Subgroups — #72317229

Order 8192.

Only the 544 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 302 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317206	C₂	1	2	yes	maximal, complemented
2	#72317188	C₂	1	2	yes	maximal, complemented
3	#72317176	C₂	1	2	yes	maximal, complemented
4	#72317168	C₂	1	2	yes	maximal, complemented
5	#72317088	C₂	1	1	yes	maximal, characteristic, complemented
6	#72314621	C₂	1	1	yes	maximal, characteristic, complemented
7	#72314618	C₂²	1	2	yes	complemented
8	#72314610	C₂²	1	2	yes	complemented
9	#72314568	C₂	1	1	yes	maximal, characteristic, complemented
10	#72314476	C₂²	1	2	yes	complemented
11	#72314447	C₂²	1	2	yes	complemented
12	#72310485	C₂	1	1	yes	maximal, characteristic, complemented
13	#72310472	C₂²	1	2	yes	complemented
14	#72310468	C₂²	1	2	yes	complemented
15	#72310091	C₂	1	1	yes	maximal, characteristic, complemented
16	#72309929	C₂²	1	1	yes	characteristic, complemented
17	#72309776	C₂²	1	1	yes	characteristic, complemented
18	#72309645	C₂²	1	1	yes	characteristic, complemented
19	#72309421	C₂²	1	1	yes	characteristic, complemented
20	#72266165	C₂²	1	2	yes	complemented
21	#72266163	C₂²	1	2	yes	complemented
22	#72265785	C₂²	1	1	yes	characteristic
23	#72265766	C₂³	1	2	yes	—
24	#72259029	C₂²	1	1	yes	characteristic
25	#72259008	C₂³	1	1	yes	characteristic
26	#72238358	C₂	1	1	yes	maximal, characteristic, complemented
27	#72237608	D₄	1	2	yes	complemented
28	#72237603	C₂²	1	2	yes	complemented
29	#72237585	C₂²	1	2	yes	complemented
30	#72235847	C₂³	1	1	yes	characteristic
31	#72234751	C₂	1	1	yes	maximal, characteristic
32	#72234688	C₂²	1	1	yes	characteristic
33	#72234683	C₂²	1	1	yes	characteristic
34	#72234672	C₄	1	2	yes	complemented
35	#72234670	C₂×C₄	1	1	yes	characteristic, complemented
36	#72234665	C₂²	1	1	yes	characteristic
37	#72234663	D₄	1	1	yes	characteristic
38	#72234639	C₂²	1	2	yes	—
39	#72234639	C₂²	1	2	yes	—
40	#72234638	D₄	1	4	yes	—
41	#72234637	D₄	1	4	yes	—
42	#72234636	C₂³	1	2	yes	—
43	#72234628	C₄	1	2	yes	complemented
44	#72234619	D₄	1	1	yes	characteristic
45	#72234619	D₄	1	1	yes	characteristic
46	#72234599	C₄	1	4	yes	complemented
47	#72234598	C₂×C₄	1	4	yes	—
48	#72234598	C₂×C₄	1	4	yes	complemented
49	#72234597	C₂×C₄	1	4	yes	—
50	#72234597	C₂×C₄	1	4	yes	complemented
51	#72234592	C₂×C₄	1	2	yes	complemented
52	#72234591	C₂×C₄	1	2	yes	—
53	#72234588	C₂×C₄	1	2	yes	complemented
54	#72232884	D₄	1	1	yes	characteristic
55	C₂×#8932885	C₂×C₄	1	1	yes	characteristic, complemented
56	#72225776	C₂×C₄	1	2	yes	complemented
57	#72225655	C₂×C₄	1	2	yes	—
58	#72224136	C₂³	1	1	yes	characteristic
59	#72224136	C₂³	1	1	yes	characteristic
60	#72219397	C₂²	1	1	yes	characteristic
61	#72219397	C₂²	1	1	yes	characteristic
62	#72219345	D₄	1	2	yes	complemented
63	#72219314	C₂³	1	2	yes	—
64	#72219311	D₄	1	1	yes	characteristic, complemented
65	#72219311	C₂³	1	1	yes	characteristic
66	#72219311	C₂³	1	2	yes	—
67	#72219310	D₄	1	1	yes	characteristic, complemented
68	#72219310	C₂³	1	1	yes	characteristic
69	C₂×#7670203	C₂³	1	1	yes	characteristic
70	#72218518	D₄	1	1	yes	characteristic, complemented
71	#72218291	D₄	1	1	yes	characteristic, complemented
72	#72217511	D₄	1	2	yes	—
73	#72216064	D₄	1	2	yes	—
74	#72211186	D₄	1	4	yes	—
75	#72210578	D₄	1	4	yes	—
76	#10586404	C₂×D₄	1	1	yes	characteristic
77	#10586404	C₂×D₄	1	1	yes	characteristic
78	C₂×#62817	C₂²⋊C₄	1	2	yes	—
79	#10586066	C₂²⋊C₄	1	2	yes	—
80	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
81	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
82	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
83	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
84	C₂×#46996	C₂²×C₄	1	2	yes	—
85	#7642244	C₂²⋊C₄	1	2	yes	—
86	#7642244	C₂²⋊C₄	1	2	yes	—
87	C₂²×#2846	#41	1	1	yes	characteristic
88	#7630939	C₂×D₄	1	2	yes	—
89	#7630939	C₂×D₄	1	2	yes	—
90	C₂×#35390	C₂×D₄	1	1	yes	characteristic
91	C₂×#35348	C₂×D₄	1	1	yes	characteristic
92	C₂×#35348	#41	1	1	yes	characteristic
93	C₂×#35346	C₂²×C₄	1	1	yes	characteristic
94	C₂×#34981	C₂×D₄	1	1	yes	characteristic
95	C₂×#34981	#41	1	1	yes	characteristic
96	C₂×#34980	C₂×D₄	1	1	yes	characteristic
97	C₂×#34980	C₂⁴	1	1	yes	characteristic
98	#7626202	C₂²⋊C₄	1	2	yes	—
99	C₂×#34198	C₂²×C₄	1	1	yes	characteristic
100	C₂×#34198	#41	1	1	yes	characteristic
101	#6499870	C₂×D₄	1	4	yes	—
102	#6499867	C₂×D₄	1	4	yes	—
103	#6374210	C₂²×C₄	1	2	yes	—
104	#6374210	C₂²×C₄	1	2	yes	—
105	#6374128	C₂×D₄	1	1	yes	characteristic
106	#6374128	C₂×D₄	1	1	yes	characteristic
107	#6371504	C₂²⋊C₄	1	2	yes	—
108	#6371504	C₂²⋊C₄	1	2	yes	—
109	#6370693	C₂²⋊C₄	1	2	yes	—
110	#5024691	C₂×D₄	1	2	yes	—
111	#5020751	C₂×D₄	1	2	yes	—
112	#4430405	C₂×D₄	1	2	yes	—
113	#648360	C₂×D₄	1	2	yes	—
114	#576160	C₂²×C₄	1	2	yes	—
115	#238077	C₂²⋊C₄	1	8	yes	complemented
116	#233541	C₂²⋊C₄	1	8	yes	complemented
117	#190116	C₂²×C₄	1	4	yes	—
118	#188109	C₂×D₄	1	2	yes	complemented
119	C₂×#8547	C₂×D₄	1	2	yes	complemented
120	#188049	C₂×D₄	1	2	yes	complemented
121	C₂×#8541	C₂×D₄	1	2	yes	complemented
122	#115690	C₂×D₄	1	4	yes	—
123	#115631	C₂×D₄	1	4	yes	—
124	#110180	C₂²⋊C₄	1	8	yes	complemented
125	#110086	C₂²⋊C₄	1	8	yes	complemented
126	C₂⁸	#121	1	1	yes	characteristic
127	C₂×H(7,𝔽₂)	#99	1	2	yes	—
128	C₂³×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂⁵×D₄	#120	1	1	yes	characteristic
130	C₂⁵×D₄	C₂²×D₄	1	1	yes	characteristic
131	#62842	#99	1	2	yes	—
132	#62842	#99	1	2	yes	—
133	#62817	#120	1	4	yes	—
134	C₂×#3484	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×#3484	C₂×C₂²⋊C₄	1	2	yes	—
136	C₂²×#534	#120	1	1	yes	characteristic
137	C₂²×#534	#121	1	1	yes	characteristic
138	C₂²×#534	#121	1	1	yes	characteristic
139	C₂²×#534	C₂²×D₄	1	1	yes	characteristic
140	C₂²×#534	C₂²×D₄	1	1	yes	characteristic
141	C₂³×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂³×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂³×#120	C₄×D₄	1	1	yes	characteristic
144	C₂³×#120	C₄×D₄	1	1	yes	characteristic
145	C₂³×#120	#120	1	1	yes	characteristic
146	C₂³×#120	#121	1	1	yes	characteristic
147	C₂³×#120	#121	1	1	yes	characteristic
148	C₂³×#120	#121	1	1	yes	characteristic
149	C₂³×#120	#121	1	1	yes	characteristic
150	C₂³×#120	#123	1	1	yes	characteristic
151	C₂³×#120	C₂²×D₄	1	1	yes	characteristic
152	#62652	#99	1	2	yes	—
153	C₂²×#518	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×#518	#123	1	1	yes	characteristic
155	C₂⁴×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
156	C₂⁴×C₂²⋊C₄	#121	1	1	yes	characteristic
157	#36714	#99	1	8	yes	—
158	C₂×#2840	C₂×C₂²⋊C₄	1	2	yes	complemented
159	C₂×#2849	H(5,𝔽₂)	1	2	yes	—
160	C₂×#2846	#120	1	4	yes	—
161	C₂×#2846	C₂²×D₄	1	2	yes	—
162	#34980	#120	1	4	yes	—
163	C₂×#2403	C₂×C₂²⋊C₄	1	2	yes	complemented
164	C₂×#2613	C₂²×D₄	1	2	yes	—
165	C₂×#2613	H(5,𝔽₂)	1	2	yes	—
166	#34171	C₂×C₂²⋊C₄	1	2	yes	—
167	#34171	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂×#2315	C₄×D₄	1	1	yes	characteristic
169	C₂×#2315	#123	1	1	yes	characteristic
170	C₂×#2308	C₄×D₄	1	1	yes	characteristic
171	C₂×#2308	C₄×D₄	1	1	yes	characteristic
172	C₂×#2308	#120	1	1	yes	characteristic
173	C₂×#2308	C₂³×C₄	1	1	yes	characteristic
174	C₂×#2308	C₂×#41	1	1	yes	characteristic
175	C₂²×#379	C₄×D₄	1	1	yes	characteristic
176	C₂²×#379	#123	1	1	yes	characteristic
177	C₂²×#379	C₂×#41	1	1	yes	characteristic
178	C₂×#2273	C₄×D₄	1	1	yes	characteristic
179	#21746	C₂×C₂²⋊C₄	1	8	yes	—
180	#21746	#120	1	4	yes	—
181	#13827	C₂×C₂²⋊C₄	1	2	yes	complemented
182	C₂²⋊C₄²	C₂×C₂²⋊C₄	1	2	yes	complemented
183	#8101	C₂×C₂²⋊C₄	1	8	yes	complemented
184	C₂⁷	#386	1	1	yes	characteristic
185	C₂⁷	#386	1	1	yes	characteristic
186	C₂⁷	B(4,𝔽₂)	1	1	yes	characteristic
187	C₂⁷	C₂×#120	1	1	yes	characteristic
188	C₂⁷	C₂×#121	1	1	yes	characteristic
189	C₂²×H(5,𝔽₂)	C₂×#99	1	2	yes	—
190	C₂²×H(5,𝔽₂)	C₂×#99	1	2	yes	—
191	C₂²×H(5,𝔽₂)	C₂×#99	1	2	yes	—
192	C₂²×H(5,𝔽₂)	C₂×#99	1	2	yes	—
193	C₂⁴×D₄	#386	1	1	yes	characteristic
194	C₂⁴×D₄	#386	1	1	yes	characteristic
195	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
196	C₂⁴×D₄	B(4,𝔽₂)	1	1	yes	characteristic
197	C₂⁴×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
198	C₂⁴×D₄	C₂×#120	1	1	yes	characteristic
199	C₂⁴×D₄	C₂×#120	1	1	yes	characteristic
200	C₂⁴×D₄	C₂×#121	1	1	yes	characteristic
201	C₂⁴×D₄	#534	1	1	yes	characteristic
202	C₂⁵×C₄	B(4,𝔽₂)	1	1	yes	characteristic
203	C₂⁵×C₄	#534	1	1	yes	characteristic
204	C₂×#534	C₂×#99	1	4	yes	—
205	C₂²×#123	B(4,𝔽₂)	1	2	yes	—
206	C₂²×#123	#534	1	2	yes	—
207	C₂²×#123	D₄²	1	2	yes	—
208	C₂²×#120	#379	1	4	yes	—
209	C₂²×#120	C₂×#99	1	4	yes	—
210	C₂²×#120	B(4,𝔽₂)	1	2	yes	—
211	C₂²×#120	C₂×#120	1	2	yes	—
212	C₂²×#120	D₄²	1	2	yes	—
213	C₂²×#120	D₄²	1	2	yes	—
214	C₂²×#120	D₄²	1	2	yes	—
215	C₂×#518	C₂×#99	1	2	yes	—
216	C₂×#518	C₂×#99	1	2	yes	—
217	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
218	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
219	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
220	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
221	C₂³×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂³×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂³×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
224	#2309	C₂×#99	1	4	yes	—
225	C₂×#381	#518	1	2	yes	—
226	C₂×#381	#546	1	2	yes	—
227	C₂×#379	#379	1	4	yes	—
228	C₂×#379	C₂²×C₂²⋊C₄	1	2	yes	—
229	C₂×#379	#546	1	2	yes	—
230	C₂×#379	#546	1	2	yes	—
231	C₂×#379	#546	1	2	yes	—
232	C₂⁶	#1889	1	1	yes	characteristic
233	C₂⁶	#1889	1	1	yes	characteristic
234	C₂⁶	#1889	1	1	yes	characteristic
235	C₂⁶	#1889	1	1	yes	characteristic
236	C₂⁶	#1898	1	2	yes	—
237	C₂⁶	C₂×#379	1	1	yes	characteristic
238	C₂⁶	C₂×#386	1	1	yes	characteristic
239	C₂⁶	#2410	1	1	yes	characteristic
240	C₂⁶	#2410	1	1	yes	characteristic
241	C₂⁶	#2410	1	1	yes	characteristic
242	C₂⁶	C₂×B(4,𝔽₂)	1	1	yes	characteristic
243	C₂³×D₄	#1781	1	4	yes	—
244	C₂³×D₄	#1898	1	1	yes	characteristic
245	C₂³×D₄	#1898	1	2	yes	—
246	C₂³×D₄	#1898	1	1	yes	characteristic
247	C₂³×D₄	#1913	1	1	yes	characteristic
248	C₂³×D₄	#1913	1	2	yes	—
249	C₂³×D₄	#1913	1	1	yes	characteristic
250	C₂³×D₄	#2403	1	2	yes	—
251	C₂³×D₄	C₂²×#99	1	2	yes	—
252	C₂³×D₄	C₂²×#99	1	2	yes	—
253	C₂³×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
254	C₂⁴×C₄	#1889	1	1	yes	characteristic
255	C₂⁴×C₄	#1889	1	1	yes	characteristic
256	C₂⁴×C₄	#2308	1	1	yes	characteristic
257	C₂⁴×C₄	#2410	1	1	yes	characteristic
258	C₂⁴×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
259	C₂²×C₂²⋊C₄	#1781	1	4	yes	—
260	C₂²×C₂²⋊C₄	#1889	1	2	yes	—
261	C₂²×C₂²⋊C₄	#1898	1	2	yes	—
262	C₂²×C₂²⋊C₄	D₄×C₂²⋊C₄	1	2	yes	—
263	C₂²×C₂²⋊C₄	D₄×C₂²⋊C₄	1	2	yes	—
264	C₂²×C₂²⋊C₄	#2405	1	2	yes	—
265	C₂⁵	#9495	1	1	yes	characteristic
266	C₂⁵	#9495	1	2	yes	—
267	C₂⁵	#9495	1	1	yes	characteristic
268	C₂⁵	#12707	1	4	yes	complemented
269	C₂⁵	#13867	1	1	yes	characteristic
270	C₂⁵	C₂×#1781	1	2	yes	—
271	C₂⁵	C₂×#1889	1	1	yes	characteristic
272	C₂⁵	C₂×#1889	1	1	yes	characteristic
273	C₂⁵	C₂×#1898	1	1	yes	characteristic
274	C₂⁵	C₂×#1898	1	1	yes	characteristic
275	C₂⁵	C₂×#1913	1	1	yes	characteristic
276	C₂⁵	C₂×#1913	1	1	yes	characteristic
277	C₂⁵	#23900	1	2	yes	—
278	C₂³×C₄	#9495	1	1	yes	characteristic
279	C₂³×C₄	#9495	1	1	yes	characteristic
280	C₂³×C₄	#9495	1	2	yes	—
281	C₂³×C₄	#12707	1	4	yes	complemented
282	C₂³×C₄	#21746	1	2	yes	—
283	C₂³×C₄	#23900	1	2	yes	—
284	C₂⁴	#139743	1	2	yes	—
285	C₂⁴	C₂×#9495	1	1	yes	characteristic
286	C₂⁴	C₂×#9495	1	1	yes	characteristic
287	C₂⁴	#201664	1	1	yes	characteristic
288	C₂⁴	#201664	1	1	yes	characteristic
289	C₂⁴	#359428	1	1	yes	characteristic
290	C₂⁴	#359442	1	1	yes	characteristic
291	C₂⁴	#359501	1	1	yes	characteristic
292	C₂⁴	C₂×#12707	1	4	yes	—
293	C₂³	#72218057	1	2	yes	—
294	C₂³	C₂×#139743	1	1	yes	characteristic
295	C₂³	#72218846	1	1	yes	characteristic
296	C₂³	#72255841	1	1	yes	characteristic
297	C₂²	#72240489	1	1	yes	characteristic
298	C₂²	#72258080	1	1	yes	characteristic
299	C₂²	#72265911	1	1	yes	characteristic
300	C₂	#72266246	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266246
commutator	C₂×#2308	C₂³×C₄
Frattini, cosocle	C₂×#34980	C₂⁴
prime radical	#72234751	C₂
LCS	G ▷ C₂×#2308 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#34980 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72317206	C₂	semidirect
2. #72317188	C₂	semidirect
3. #72317176	C₂	semidirect
4. #72317168	C₂	semidirect
5. #72317088	C₂	semidirect
6. #72314621	C₂	semidirect
7. #72314618	C₂²	semidirect
8. #72314610	C₂²	semidirect
9. #72314568	C₂	semidirect
10. #72314476	C₂²	semidirect
11. #72314447	C₂²	semidirect
12. #72310485	C₂	semidirect
13. #72310472	C₂²	semidirect
14. #72310468	C₂²	semidirect
15. #72310091	C₂	semidirect
16. #72309929	C₂²	semidirect
17. #72309776	C₂²	semidirect
18. #72309645	C₂²	semidirect
19. #72309421	C₂²	semidirect
20. #72266165	C₂²	semidirect
21. #72266163	C₂²	semidirect
26. #72238358	C₂	semidirect
27. #72237608	D₄	semidirect
28. #72237603	C₂²	semidirect
29. #72237585	C₂²	semidirect
34. #72234672	C₄	semidirect
35. #72234670	C₂×C₄	semidirect
43. #72234628	C₄	semidirect
46. #72234599	C₄	semidirect
48. #72234598	C₂×C₄	semidirect
50. #72234597	C₂×C₄	semidirect
51. #72234592	C₂×C₄	semidirect
53. #72234588	C₂×C₄	semidirect
55. C₂×#8932885	C₂×C₄	semidirect
56. #72225776	C₂×C₄	semidirect
62. #72219345	D₄	semidirect
64. #72219311	D₄	semidirect
67. #72219310	D₄	semidirect
70. #72218518	D₄	semidirect
71. #72218291	D₄	semidirect
115. #238077	C₂²⋊C₄	semidirect
116. #233541	C₂²⋊C₄	semidirect
118. #188109	C₂×D₄	semidirect
119. C₂×#8547	C₂×D₄	semidirect
120. #188049	C₂×D₄	semidirect
121. C₂×#8541	C₂×D₄	semidirect
124. #110180	C₂²⋊C₄	semidirect
125. #110086	C₂²⋊C₄	semidirect
158. C₂×#2840	C₂×C₂²⋊C₄	semidirect
163. C₂×#2403	C₂×C₂²⋊C₄	semidirect
181. #13827	C₂×C₂²⋊C₄	semidirect
182. C₂²⋊C₄²	C₂×C₂²⋊C₄	semidirect
183. #8101	C₂×C₂²⋊C₄	semidirect
268. C₂⁵	#12707	semidirect
281. C₂³×C₄	#12707	semidirect

Maximal subgroups

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

subgroup	index
#72317206	2
#72317188	2
#72317176	2
#72317168	2
#72317088	2
#72314621	2
#72314568	2
#72310485	2
#72310091	2
#72238358	2
#72234751	2