Subgroups — #72232011

Order 1024.

639 conjugacy classes of subgroups.

The subgroup lattice has 285 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512761	C₂	1	1	yes	maximal, characteristic
2	#483508	C₂	1	2	yes	maximal, complemented
3	#151085	C₂	1	2	yes	maximal, complemented
4	#151084	C₂	1	2	yes	maximal, complemented
5	#151083	C₂	1	4	yes	maximal, complemented
6	C₂×#7463	C₂	1	1	yes	maximal, characteristic, complemented
7	#150063	C₂	1	2	yes	maximal, complemented
8	#149496	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#2262	C₄	1	1	yes	characteristic
10	C₄×#508	C₂²	1	2	yes	—
11	#17857	—	2	2	no	—
12	#13654	C₄	1	1	yes	characteristic
13	#13654	C₂²	1	1	yes	characteristic
14	#13653	C₄	1	1	yes	characteristic
15	#13653	C₂²	1	1	yes	characteristic
16	#13652	C₄	1	2	yes	—
17	#13652	C₂²	1	2	yes	—
18	#13633	C₄	1	1	yes	characteristic
19	#13615	C₄	1	2	yes	—
20	C₄×M₆₄	C₂²	1	1	yes	characteristic
21	C₂×#1397	C₂²	1	1	yes	characteristic
22	#12045	C₂²	1	4	yes	—
23	#12044	C₂²	1	4	yes	—
24	C₈×SD₃₂	—	2	4	no	—
25	C₂×#1372	C₂²	1	2	yes	—
26	#11592	—	2	2	no	—
27	#11399	C₂²	1	1	yes	characteristic
28	#7463	C₂²	1	2	yes	—
29	#7463	C₂²	1	2	yes	—
30	#7386	C₂²	1	4	yes	—
31	#7385	C₂²	1	4	yes	—
32	#7330	C₂²	1	2	yes	—
33	#7330	C₂²	1	2	yes	—
34	C₂×#508	C₂×C₄	1	2	yes	—
35	#2988	—	4	2	no	—
36	C₄×#135	—	2	2	no	—
37	#2262	C₈	1	8	yes	complemented
38	C₂×Q₆₄	C₂×C₄	1	1	yes	characteristic
39	C₂×SD₆₄	C₂×C₄	1	2	yes	—
40	C₂×D₃₂	C₂×C₄	1	1	yes	characteristic
41	C₂×M₆₄	—	2	1	no	—
42	C₂²×C₃₂	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₃₂	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₃₂	C₂×C₄	1	2	yes	—
45	C₄×Q₃₂	C₂³	1	2	yes	—
46	C₄×SD₃₂	—	2	2	no	—
47	C₄×SD₃₂	—	2	2	no	—
48	C₄×D₁₆	C₂×C₄	1	2	yes	—
49	C₄×D₁₆	C₂³	1	2	yes	—
50	#2158	C₂×C₄	1	2	yes	—
51	#2138	—	2	2	no	—
52	C₄×M₃₂	D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₆	C₂³	1	1	yes	characteristic
54	SD₁₆⋊C₈	—	4	4	no	—
55	C₈×Q₁₆	—	2	4	no	—
56	C₈×D₈	—	2	4	no	—
57	C₂×#335	—	2	2	no	—
58	#1477	—	4	2	no	—
59	#1450	D₄	1	1	yes	characteristic
60	#1424	C₂×C₄	1	1	yes	characteristic
61	#1424	C₂×C₄	1	1	yes	characteristic
62	#1423	C₂×C₄	1	2	yes	—
63	#1416	C₂×C₄	1	2	yes	—
64	#1416	C₂×C₄	1	2	yes	—
65	#1415	C₂×C₄	1	2	yes	—
66	#1415	C₂×C₄	1	2	yes	—
67	#1397	C₂×C₄	1	1	yes	characteristic
68	#1397	C₂×C₄	1	1	yes	characteristic
69	#1397	C₂³	1	2	yes	—
70	#1397	C₂³	1	1	yes	characteristic
71	#1397	C₂³	1	1	yes	characteristic
72	C₄×C₃₂	—	2	1	no	—
73	#1372	—	2	2	no	—
74	#1372	C₂³	1	4	yes	—
75	#1332	—	2	4	no	—
76	#1331	—	2	4	no	—
77	#1311	C₂³	1	2	yes	—
78	C₈×C₁₆	D₄	1	2	yes	—
79	C₂×#135	—	2	2	no	—
80	C₄×#41	—	4	2	no	—
81	#508	—	2	4	no	—
82	#508	C₂×C₈	1	8	yes	complemented
83	C₂×Q₃₂	C₂²×C₄	1	2	yes	—
84	C₂×SD₃₂	—	2	2	no	—
85	C₂×D₁₆	C₂²×C₄	1	2	yes	—
86	C₂×M₃₂	—	2	1	no	—
87	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₁₆	#41	1	1	yes	characteristic
89	C₈×Q₈	—	4	4	no	—
90	C₄×Q₁₆	—	2	2	no	—
91	C₄×Q₁₆	—	2	2	no	—
92	C₄×SD₁₆	—	4	2	no	—
93	C₄×SD₁₆	—	4	2	no	—
94	C₄×D₈	—	2	2	no	—
95	C₄×D₈	—	2	2	no	—
96	C₈×D₄	—	4	4	no	—
97	C₈×D₄	—	8	4	no	—
98	#431	—	8	2	no	—
99	#431	D₈	1	2	yes	—
100	#427	—	2	2	no	—
101	C₂×C₄⋊C₈	—	4	2	no	—
102	C₂×C₄⋊C₈	—	8	2	no	—
103	#416	—	4	2	no	—
104	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
105	Q₆₄	C₂×C₈	1	4	yes	complemented
106	SD₆₄	C₂×C₈	1	8	yes	complemented
107	D₃₂	C₂×C₈	1	4	yes	complemented
108	M₆₄	—	2	4	no	—
109	C₂×C₃₂	—	2	2	no	—
110	C₂×C₃₂	—	2	2	no	—
111	C₂×C₃₂	C₂×C₈	1	4	yes	complemented
112	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
116	#367	—	2	2	no	—
117	#366	C₂²×C₄	1	2	yes	—
118	#366	C₂²×C₄	1	2	yes	—
119	#358	—	2	2	no	—
120	#358	—	2	2	no	—
121	#357	—	2	2	no	—
122	#357	—	2	2	no	—
123	#346	—	2	1	no	—
124	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
125	C₄×C₁₆	C₂×D₄	1	2	yes	—
126	C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
127	C₄×C₁₆	#41	1	1	yes	characteristic
128	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
129	#335	—	2	4	no	—
130	#335	—	4	2	no	—
131	Q₈⋊C₈	—	4	4	no	—
132	D₄⋊C₈	—	4	4	no	—
133	#322	D₈	1	2	yes	—
134	C₈²	C₂×D₄	1	2	yes	—
135	C₂×#41	—	4	2	no	—
136	#135	—	2	8	no	—
137	#135	—	4	4	no	—
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×SD₁₆	—	4	2	no	—
140	C₂×D₈	—	2	2	no	—
141	C₂²×C₈	—	8	4	no	—
142	C₂²×C₈	—	8	2	no	—
143	C₂²×C₈	C₄×D₄	1	1	yes	characteristic
144	C₂²×C₈	#135	1	1	yes	characteristic
145	C₄×Q₈	—	4	2	no	—
146	C₄×Q₈	—	4	2	no	—
147	C₄×D₄	—	4	2	no	—
148	C₄×D₄	—	4	2	no	—
149	C₄×D₄	—	8	2	no	—
150	C₄×D₄	—	8	2	no	—
151	#117	—	4	2	no	—
152	#117	—	8	2	no	—
153	C₂×C₄²	—	8	2	no	—
154	C₂×C₄²	C₂×D₈	1	1	yes	characteristic
155	Q₃₂	—	2	2	no	—
156	Q₃₂	C₂²×C₈	1	4	yes	—
157	SD₃₂	—	2	8	no	—
158	D₁₆	—	2	2	no	—
159	D₁₆	C₂²×C₈	1	4	yes	—
160	M₃₂	—	4	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	no	—
163	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
164	C₂×C₁₆	C₄×D₄	1	1	yes	characteristic
165	C₂×C₁₆	C₂²×C₈	1	2	yes	—
166	C₂×C₁₆	#131	1	2	yes	—
167	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
168	C₂×C₁₆	C₂×#41	1	1	yes	characteristic
169	#107	—	2	2	no	—
170	#107	—	2	2	no	—
171	#106	—	4	2	no	—
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	4	4	no	—
174	C₄⋊C₈	—	8	4	no	—
175	C₄⋊C₈	—	8	2	no	—
176	#103	—	4	2	no	—
177	#103	—	4	2	no	—
178	#102	—	4	2	no	—
179	#102	—	4	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	8	4	no	—
182	C₄×C₈	—	8	4	no	—
183	C₄×C₈	C₄×D₄	1	1	yes	characteristic
184	C₄×C₈	C₂×D₈	1	4	yes	—
185	C₄×C₈	C₂×D₈	1	1	yes	characteristic
186	C₄×C₈	#135	1	1	yes	characteristic
187	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
188	C₃₂	—	2	4	no	—
189	C₃₂	—	2	1	no	—
190	C₃₂	—	2	1	no	—
191	C₃₂	C₂²×C₈	1	2	yes	—
192	C₃₂	C₂²×C₈	1	2	yes	—
193	#41	—	4	8	no	—
194	#41	—	8	4	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	8	2	no	—
198	C₂²×C₄	—	8	2	no	—
199	C₂²×C₄	—	8	2	no	—
200	C₂²×C₄	—	8	2	no	—
201	C₂²×C₄	C₄×D₈	1	1	yes	characteristic
202	C₂²×C₄	#509	1	2	yes	—
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	4	2	no	—
205	SD₁₆	—	4	8	no	—
206	D₈	—	2	4	no	—
207	D₈	—	4	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	8	4	no	—
211	C₂×C₈	—	32	4	no	—
212	C₂×C₈	—	32	2	no	—
213	C₂×C₈	C₈×D₄	1	2	yes	—
214	C₂×C₈	C₄×D₈	1	1	yes	characteristic
215	C₂×C₈	C₄×D₈	1	1	yes	characteristic
216	C₂×C₈	#443	1	2	yes	—
217	C₂×C₈	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂×C₈	C₂×#135	1	1	yes	characteristic
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	8	2	no	—
222	C₄⋊C₄	—	8	2	no	—
223	C₂²⋊C₄	—	8	2	no	—
224	C₂²⋊C₄	—	8	2	no	—
225	C₂²⋊C₄	—	8	2	no	—
226	C₄²	—	8	2	no	—
227	C₄²	—	8	2	no	—
228	C₄²	—	8	2	no	—
229	C₄²	C₄×D₈	1	1	yes	characteristic
230	C₄²	#509	1	2	yes	—
231	C₄²	C₂²×D₈	1	1	yes	characteristic
232	C₁₆	—	2	1	no	—
233	C₁₆	—	2	1	no	—
234	C₁₆	—	4	2	no	—
235	C₁₆	C₈×D₄	1	2	yes	—
236	C₁₆	C₂³×C₈	1	1	yes	characteristic
237	C₁₆	C₂×#131	1	1	yes	characteristic
238	C₂³	—	8	2	no	—
239	C₂³	#2177	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	8	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	8	8	no	—
244	D₄	—	8	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	—	8	8	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	16	2	no	—
251	C₂×C₄	—	16	2	no	—
252	C₂×C₄	—	32	2	no	—
253	C₂×C₄	C₈×D₈	1	2	yes	—
254	C₂×C₄	#2177	1	2	yes	—
255	C₂×C₄	#2177	1	1	yes	characteristic
256	C₂×C₄	#2179	1	2	yes	—
257	C₂×C₄	C₂×C₄×D₈	1	1	yes	characteristic
258	C₂×C₄	C₂×#509	1	2	yes	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	4	4	no	—
262	C₈	—	16	4	no	—
263	C₈	C₈×D₈	1	2	yes	—
264	C₈	C₂×C₈×D₄	1	1	yes	characteristic
265	C₈	C₂×#443	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	16	2	no	—
269	C₂²	—	16	2	no	—
270	C₂²	#12045	1	2	yes	—
271	C₂²	C₂×#2177	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	8	4	no	—
275	C₄	—	16	2	no	—
276	C₄	—	16	2	no	—
277	C₄	#12045	1	2	yes	—
278	C₄	C₂×C₈×D₈	1	1	yes	characteristic
279	C₄	C₂×#2179	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	16	4	no	—
282	C₂	C₂×#12045	1	1	yes	characteristic
283	C₂	#512779	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#2177
commutator	C₁₆	C₂³×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#2262	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ #11399 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #483508	C₂	semidirect
3. #151085	C₂	semidirect
4. #151084	C₂	semidirect
5. #151083	C₂	semidirect
6. C₂×#7463	C₂	semidirect
7. #150063	C₂	semidirect
8. #149496	C₂	semidirect
37. #2262	C₈	semidirect
82. #508	C₂×C₈	semidirect
105. Q₆₄	C₂×C₈	semidirect
106. SD₆₄	C₂×C₈	semidirect
107. D₃₂	C₂×C₈	semidirect
111. C₂×C₃₂	C₂×C₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#512761	2
#483508	2
#151085	2
#151084	2
#151083	2
C₂×#7463	2
#150063	2
#149496	2