Subgroups — #72231905

Order 1024.

719 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512731	C₂	1	1	yes	maximal, characteristic
2	#512545	C₂	1	1	yes	maximal, characteristic
3	#479917	C₂	1	1	yes	maximal, characteristic
4	#149509	C₂	1	4	yes	maximal, complemented
5	#149508	C₂	1	4	yes	maximal, complemented
6	#149501	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#7332	C₂	1	1	yes	maximal, characteristic, complemented
8	#149496	C₂	1	1	yes	maximal, characteristic, complemented
9	#149495	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#2258	C₄	1	2	yes	—
11	#33420	C₄	1	1	yes	characteristic
12	C₄×#504	C₂²	1	1	yes	characteristic
13	#17335	C₄	1	1	yes	characteristic
14	#13641	C₄	1	4	yes	—
15	#13640	C₂²	1	4	yes	—
16	#13628	C₄	1	2	yes	—
17	#13554	C₄	1	2	yes	—
18	#13549	C₄	1	4	yes	—
19	#13549	C₄	1	2	yes	—
20	C₄×M₆₄	C₂²	1	2	yes	—
21	C₄×M₆₄	C₂²	1	1	yes	characteristic
22	C₄×M₆₄	C₂²	1	1	yes	characteristic
23	C₂×#1397	C₂²	1	2	yes	—
24	C₂×#1397	C₂²	1	1	yes	characteristic
25	C₂×#1397	C₂²	1	1	yes	characteristic
26	#11409	C₄	1	4	yes	—
27	#11408	C₂²	1	4	yes	—
28	#11402	C₄	1	1	yes	characteristic
29	#11401	C₄	1	1	yes	characteristic
30	#11399	C₂²	1	1	yes	characteristic
31	C₂×#1311	C₂²	1	1	yes	characteristic
32	#7332	C₂²	1	2	yes	—
33	#7332	C₂²	1	2	yes	—
34	#7332	C₂²	1	2	yes	—
35	#7332	C₂²	1	2	yes	—
36	#7330	C₂²	1	4	yes	—
37	#7330	C₂²	1	4	yes	—
38	C₂×#504	C₂×C₄	1	1	yes	characteristic
39	C₂×#504	C₂×C₄	1	1	yes	characteristic
40	#2965	C₈	1	2	yes	—
41	C₄×#131	C₂×C₄	1	1	yes	characteristic
42	#2258	—	2	8	no	—
43	C₂×M₆₄	C₂×C₄	1	2	yes	—
44	C₂×M₆₄	C₂×C₄	1	4	yes	—
45	C₂×M₆₄	C₂×C₄	1	2	yes	—
46	C₂²×C₃₂	C₂×C₄	1	2	yes	—
47	C₂²×C₃₂	C₂×C₄	1	2	yes	—
48	C₂²×C₃₂	C₂×C₄	1	4	yes	—
49	#2164	C₈	1	4	yes	—
50	#2162	C₂×C₄	1	4	yes	—
51	#2151	C₈	1	2	yes	—
52	#2113	C₈	1	2	yes	—
53	#2108	C₂×C₄	1	1	yes	characteristic
54	#2108	C₂×C₄	1	2	yes	—
55	#2108	C₂×C₄	1	2	yes	—
56	#2108	C₂×C₄	1	1	yes	characteristic
57	C₄×M₃₂	C₂×C₄	1	1	yes	characteristic
58	C₄×M₃₂	C₂×C₄	1	2	yes	—
59	C₄×M₃₂	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₆	C₂³	1	2	yes	—
62	C₂×C₄×C₁₆	C₂³	1	1	yes	characteristic
63	#1455	C₈	1	2	yes	—
64	#1453	C₈	1	2	yes	—
65	#1451	C₈	1	2	yes	—
66	#1450	C₂×C₄	1	1	yes	characteristic
67	C₈×M₁₆	C₂×C₄	1	4	yes	—
68	C₂×C₈²	C₂×C₄	1	1	yes	characteristic
69	C₄⋊C₃₂	C₂×C₄	1	8	yes	—
70	C₂²⋊C₃₂	C₂×C₄	1	8	yes	—
71	#1397	C₂×C₄	1	2	yes	—
72	#1397	C₂×C₄	1	2	yes	—
73	#1397	C₂×C₄	1	4	yes	—
74	#1397	C₂³	1	2	yes	—
75	#1397	C₂³	1	2	yes	—
76	#1397	C₂³	1	2	yes	—
77	#1397	C₂³	1	1	yes	characteristic
78	#1397	C₂³	1	1	yes	characteristic
79	C₄×C₃₂	C₂×C₄	1	2	yes	—
80	C₄×C₃₂	C₂×C₄	1	4	yes	—
81	C₄×C₃₂	C₂×C₄	1	2	yes	—
82	#1312	C₂×C₄	1	4	yes	—
83	#1312	C₂×C₄	1	4	yes	—
84	#1311	C₂×C₄	1	2	yes	—
85	#1311	C₂³	1	2	yes	—
86	#1311	C₂³	1	2	yes	—
87	C₈×C₁₆	C₂×C₄	1	2	yes	—
88	C₂×#131	C₄²	1	1	yes	characteristic
89	C₂×#131	C₂×C₈	1	1	yes	characteristic
90	C₄×#41	C₂×C₈	1	1	yes	characteristic
91	#504	—	2	4	no	—
92	#504	—	2	4	no	—
93	C₂×M₃₂	C₄²	1	1	yes	characteristic
94	C₂×M₃₂	C₄²	1	1	yes	characteristic
95	C₂×M₃₂	C₄²	1	2	yes	—
96	C₂×M₃₂	C₄²	1	2	yes	—
97	C₂×M₃₂	C₂×C₈	1	2	yes	—
98	C₂²×C₁₆	C₂×C₈	1	2	yes	—
99	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
100	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
101	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
103	#446	—	2	2	no	—
104	#436	—	2	2	no	—
105	#435	—	2	4	no	—
106	#431	C₂×C₈	1	4	yes	—
107	#431	C₂×C₈	1	2	yes	—
108	#405	C₄²	1	2	yes	—
109	#405	C₄²	1	1	yes	characteristic
110	#405	C₂×C₈	1	1	yes	characteristic
111	#405	C₂×C₈	1	2	yes	—
112	C₄×M₁₆	C₂×C₈	1	4	yes	—
113	C₄×M₁₆	C₂×C₈	1	1	yes	characteristic
114	C₄×M₁₆	C₂×C₈	1	2	yes	—
115	C₂×C₄×C₈	C₂×C₈	1	2	yes	—
116	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₄	1	2	yes	—
118	M₆₄	—	2	4	no	—
119	M₆₄	—	2	8	no	—
120	M₆₄	M₁₆	1	8	yes	complemented
121	C₂×C₃₂	—	2	4	no	—
122	C₂×C₃₂	—	2	4	no	—
123	C₂×C₃₂	—	2	8	no	—
124	C₂×C₃₂	C₂²×C₄	1	2	yes	—
125	C₂×C₃₂	C₂²×C₄	1	4	yes	—
126	C₂×C₃₂	C₂²×C₄	1	2	yes	—
127	C₂×C₃₂	C₂²×C₄	1	4	yes	—
128	C₂×C₃₂	C₂²×C₄	1	4	yes	—
129	C₄⋊C₁₆	C₂×C₈	1	8	yes	—
130	C₂²⋊C₁₆	C₂×C₈	1	8	yes	—
131	#346	C₄²	1	2	yes	—
132	#346	C₄²	1	2	yes	—
133	#346	C₄²	1	2	yes	—
134	#346	C₂×C₈	1	2	yes	—
135	C₄×C₁₆	C₂×C₈	1	2	yes	—
136	C₄×C₁₆	C₂²×C₄	1	2	yes	—
137	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
138	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
139	C₄×C₁₆	C₂²×C₄	1	2	yes	—
140	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
141	C₄×C₁₆	C₂²×C₄	1	2	yes	—
142	C₄×C₁₆	C₂²×C₄	1	2	yes	—
143	C₄×C₁₆	C₂²×C₄	1	2	yes	—
144	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
145	#322	C₂×C₈	1	4	yes	—
146	#322	C₂×C₈	1	4	yes	—
147	#322	C₂×C₈	1	2	yes	—
148	C₈²	C₂×C₈	1	1	yes	characteristic
149	C₈²	C₂×C₈	1	1	yes	characteristic
150	C₈²	C₂²×C₄	1	2	yes	—
151	C₈²	C₂²×C₄	1	2	yes	—
152	C₂×#41	C₄×C₈	1	1	yes	characteristic
153	#131	—	2	2	no	—
154	#131	—	2	2	no	—
155	#131	—	2	4	no	—
156	C₂×M₁₆	—	2	4	no	—
157	C₂×M₁₆	C₄×C₈	1	1	yes	characteristic
158	C₂×M₁₆	C₄×C₈	1	2	yes	—
159	C₂×M₁₆	C₄×C₈	1	1	yes	characteristic
160	C₂×M₁₆	C₄×C₈	1	2	yes	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	C₂×C₄²	1	2	yes	—
163	C₂²×C₈	C₂×C₄²	1	1	yes	characteristic
164	C₂²×C₈	C₂²×C₈	1	2	yes	—
165	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
166	C₄×Q₈	M₃₂	1	2	yes	—
167	C₄×D₄	M₃₂	1	4	yes	—
168	C₄×D₄	M₃₂	1	2	yes	—
169	#117	C₄×C₈	1	2	yes	—
170	#117	C₄×C₈	1	1	yes	characteristic
171	C₂×C₄²	C₂²×C₈	1	1	yes	characteristic
172	C₂×C₄²	C₂²×C₈	1	2	yes	—
173	M₃₂	—	2	2	no	—
174	M₃₂	—	2	2	no	—
175	M₃₂	—	2	4	no	—
176	M₃₂	—	2	4	no	—
177	M₃₂	—	2	4	no	—
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	C₂×C₄²	1	2	yes	—
182	C₂×C₁₆	C₂×C₄²	1	2	yes	—
183	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
184	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
185	C₂×C₁₆	C₂²×C₈	1	4	yes	—
186	C₂×C₁₆	C₂²×C₈	1	4	yes	—
187	C₂×C₁₆	C₂×M₁₆	1	4	yes	—
188	C₂×C₁₆	#131	1	4	yes	—
189	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	2	no	—
193	C₂²⋊C₈	—	2	2	no	—
194	C₂²⋊C₈	—	2	4	no	—
195	#97	—	2	4	no	—
196	#97	C₄×C₈	1	2	yes	—
197	#97	C₄×C₈	1	2	yes	—
198	#97	C₄×C₈	1	2	yes	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	C₂×C₄²	1	2	yes	—
201	C₄×C₈	C₂×C₄²	1	1	yes	characteristic
202	C₄×C₈	C₂²×C₈	1	2	yes	—
203	C₄×C₈	C₂²×C₈	1	2	yes	—
204	C₄×C₈	C₂²×C₈	1	4	yes	—
205	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
206	C₄×C₈	C₂²×C₈	1	2	yes	—
207	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
208	C₄×C₈	C₂²×C₈	1	2	yes	—
209	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
210	C₃₂	—	2	2	no	—
211	C₃₂	—	2	2	no	—
212	C₃₂	—	2	4	no	—
213	C₃₂	—	2	4	no	—
214	C₃₂	C₂×M₁₆	1	8	yes	—
215	#41	—	2	2	no	—
216	#41	—	2	2	no	—
217	C₂×Q₈	#346	1	1	yes	characteristic
218	C₂×D₄	#346	1	2	yes	—
219	C₂×D₄	#346	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×C₈	1	2	yes	—
221	C₂²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
222	C₂²×C₄	C₂×M₃₂	1	2	yes	—
223	C₂²×C₄	C₂×M₃₂	1	4	yes	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	8	4	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	8	2	no	—
236	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
237	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
238	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
240	C₂×C₈	C₄×M₁₆	1	2	yes	—
241	C₂×C₈	#405	1	2	yes	—
242	C₂×C₈	#504	1	4	yes	—
243	C₂×C₈	C₂²×C₄²	1	1	yes	characteristic
244	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
245	C₄⋊C₄	#346	1	4	yes	—
246	C₄⋊C₄	#346	1	2	yes	—
247	C₂²⋊C₄	#346	1	4	yes	—
248	C₂²⋊C₄	#346	1	2	yes	—
249	C₄²	C₂×C₄×C₈	1	1	yes	characteristic
250	C₄²	C₂×C₄×C₈	1	2	yes	—
251	C₄²	C₂×M₃₂	1	2	yes	—
252	C₄²	C₂×M₃₂	1	4	yes	—
253	C₄²	C₂³×C₈	1	1	yes	characteristic
254	C₁₆	—	2	2	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	4	no	—
257	C₁₆	—	2	4	no	—
258	C₁₆	C₄×M₁₆	1	2	yes	—
259	C₁₆	#405	1	2	yes	—
260	C₁₆	C₂²×M₁₆	1	2	yes	—
261	C₁₆	C₂×#131	1	2	yes	—
262	C₂³	C₂×#346	1	2	yes	—
263	C₂³	C₂×#346	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	C₈×M₁₆	1	2	yes	—
272	C₂×C₄	C₂×#346	1	2	yes	—
273	C₂×C₄	C₂×#346	1	4	yes	—
274	C₂×C₄	C₂×#346	1	2	yes	—
275	C₂×C₄	C₂×#346	1	1	yes	characteristic
276	C₂×C₄	#2108	1	2	yes	—
277	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
278	C₂×C₄	C₂²×M₃₂	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	4	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	4	no	—
284	C₈	C₈×M₁₆	1	2	yes	—
285	C₈	#2108	1	2	yes	—
286	C₈	C₂×C₄×M₁₆	1	1	yes	characteristic
287	C₈	C₂×#405	1	1	yes	characteristic
288	C₈	C₂×#504	1	2	yes	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	#11408	1	2	yes	—
293	C₂²	C₂²×#346	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	4	no	—
297	C₄	—	2	2	no	—
298	C₄	#11408	1	2	yes	—
299	C₄	C₂×C₈×M₁₆	1	1	yes	characteristic
300	C₄	C₂×#2108	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	4	2	no	—
303	C₂	C₂×#11408	1	1	yes	characteristic
304	C₂	#512548	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂×#41	C₄×C₈
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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
4. #149509	C₂	semidirect
5. #149508	C₂	semidirect
6. #149501	C₂	semidirect
7. C₂×#7332	C₂	semidirect
8. #149496	C₂	semidirect
9. #149495	C₂	semidirect
120. M₆₄	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#512731	2
#512545	2
#479917	2
#149509	2
#149508	2
#149501	2
C₂×#7332	2
#149496	2
#149495	2