}{m}$	C_2h
16 P222	Simple Orthorhombic	222	V = D₂
17 P222₁	Simple Orthorhombic	222	V = D₂
18 P2₁2₁2	Simple Orthorhombic	222	V = D₂
19 P2₁2₁2₁	Simple Orthorhombic	222	V = D₂
20 C222₁	Base-Centered Orthorhombic	222	V = D₂
21 C222	Base-Centered Orthorhombic	222	V = D₂
22 F222	Face-Centered Orthorhombic	222	V = D₂
23 I222	Body-Centered Orthorhombic	222	V = D₂
24 I2₁2₁2₁	Body-Centered Orthorhombic	222	V = D₂
25 Pmm2	Simple Orthorhombic	mm2	C_2v
26 Pmc2₁	Simple Orthorhombic	mm2	C_2v
27 Pcc2	Simple Orthorhombic	mm2	C_2v
28 Pma2	Simple Orthorhombic	mm2	C_2v
29 Pca2₁	Simple Orthorhombic	mm2	C_2v
30 Pnc2	Simple Orthorhombic	mm2	C_2v
31 Pmn2₁	Simple Orthorhombic	mm2	C_2v
32 Pba2	Simple Orthorhombic	mm2	C_2v
33 Pna2₁	Simple Orthorhombic	mm2	C_2v
34 Pnn2	Simple Orthorhombic	mm2	C_2v
35 Cmm2	Base-Centered Orthorhombic	mm2	C_2v
36 Cmc2₁	Base-Centered Orthorhombic	mm2	C_2v
37 Ccc2	Base-Centered Orthorhombic	mm2	C_2v
38 Amm2	Base-Centered Orthorhombic	mm2	C_2v
39 Aem2	Base-Centered Orthorhombic	mm2	C_2v
40 Ama2	Base-Centered Orthorhombic	mm2	C_2v
41 Aea2	Base-Centered Orthorhombic	mm2	C_2v
42 Fmm2	Face-Centered Orthorhombic	mm2	C_2v
43 Fdd2	Face-Centered Orthorhombic	mm2	C_2v
44 Imm2	Body-Centered Orthorhombic	mm2	C_2v
45 Iba2	Body-Centered Orthorhombic	mm2	C_2v
46 Ima2	Body-Centered Orthorhombic	mm2	C_2v
47 Pmmm	Simple Orthorhombic	mmm	V_h = D_2h
48 Pnnn	Simple Orthorhombic	mmm	V_h = D_2h
49 Pccm	Simple Orthorhombic	mmm	V_h = D_2h
50 Pban	Simple Orthorhombic	mmm	V_h = D_2h
51 Pmma	Simple Orthorhombic	mmm	V_h = D_2h
52 Pnna	Simple Orthorhombic	mmm	V_h = D_2h
53 Pmna	Simple Orthorhombic	mmm	V_h = D_2h
54 Pcca	Simple Orthorhombic	mmm	V_h = D_2h
55 Pbam	Simple Orthorhombic	mmm	V_h = D_2h
56 Pccn	Simple Orthorhombic	mmm	V_h = D_2h
57 Pbcm	Simple Orthorhombic	mmm	V_h = D_2h
58 Pnnm	Simple Orthorhombic	mmm	V_h = D_2h
59 Pmmn	Simple Orthorhombic	mmm	V_h = D_2h
60 Pbcn	Simple Orthorhombic	mmm	V_h = D_2h
61 Pbca	Simple Orthorhombic	mmm	V_h = D_2h
62 Pnma	Simple Orthorhombic	mmm	V_h = D_2h
63 Cmcm	Base-Centered Orthorhombic	mmm	V_h = D_2h
64 Cmce	Base-Centered Orthorhombic	mmm	V_h = D_2h
65 Cmmm	Base-Centered Orthorhombic	mmm	V_h = D_2h
66 Cccm	Base-Centered Orthorhombic	mmm	V_h = D_2h
67 Cmme	Base-Centered Orthorhombic	mmm	V_h = D_2h
68 Ccce	Base-Centered Orthorhombic	mmm	V_h = D_2h
69 Fmmm	Face-Centered Orthorhombic	mmm	V_h = D_2h
70 Fddd	Face-Centered Orthorhombic	mmm	V_h = D_2h
71 Immm	Body-Centered Orthorhombic	mmm	V_h = D_2h
72 Ibam	Body-Centered Orthorhombic	mmm	V_h = D_2h
73 Ibca	Body-Centered Orthorhombic	mmm	V_h = D_2h
74 Imma	Body-Centered Orthorhombic	mmm	V_h = D_2h
75 P4	Simple Tetragonal	4	C₄
76 P4₁	Simple Tetragonal	4	C₄
77 P4₂	Simple Tetragonal	4	C₄
78 P4₃	Simple Tetragonal	4	C₄
79 I4	Body-Centered Tetragonal	4	C₄
80 I4₁	Body-Centered Tetragonal	4	C₄
81 P4	Simple Tetragonal	4	S₄
82 I4	Body-Centered Tetragonal	4	S₄
83 P4/m	Simple Tetragonal	$\frac{4}{m}$	C_4h
84 P4₂/m	Simple Tetragonal	$\frac{4}{m}$	C_4h
85 P4/n	Simple Tetragonal	$\frac{4}{m}$	C_4h
86 P4₂/n	Simple Tetragonal	$\frac{4}{m}$	C_4h
87 I4/m	Body-Centered Tetragonal	$\frac{4}{m}$	C_4h
88 I4₁/a	Body-Centered Tetragonal	$\frac{4}{m}$	C_4h
89 P422	Simple Tetragonal	422	D₄
90 P42₁2	Simple Tetragonal	422	D₄
91 P4₁22	Simple Tetragonal	422	D₄
92 P4₁2₁2	Simple Tetragonal	422	D₄
93 P4₂22	Simple Tetragonal	422	D₄
94 P4₂2₁2	Simple Tetragonal	422	D₄
95 P4₃22	Simple Tetragonal	422	D₄
96 P4₃2₁2	Simple Tetragonal	422	D₄
97 I422	Body-Centered Tetragonal	422	D₄
98 I4₁22	Body-Centered Tetragonal	4mm	C_4v
99 P4mm	Simple Tetragonal	4mm	C_4v
100 P4bm	Simple Tetragonal	4mm	C_4v
101 P4₂cm	Simple Tetragonal	4mm	C_4v
102 P4₂nm	Simple Tetragonal	4mm	C_4v
103 P4cc	Simple Tetragonal	4mm	C_4v
104 P4nc	Simple Tetragonal	4mm	C_4v
105 P4₂mc	Simple Tetragonal	4mm	C_4v
106 P4₂bc	Simple Tetragonal	4mm	C_4v
107 I4mm	Body-Centered Tetragonal	4mm	C_4v
108 I4cm	Body-Centered Tetragonal	4mm	C_4v
109 I4₁md	Body-Centered Tetragonal	4mm	C_4v
110 I4₁cd	Body-Centered Tetragonal	4mm	C_4v
111 P42m	Simple Tetragonal	42m	V_d = D_2d
112 P42c	Simple Tetragonal	42m	V_d = D_2d
113 P42₁m	Simple Tetragonal	42m	V_d = D_2d
114 P42₁c	Simple Tetragonal	42m	V_d = D_2d
115 P4m2	Simple Tetragonal	42m	V_d = D_2d
116 P4c2	Simple Tetragonal	42m	V_d = D_2d
117 P4b2	Simple Tetragonal	42m	V_d = D_2d
118 P4n2	Simple Tetragonal	42m	V_d = D_2d
119 I4m2	Body-Centered Tetragonal	42m	V_d = D_2d
120 I4c2	Body-Centered Tetragonal	42m	V_d = D_2d
121 I42m	Body-Centered Tetragonal	42m	V_d = D_2d
122 I42d	Body-Centered Tetragonal	42m	V_d = D_2d
123 P4/mmm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
124 P4/mcc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
125 P4/nbm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
126 P4/nnc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
127 P4/mbm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
128 P4/mnc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
129 P4/nmm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
130 P4/ncc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
131 P4₂/mmc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
132 P4₂/mcm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
133 P4₂/nbc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
134 P4₂/nnm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
135 P4₂/mbc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
136 P4₂/mnm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
137 P4₂/nmc	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
138 P4₂/ncm	Simple Tetragonal	$\frac{4}{m}mm$	D_4h
139 I4/mmm	Body-Centered Tetragonal	$\frac{4}{m}mm$	D_4h
140 I4/mcm	Body-Centered Tetragonal	$\frac{4}{m}mm$	D_4h
141 I4₁/amd	Body-Centered Tetragonal	$\frac{4}{m}mm$	D_4h
142 I4₁/acd	Body-Centered Tetragonal	$\frac{4}{m}mm$	D_4h
143 P3	Hexagonal	3	C₃
144 P3₁	Hexagonal	3	C₃
145 P3₂	Hexagonal	3	C₃
146 R3	Rhombohedral	3	C₃
147 P3	Hexagonal	3	S₆ = C_6i
148 R3	Rhombohedral	3	S₆ = C_6i
149 P312	Hexagonal	32	D₃
150 P321	Hexagonal	32	D₃
151 P3₁12	Hexagonal	32	D₃
152 P3₁21	Hexagonal	32	D₃
153 P3₂12	Hexagonal	32	D₃
154 P3₂21	Hexagonal	32	D₃
155 R32	Rhombohedral	32	D₃
156 P3m1	Hexagonal	3m	C_3v
157 P31m	Hexagonal	3m	C_3v
158 P3c1	Hexagonal	3m	C_3v
159 P31c	Hexagonal	3m	C_3v
160 R3m	Rhombohedral	3m	C_3v
161 R3c	Rhombohedral	3m	C_3v
162 P31m	Hexagonal	3m	D_3d
163 P31c	Hexagonal	3m	D_3d
164 P3m1	Hexagonal	3m	D_3d
165 P3c1	Hexagonal	3m	D_3d
166 R3m	Rhombohedral	3m	D_3d
167 R3c	Rhombohedral	3m	D_3d
168 P6	Hexagonal	6	C₆
169 P6₁	Hexagonal	6	C₆
170 P6₅	Hexagonal	6	C₆
171 P6₂	Hexagonal	6	C₆
172 P6₄	Hexagonal	6	C₆
173 P6₃	Hexagonal	6	C₆
174 P6	Hexagonal	6	C_3h = S₃
175 P6/m	Hexagonal	$\frac{6}{m}$	C_6h
176 P6₃/m	Hexagonal	$\frac{6}{m}$	C_6h
177 P622	Hexagonal	622	D₆
178 P6₁22	Hexagonal	622	D₆
179 P6₅22	Hexagonal	622	D₆
180 P6₂22	Hexagonal	622	D₆
181 P6₄22	Hexagonal	622	D₆
182 P6₃22	Hexagonal	622	D₆
183 P6mm	Hexagonal	6mm	C_6v
184 P6cc	Hexagonal	6mm	C_6v
185 P6₃cm	Hexagonal	6mm	C_6v
186 P6₃mc	Hexagonal	6mm	C_6v
187 P6m2	Hexagonal	6m2	D_3h
188 P6c2	Hexagonal	6m2	D_3h
189 P62m	Hexagonal	6m2	D_3h
190 P62c	Hexagonal	6m2	D_3h
191 P6/mmm	Hexagonal	$\frac{6}{m}mm$	D_6h
192 P6/mcc	Hexagonal	$\frac{6}{m}mm$	D_6h
193 P6₃/mcm	Hexagonal	$\frac{6}{m}mm$	D_6h
194 P6₃/mmc	Hexagonal	$\frac{6}{m}mm$	D_6h
195 P23	Simple Cubic	23	T
196 F23	Face-Centered Cubic	23	T
197 I23	Body-Centered Cubic	23	T
198 P2₁3	Simple Cubic	23	T
199 I2₁3	Body-Centered Cubic	23	T
200 Pm3	Simple Cubic	m3	T_h
201 Pn3	Simple Cubic	m3	T_h
202 Fm3	Face-Centered Cubic	m3	T_h
203 Fd3	Face-Centered Cubic	m3	T_h
204 Im3	Body-Centered Cubic	m3	T_h
205 Pa3	Simple Cubic	m3	T_h
206 Ia3	Body-Centered Cubic	m3	T_h
207 P432	Simple Cubic	432	O
208 P4₂32	Simple Cubic	432	O
209 F432	Face-Centered Cubic	432	O
210 F4₁32	Face-Centered Cubic	432	O
211 I432	Body-Centered Cubic	432	O
212 P4₃32	Simple Cubic	432	O
213 P4₁32	Simple Cubic	432	O
214 I4₁32	Body-Centered Cubic	432	O
215 P43m	Simple Cubic	43m	T_d
216 F43m	Face-Centered Cubic	43m	T_d
217 I43m	Body-Centered Cubic	43m	T_d
218 P43n	Simple Cubic	43m	T_d
219 F43c	Face-Centered Cubic	43m	T_d
220 I43d	Body-Centered Cubic	43m	T_d
221 Pm3m	Simple Cubic	m3m	O_h
222 Pn3n	Simple Cubic	m3m	O_h
223 Pm3n	Simple Cubic	m3m	O_h
224 Pn3m	Simple Cubic	m3m	O_h
225 Fm3m	Face-Centered Cubic	m3m	O_h
226 Fm3c	Face-Centered Cubic	m3m	O_h
227 Fd3m	Face-Centered Cubic	m3m	O_h
228 Fd3c	Face-Centered Cubic	m3m	O_h
229 Im3m	Body-Centered Cubic	m3m	O_h
230 Ia3d	Body-Centered Cubic	m3m	O_h

P - Primitive
I - Body centered (Innenzentriert)
F - Face centered
C - Centered on C faces
A - Centered on A faces
R - Rhombohedral