Subgroups — #72323237

Order 8192.

Only the 340 normal subgroups are shown; non-normal subgroups are omitted.

The subgroup lattice has 290 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323220	C₂	1	1	yes	maximal, characteristic, complemented
2	#72316216	C₂	1	1	yes	maximal, characteristic, complemented
3	#72312950	C₂	1	1	yes	maximal, characteristic
4	#72310419	C₄	1	1	yes	characteristic, complemented
5	#72309403	C₂	1	1	yes	maximal, characteristic, complemented
6	#72266935	C₂	1	1	yes	maximal, characteristic
7	#72266926	C₄	1	1	yes	characteristic, complemented
8	#72266923	C₄	1	1	yes	characteristic, complemented
9	#72266913	C₄	1	1	yes	characteristic, complemented
10	#72266875	C₄	1	1	yes	characteristic, complemented
11	#72265785	C₂²	1	1	yes	characteristic
12	#72238419	C₂	1	1	yes	maximal, characteristic
13	#72237812	C₄	1	2	yes	complemented
14	#72237812	C₄	1	2	yes	complemented
15	#72237630	C₂²	1	1	yes	characteristic
16	#72237622	C₂×C₄	1	2	yes	complemented
17	#72237032	C₄	1	1	yes	characteristic, complemented
18	#72237027	C₄	1	1	yes	characteristic, complemented
19	#72234688	C₂²	1	1	yes	characteristic
20	#72234665	C₂²	1	1	yes	characteristic
21	#72234663	C₂×C₄	1	1	yes	characteristic
22	#72234619	Q₈	1	1	yes	characteristic
23	#72234619	Q₈	1	1	yes	characteristic
24	#72234591	C₂×C₄	1	1	yes	characteristic
25	#72234591	C₂×C₄	1	1	yes	characteristic
26	#72232884	C₂×C₄	1	1	yes	characteristic
27	#72225910	C₂	1	1	yes	maximal, characteristic, complemented
28	#72225883	C₂²	1	1	yes	characteristic
29	#72225868	C₂×C₄	1	2	yes	complemented
30	#72225655	D₄	1	1	yes	characteristic
31	#72225655	D₄	1	1	yes	characteristic
32	#72225629	C₂²	1	1	yes	characteristic
33	#72224136	D₄	1	1	yes	characteristic
34	#72224136	D₄	1	1	yes	characteristic
35	#72219311	D₄	1	1	yes	characteristic
36	#72219311	D₄	1	1	yes	characteristic
37	#72219309	C₂²	1	1	yes	characteristic
38	#72219305	D₄	1	2	yes	—
39	#72219304	D₄	1	2	yes	—
40	#72219301	C₂×C₄	1	1	yes	characteristic
41	#72219301	C₂×C₄	1	1	yes	characteristic
42	#72219301	C₂×C₄	1	1	yes	characteristic
43	#72219299	C₂×C₄	1	1	yes	characteristic
44	#72219247	C₄	1	1	yes	characteristic, complemented
45	#72219227	C₂×C₄	1	1	yes	characteristic, complemented
46	C₂×#7670203	C₂³	1	1	yes	characteristic
47	#72218518	C₂×C₄	1	1	yes	characteristic, complemented
48	#72218518	D₄	1	1	yes	characteristic
49	#72218518	D₄	1	1	yes	characteristic
50	#72218375	C₂×C₄	1	1	yes	characteristic, complemented
51	#72218291	C₂×C₄	1	1	yes	characteristic, complemented
52	#72217511	C₂×C₄	1	1	yes	characteristic
53	#72217511	C₂×C₄	1	1	yes	characteristic
54	#72216064	D₄	1	1	yes	characteristic
55	#72216064	D₄	1	1	yes	characteristic
56	#10586404	C₄⋊C₄	1	1	yes	characteristic
57	#10586404	C₄⋊C₄	1	1	yes	characteristic
58	C₂×#62817	C₄²	1	1	yes	characteristic
59	C₂×#62817	C₄²	1	1	yes	characteristic
60	#10586066	C₂²⋊C₄	1	1	yes	characteristic
61	#10586066	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×#3484	C₂²×C₄	1	1	yes	characteristic
63	C₂²×#3484	C₂×D₄	1	1	yes	characteristic
64	C₂⁴×#120	C₂²×C₄	1	1	yes	characteristic
65	C₂⁴×#120	C₂×D₄	1	1	yes	characteristic
66	#7642244	C₄⋊C₄	1	1	yes	characteristic
67	#7642244	C₄⋊C₄	1	1	yes	characteristic
68	#7642244	C₄⋊C₄	1	1	yes	characteristic
69	#7642244	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×#2846	#41	1	1	yes	characteristic
71	#7630939	C₂²⋊C₄	1	1	yes	characteristic
72	#7630939	C₂²⋊C₄	1	1	yes	characteristic
73	#7630939	C₂²⋊C₄	1	1	yes	characteristic
74	#7630939	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×#35390	#41	1	1	yes	characteristic
76	C₂×#35348	C₂×D₄	1	1	yes	characteristic
77	C₂×#35348	C₂×Q₈	1	1	yes	characteristic
78	C₂×#35346	C₂×D₄	1	1	yes	characteristic
79	C₂×#34981	#41	1	1	yes	characteristic
80	C₂×#34981	#41	1	1	yes	characteristic
81	C₂×#34980	C₂×D₄	1	1	yes	characteristic
82	C₂×#34980	C₂×D₄	1	1	yes	characteristic
83	#7626202	C₄²	1	1	yes	characteristic
84	#7626202	C₄²	1	1	yes	characteristic
85	C₂×#34198	C₂²×C₄	1	1	yes	characteristic
86	C₂×#34198	C₂×D₄	1	1	yes	characteristic
87	#6374210	C₂²⋊C₄	1	1	yes	characteristic
88	#6374210	C₂²⋊C₄	1	1	yes	characteristic
89	#6374210	C₂²⋊C₄	1	1	yes	characteristic
90	#6374210	C₂²⋊C₄	1	1	yes	characteristic
91	#6374128	C₄⋊C₄	1	1	yes	characteristic
92	#6374128	C₄⋊C₄	1	1	yes	characteristic
93	#6371504	C₄⋊C₄	1	1	yes	characteristic
94	#6371504	C₄⋊C₄	1	1	yes	characteristic
95	#6371504	C₄⋊C₄	1	1	yes	characteristic
96	#6371504	C₄⋊C₄	1	1	yes	characteristic
97	#6370693	C₂²⋊C₄	1	1	yes	characteristic
98	#6370693	C₂²⋊C₄	1	1	yes	characteristic
99	#392233	C₂²⋊C₄	1	2	yes	—
100	#392231	C₂²⋊C₄	1	2	yes	—
101	#392109	C₂²⋊C₄	1	1	yes	characteristic
102	#392109	C₂²⋊C₄	1	1	yes	characteristic
103	#392109	C₂²⋊C₄	1	1	yes	characteristic
104	#392109	C₂²⋊C₄	1	1	yes	characteristic
105	C₂⁸	C₄×D₄	1	1	yes	characteristic
106	C₂×H(7,𝔽₂)	#99	1	1	yes	characteristic
107	C₂×H(7,𝔽₂)	#99	1	1	yes	characteristic
108	C₂³×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂⁵×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂⁵×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	#62842	#99	1	1	yes	characteristic
112	#62842	#99	1	1	yes	characteristic
113	#62842	#99	1	1	yes	characteristic
114	#62842	#99	1	1	yes	characteristic
115	C₂×#3484	#95	1	1	yes	characteristic
116	C₂×#3484	#95	1	1	yes	characteristic
117	C₂×#3484	#95	1	1	yes	characteristic
118	C₂×#3484	#95	1	1	yes	characteristic
119	C₂²×#534	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×#534	#117	1	1	yes	characteristic
121	C₂²×#534	#120	1	1	yes	characteristic
122	C₂²×#534	#121	1	1	yes	characteristic
123	C₂²×#534	#123	1	1	yes	characteristic
124	C₂³×#120	C₂×C₄²	1	1	yes	characteristic
125	C₂³×#120	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂³×#120	#117	1	1	yes	characteristic
127	C₂³×#120	C₄×D₄	1	1	yes	characteristic
128	C₂³×#120	C₄×D₄	1	1	yes	characteristic
129	C₂³×#120	#120	1	1	yes	characteristic
130	C₂³×#120	#120	1	1	yes	characteristic
131	C₂³×#120	#121	1	1	yes	characteristic
132	C₂³×#120	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂³×#120	#123	1	1	yes	characteristic
134	C₂³×#120	#123	1	1	yes	characteristic
135	#62652	#99	1	1	yes	characteristic
136	#62652	#99	1	1	yes	characteristic
137	C₂²×#518	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×#518	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂⁴×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
140	C₂⁴×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
141	#36614	#99	1	2	yes	—
142	C₂×#2846	#99	1	2	yes	—
143	#35222	#99	1	2	yes	—
144	#34980	#99	1	2	yes	—
145	#34171	#95	1	1	yes	characteristic
146	#34171	#95	1	1	yes	characteristic
147	#34171	#95	1	1	yes	characteristic
148	#34171	#95	1	1	yes	characteristic
149	C₂×#2315	C₄×D₄	1	1	yes	characteristic
150	C₂×#2315	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×#2308	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂×#2308	#120	1	1	yes	characteristic
153	C₂×#2308	#121	1	1	yes	characteristic
154	C₂×#2308	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×#2308	#123	1	1	yes	characteristic
156	C₂²×#379	C₄×D₄	1	1	yes	characteristic
157	C₂²×#379	#121	1	1	yes	characteristic
158	C₂²×#379	C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×#2273	C₄×D₄	1	1	yes	characteristic
160	#21750	#99	1	4	yes	—
161	#21749	#99	1	4	yes	—
162	C₂×#1781	#99	1	2	yes	complemented
163	#14679	#99	1	2	yes	complemented
164	#10768	#99	1	4	yes	complemented
165	C₂⁷	C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₂⁷	#385	1	1	yes	characteristic
167	C₂⁷	#386	1	1	yes	characteristic
168	C₂⁷	C₂×#99	1	1	yes	characteristic
169	C₂⁷	C₂×#99	1	1	yes	characteristic
170	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
171	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
172	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
173	C₂²×H(5,𝔽₂)	#342	1	1	yes	characteristic
174	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
175	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
176	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
177	C₂²×H(5,𝔽₂)	C₂×#99	1	1	yes	characteristic
178	C₂⁴×D₄	C₂×#95	1	1	yes	characteristic
179	C₂⁴×D₄	#379	1	1	yes	characteristic
180	C₂⁴×D₄	#380	1	1	yes	characteristic
181	C₂⁴×D₄	#381	1	1	yes	characteristic
182	C₂⁴×D₄	#386	1	1	yes	characteristic
183	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
184	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
185	C₂⁴×D₄	C₂×#99	1	1	yes	characteristic
186	C₂⁴×D₄	#410	1	1	yes	characteristic
187	C₂⁵×C₄	C₂×#99	1	1	yes	characteristic
188	C₂⁵×C₄	#410	1	1	yes	characteristic
189	C₂×#535	#342	1	2	yes	—
190	C₂×#534	#342	1	2	yes	—
191	C₂×#525	#342	1	2	yes	—
192	C₂²×#120	#342	1	2	yes	—
193	C₂×#518	#342	1	1	yes	characteristic
194	C₂×#518	#342	1	1	yes	characteristic
195	C₂×#518	#342	1	1	yes	characteristic
196	C₂×#518	#342	1	1	yes	characteristic
197	C₂×#518	C₂×#99	1	2	yes	—
198	C₂×#518	#410	1	2	yes	—
199	C₂³×C₂²⋊C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
200	C₂³×C₂²⋊C₄	#380	1	1	yes	characteristic
201	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
202	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
203	C₂³×C₂²⋊C₄	#385	1	1	yes	characteristic
204	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
205	C₂³×C₂²⋊C₄	#386	1	1	yes	characteristic
206	#2309	#342	1	2	yes	—
207	#2308	#342	1	2	yes	—
208	#2308	C₂×#99	1	2	yes	—
209	#2308	C₂×#99	1	2	yes	—
210	#2280	#342	1	2	yes	—
211	C₂×#379	#342	1	2	yes	—
212	C₂⁶	C₂²⋊#95	1	1	yes	characteristic
213	C₂⁶	C₂×#342	1	1	yes	characteristic
214	C₂⁶	C₄×#99	1	1	yes	characteristic
215	C₂⁶	C₄×#99	1	1	yes	characteristic
216	C₂⁶	#1781	1	1	yes	characteristic
217	C₂⁶	#1799	1	1	yes	characteristic
218	C₂⁶	#1828	1	1	yes	characteristic
219	C₂⁶	#1898	1	1	yes	characteristic
220	C₂⁶	#1898	1	1	yes	characteristic
221	C₂⁶	#1898	1	1	yes	characteristic
222	C₂²×#41	#1304	1	2	yes	—
223	C₂³×D₄	#1304	1	2	yes	—
224	C₂³×D₄	C₂×#342	1	1	yes	characteristic
225	C₂³×D₄	C₂×#342	1	1	yes	characteristic
226	C₂³×D₄	C₂×#342	1	1	yes	characteristic
227	C₂³×D₄	C₂×#342	1	1	yes	characteristic
228	C₂³×D₄	#1781	1	1	yes	characteristic
229	C₂³×D₄	#1781	1	1	yes	characteristic
230	C₂³×D₄	#1781	1	1	yes	characteristic
231	C₂³×D₄	#1782	1	1	yes	characteristic
232	C₂³×D₄	#1819	1	1	yes	characteristic
233	C₂³×D₄	#1819	1	1	yes	characteristic
234	C₂³×D₄	#1828	1	1	yes	characteristic
235	C₂³×D₄	#1828	1	1	yes	characteristic
236	C₂⁴×C₄	C₂×#342	1	1	yes	characteristic
237	C₂⁴×C₄	#1782	1	1	yes	characteristic
238	C₂⁴×C₄	#1799	1	1	yes	characteristic
239	C₂⁴×C₄	#1828	1	1	yes	characteristic
240	C₂⁴×C₄	#1898	1	1	yes	characteristic
241	C₂×#117	#1304	1	2	yes	—
242	C₂²×C₂²⋊C₄	#1304	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₄×#99	1	2	yes	—
244	C₂²×C₂²⋊C₄	#1799	1	2	yes	—
245	C₂²×C₂²⋊C₄	#1898	1	2	yes	—
246	C₂²×C₂²⋊C₄	#1898	1	2	yes	—
247	C₂⁵	#7316	1	2	yes	complemented
248	C₂⁵	C₂×#1304	1	1	yes	characteristic
249	C₂⁵	#8101	1	1	yes	characteristic
250	C₂⁵	#8541	1	1	yes	characteristic
251	C₂⁵	#8542	1	1	yes	characteristic
252	C₂⁵	#8746	1	1	yes	characteristic
253	C₂⁵	#8747	1	1	yes	characteristic
254	C₂⁵	#10725	1	1	yes	characteristic
255	C₂⁵	#10735	1	1	yes	characteristic
256	C₂⁵	#10735	1	1	yes	characteristic
257	C₂⁵	#10761	1	1	yes	characteristic
258	C₂⁵	#10761	1	1	yes	characteristic
259	C₂⁵	#10768	1	1	yes	characteristic
260	C₂⁵	#10768	1	1	yes	characteristic
261	C₂²×Q₈	#7316	1	2	yes	complemented
262	C₂²×D₄	#7316	1	2	yes	complemented
263	C₂³×C₄	#7316	1	2	yes	complemented
264	C₂³×C₄	C₂×#1304	1	1	yes	characteristic
265	C₂³×C₄	#8101	1	1	yes	characteristic
266	C₂³×C₄	#8542	1	1	yes	characteristic
267	C₂³×C₄	#8543	1	1	yes	characteristic
268	C₂³×C₄	#8746	1	1	yes	characteristic
269	C₂³×C₄	#8747	1	1	yes	characteristic
270	C₂⁴	C₂²⋊#1304	1	1	yes	characteristic
271	C₂⁴	#105808	1	1	yes	characteristic
272	C₂⁴	#106132	1	1	yes	characteristic
273	C₂⁴	#106654	1	1	yes	characteristic
274	C₂⁴	#106654	1	1	yes	characteristic
275	C₂⁴	#106654	1	1	yes	characteristic
276	C₂⁴	#106654	1	1	yes	characteristic
277	C₂⁴	C₂×#7316	1	2	yes	—
278	C₂⁴	#133839	1	2	yes	—
279	C₂²×C₄	C₂×#7316	1	2	yes	—
280	C₂³	#72219414	1	1	yes	characteristic
281	C₂³	#72219414	1	1	yes	characteristic
282	C₂³	#72219423	1	1	yes	characteristic
283	C₂³	#72246292	1	1	yes	characteristic
284	C₂³	#72251550	1	1	yes	characteristic
285	C₂²	#72265763	1	1	yes	characteristic
286	C₂²	#72265788	1	1	yes	characteristic
287	C₂²	#72265897	1	1	yes	characteristic
288	C₂	#72266300	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266300
commutator	C₂³×#120	C₂×C₄²
Frattini, cosocle	C₂×#7670203	C₂³
prime radical	#72234665	C₂²
LCS	G ▷ C₂³×#120 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#7670203 ◁ G
LECS	G ▷ C₂×#7670203 ▷ C₂⁶ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323220	C₂	semidirect
2. #72316216	C₂	semidirect
4. #72310419	C₄	semidirect
5. #72309403	C₂	semidirect
7. #72266926	C₄	semidirect
8. #72266923	C₄	semidirect
9. #72266913	C₄	semidirect
10. #72266875	C₄	semidirect
13. #72237812	C₄	semidirect
14. #72237812	C₄	semidirect
16. #72237622	C₂×C₄	semidirect
17. #72237032	C₄	semidirect
18. #72237027	C₄	semidirect
27. #72225910	C₂	semidirect
29. #72225868	C₂×C₄	semidirect
44. #72219247	C₄	semidirect
45. #72219227	C₂×C₄	semidirect
47. #72218518	C₂×C₄	semidirect
50. #72218375	C₂×C₄	semidirect
51. #72218291	C₂×C₄	semidirect
162. C₂×#1781	#99	semidirect
163. #14679	#99	semidirect
164. #10768	#99	semidirect
247. C₂⁵	#7316	semidirect
261. C₂²×Q₈	#7316	semidirect
262. C₂²×D₄	#7316	semidirect
263. C₂³×C₄	#7316	semidirect

Maximal subgroups

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

subgroup	index
#72323220	2
#72316216	2
#72312950	2
#72309403	2
#72266935	2
#72238419	2
#72225910	2