Subgroups — #72262722

Order 4096.

At least 10448 conjugacy classes of subgroups; only the 849 normal subgroups are shown.

The subgroup lattice has 300 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262720	C₂	1	2	yes	maximal, complemented
2	#72262718	C₂	1	1	yes	maximal, characteristic, complemented
3	#72262702	C₂	1	1	yes	maximal, characteristic, complemented
4	#72262701	C₂²	1	1	yes	characteristic, complemented
5	#72262683	C₂	1	2	yes	maximal, complemented
6	#72262641	C₂	1	2	yes	maximal, complemented
7	#72262637	C₂²	1	1	yes	characteristic, complemented
8	#72256281	C₂	1	4	yes	maximal, complemented
9	#72256233	C₂	1	4	yes	maximal, complemented
10	#72256222	C₂²	1	4	yes	complemented
11	#72256024	C₂	1	4	yes	maximal, complemented
12	#72256001	C₂²	1	4	yes	complemented
13	#72255986	C₂²	1	4	yes	complemented
14	#72255893	C₂²	1	4	yes	complemented
15	#72255857	C₂	1	4	yes	maximal, complemented
16	#72255853	C₂²	1	4	yes	complemented
17	#72255849	C₂²	1	4	yes	complemented
18	#72255803	C₂²	1	4	yes	complemented
19	#72255785	C₂²	1	4	yes	complemented
20	C₂×#144380	C₂²	1	2	yes	complemented
21	#72251552	C₂²	1	4	yes	complemented
22	#72251475	C₂²	1	4	yes	complemented
23	#72249634	C₂	1	2	yes	maximal, complemented
24	#72249626	C₂²	1	2	yes	complemented
25	#72249624	C₂²	1	2	yes	complemented
26	#72249612	C₂²	1	4	yes	complemented
27	#72249593	C₂²	1	2	yes	complemented
28	#72240083	C₂²	1	2	yes	complemented
29	#72238974	C₂²	1	2	yes	complemented
30	#72238598	C₂	1	1	yes	maximal, characteristic, complemented
31	#72237900	C₂²	1	4	yes	complemented
32	#72237381	C₂	1	2	yes	maximal, complemented
33	#72237374	C₂²	1	4	yes	complemented
34	#72237039	C₂	1	2	yes	maximal, complemented
35	#72237037	C₂²	1	4	yes	complemented
36	#72237034	C₂²	1	2	yes	complemented
37	#72237033	C₂²	1	2	yes	complemented
38	#72236928	C₂²	1	4	yes	complemented
39	#72236631	C₂²	1	4	yes	complemented
40	#72236521	C₂²	1	4	yes	complemented
41	#72236508	C₂²	1	4	yes	complemented
42	#72235238	C₂²	1	4	yes	complemented
43	#72235234	C₂²	1	2	yes	complemented
44	#72235130	C₂²	1	4	yes	complemented
45	#72234953	C₂²	1	4	yes	complemented
46	#72234675	C₂²	1	1	yes	characteristic, complemented
47	#72234668	C₂²	1	1	yes	characteristic
48	#72234625	C₂²	1	1	yes	characteristic
49	C₂×#511771	C₂²	1	4	yes	complemented
50	C₂×#188035	C₂²	1	1	yes	characteristic, complemented
51	#72225744	C₂²	1	2	yes	—
52	#72225712	C₂²	1	2	yes	complemented
53	#72225665	C₂²	1	1	yes	characteristic
54	#72225644	C₂²	1	2	yes	complemented
55	#72225635	C₂²	1	2	yes	—
56	#72225610	C₂²	1	4	yes	complemented
57	#72224136	C₂²	1	2	yes	—
58	#72219649	C₂²	1	2	yes	complemented
59	#72219424	C₂²	1	2	yes	complemented
60	#72219314	C₂²	1	2	yes	—
61	#72219158	C₂²	1	4	yes	complemented
62	#72218838	C₂²	1	4	yes	complemented
63	#72218114	C₂²	1	4	yes	complemented
64	C₂×#144378	C₂²	1	2	yes	complemented
65	C₂×#139731	C₂²	1	4	yes	complemented
66	C₂×#511790	C₂²	1	4	yes	complemented
67	#10586404	C₂³	1	2	yes	—
68	C₂×#62817	C₂³	1	1	yes	characteristic
69	C₂³×D₄²	C₂³	1	1	yes	characteristic
70	#7643002	C₂³	1	2	yes	—
71	#7642244	C₂³	1	2	yes	—
72	C₂×#34980	C₂³	1	1	yes	characteristic
73	#7626202	C₂³	1	1	yes	characteristic
74	#6442728	D₄	1	4	yes	complemented
75	C₂×#23895	C₂³	1	4	yes	—
76	C₂×#23895	C₂³	1	2	yes	—
77	#6437621	C₂³	1	4	yes	—
78	#6437621	C₂³	1	4	yes	—
79	C₂×#23900	C₂³	1	4	yes	—
80	C₂×#23900	C₂³	1	4	yes	—
81	C₂×#23897	C₂³	1	4	yes	—
82	C₂×#23906	C₂³	1	4	yes	—
83	#6431795	C₂³	1	2	yes	—
84	#6374128	C₂³	1	2	yes	—
85	C₂×#21746	C₂³	1	1	yes	characteristic
86	C₂×#21746	C₂³	1	1	yes	characteristic
87	C₂²×#1781	D₄	1	2	yes	complemented
88	#617710	C₂³	1	2	yes	—
89	#616960	C₂³	1	2	yes	—
90	#556752	C₂³	1	1	yes	characteristic
91	C₂×#12707	C₂³	1	4	yes	—
92	C₂×#12707	C₂³	1	4	yes	—
93	C₂×#12705	C₂³	1	4	yes	—
94	C₂×#12705	C₂³	1	4	yes	—
95	C₂×#12693	C₂³	1	4	yes	—
96	#493357	C₂³	1	4	yes	—
97	C₂×#12687	C₂³	1	4	yes	—
98	C₂×#12687	C₂³	1	4	yes	—
99	#392030	C₂³	1	2	yes	—
100	#382544	C₂³	1	2	yes	—
101	#382434	C₂³	1	4	yes	—
102	#364532	D₄	1	8	yes	complemented
103	#364173	D₄	1	4	yes	complemented
104	#359705	D₄	1	4	yes	complemented
105	#359549	C₂³	1	4	yes	—
106	#359442	C₂³	1	4	yes	—
107	#355913	D₄	1	2	yes	complemented
108	#355098	C₂³	1	2	yes	—
109	#355096	C₂³	1	2	yes	—
110	#188109	C₂³	1	2	yes	—
111	#188049	C₂³	1	2	yes	—
112	C₂×#8546	C₂³	1	2	yes	—
113	C₂×#8541	C₂³	1	2	yes	—
114	#187944	C₂³	1	4	yes	—
115	#187944	C₂³	1	4	yes	—
116	#186835	C₂³	1	4	yes	—
117	#146475	C₂³	1	4	yes	—
118	#133826	D₄	1	4	yes	complemented
119	#133825	D₄	1	4	yes	complemented
120	#133824	D₄	1	4	yes	complemented
121	#133822	D₄	1	4	yes	complemented
122	#128033	C₂³	1	4	yes	—
123	C₂×#7316	C₂³	1	4	yes	—
124	C₂×#7314	C₂³	1	4	yes	—
125	C₂×#7311	C₂³	1	2	yes	—
126	C₂×#7310	C₂³	1	4	yes	—
127	C₂×#7310	C₂³	1	2	yes	—
128	#128009	C₂³	1	4	yes	—
129	#107114	C₂³	1	4	yes	—
130	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
131	#62842	C₂×D₄	1	2	yes	complemented
132	C₂×#3484	C₂⁴	1	4	yes	—
133	C₂×#3484	C₂⁴	1	2	yes	—
134	C₂²×#534	C₂⁴	1	2	yes	—
135	C₂³×#127	C₂⁴	1	1	yes	characteristic
136	C₂³×#120	C₂⁴	1	1	yes	characteristic
137	C₂³×#120	C₂⁴	1	1	yes	characteristic
138	C₂²×B(4,𝔽₂)	C₂×D₄	1	8	yes	—
139	C₂×#2883	C₂×D₄	1	4	yes	complemented
140	C₂×#2849	C₂×D₄	1	2	yes	—
141	C₂×#2846	C₂×D₄	1	2	yes	—
142	C₂×#2613	C₂×D₄	1	2	yes	—
143	C₂×#2613	C₂×D₄	1	2	yes	—
144	#34198	C₂×D₄	1	4	yes	—
145	#34198	C₂×D₄	1	4	yes	—
146	#34176	C₂×D₄	1	2	yes	—
147	#34171	C₂⁴	1	2	yes	—
148	C₂×#2308	C₂×D₄	1	2	yes	—
149	C₂²×#379	C₂×D₄	1	2	yes	—
150	C₂²×#379	C₂×D₄	1	2	yes	—
151	C₂²×#379	C₂×D₄	1	1	yes	characteristic
152	C₂×#1889	C₂×D₄	1	8	yes	—
153	C₂×#1819	C₂×D₄	1	2	yes	—
154	#20633	C₂×D₄	1	4	yes	complemented
155	#20609	C₂×D₄	1	8	yes	complemented
156	C₂×#1783	C₂⁴	1	4	yes	—
157	C₂×#1781	C₂×D₄	1	8	yes	—
158	C₂×#1781	C₂⁴	1	4	yes	—
159	C₂×#1781	C₂⁴	1	2	yes	—
160	C₂×#1781	C₂⁴	1	2	yes	—
161	C₂×#1781	C₂⁴	1	4	yes	—
162	#20354	C₂×D₄	1	2	yes	complemented
163	#14335	C₂⁴	1	2	yes	—
164	#12688	C₂×D₄	1	8	yes	complemented
165	#12687	C₂×D₄	1	8	yes	—
166	#12687	C₂×D₄	1	8	yes	—
167	#12687	C₂×D₄	1	8	yes	complemented
168	#10768	C₂×D₄	1	4	yes	complemented
169	#10754	C₂×D₄	1	8	yes	complemented
170	#10735	C₂×D₄	1	8	yes	—
171	#8548	C₂×D₄	1	4	yes	complemented
172	#8547	C₂×D₄	1	4	yes	complemented
173	#8543	C₂×D₄	1	4	yes	complemented
174	#8541	C₂×D₄	1	4	yes	complemented
175	C₂⁷	#127	1	1	yes	characteristic, complemented
176	C₂⁷	C₂²×D₄	1	1	yes	characteristic
177	C₂⁷	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂²×H(5,𝔽₂)	C₂²×D₄	1	4	yes	—
179	C₂²×H(5,𝔽₂)	H(5,𝔽₂)	1	2	yes	—
180	C₂⁴×D₄	#127	1	2	yes	complemented
181	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
182	C₂⁴×D₄	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂⁴×D₄	C₂⁵	1	1	yes	characteristic
184	C₂⁵×C₄	#127	1	1	yes	characteristic, complemented
185	C₂²×#123	C₂²×D₄	1	2	yes	—
186	C₂²×#123	C₂²×D₄	1	2	yes	—
187	C₂²×#120	C₂²×D₄	1	4	yes	—
188	C₂²×#120	C₂²×D₄	1	2	yes	—
189	C₂²×#120	C₂²×D₄	1	2	yes	—
190	C₂²×#120	C₂²×D₄	1	2	yes	—
191	C₂²×#120	C₂²×D₄	1	2	yes	—
192	C₂²×#120	H(5,𝔽₂)	1	4	yes	—
193	C₂×#518	C₂²×D₄	1	4	yes	—
194	C₂×#518	C₂²×D₄	1	2	yes	—
195	C₂×#518	C₂²×D₄	1	2	yes	—
196	C₂×#518	H(5,𝔽₂)	1	2	yes	—
197	C₂³×C₂²⋊C₄	#120	1	4	yes	—
198	C₂³×C₂²⋊C₄	C₂²×D₄	1	2	yes	—
199	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
200	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂³×C₄²	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂²×#99	C₂²×D₄	1	8	yes	—
203	#2315	#120	1	4	yes	—
204	#2308	C₂²×D₄	1	4	yes	—
205	C₂×#381	#127	1	2	yes	complemented
206	C₂×#381	#127	1	2	yes	complemented
207	C₂×#379	#120	1	2	yes	—
208	C₂×#379	#120	1	2	yes	—
209	C₂×#379	#127	1	2	yes	complemented
210	C₂×#379	#127	1	2	yes	complemented
211	C₂×#379	#127	1	2	yes	complemented
212	C₂×#379	#127	1	1	yes	characteristic, complemented
213	C₂×#379	#127	1	1	yes	characteristic, complemented
214	C₂×#379	C₂²×D₄	1	4	yes	—
215	C₂×#379	C₂²×D₄	1	4	yes	—
216	#2273	#120	1	2	yes	—
217	#2273	#120	1	2	yes	—
218	#1783	C₂²×D₄	1	8	yes	—
219	#1781	C₂²×D₄	1	8	yes	—
220	#1781	H(5,𝔽₂)	1	4	yes	—
221	#1438	H(5,𝔽₂)	1	4	yes	—
222	C₂⁶	C₂×#120	1	2	yes	—
223	C₂⁶	C₂×#127	1	1	yes	characteristic
224	C₂⁶	C₂×#127	1	1	yes	characteristic
225	C₂⁶	C₂×#127	1	2	yes	—
226	C₂⁶	#535	1	2	yes	—
227	C₂⁶	C₂³×D₄	1	1	yes	characteristic
228	C₂⁶	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂³×D₄	C₂³×D₄	1	2	yes	—
230	C₂³×D₄	C₂³×D₄	1	2	yes	—
231	C₂³×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
232	C₂³×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
233	C₂³×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
234	C₂³×D₄	C₂×H(5,𝔽₂)	1	2	yes	—
235	C₂⁴×C₄	C₂×#127	1	2	yes	—
236	C₂⁴×C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
237	C₂×#120	D₄²	1	8	yes	—
238	C₂×#120	D₄²	1	8	yes	—
239	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
240	C₂²×C₂²⋊C₄	C₂×#120	1	4	yes	—
241	C₂²×C₂²⋊C₄	C₂×#120	1	4	yes	—
242	C₂²×C₂²⋊C₄	C₂×#127	1	2	yes	—
243	C₂²×C₂²⋊C₄	C₂×#127	1	2	yes	—
244	C₂²×C₂²⋊C₄	C₂×#127	1	2	yes	—
245	C₂²×C₂²⋊C₄	#535	1	2	yes	—
246	#379	B(4,𝔽₂)	1	4	yes	complemented
247	#379	B(4,𝔽₂)	1	4	yes	complemented
248	C₂⁵	#2433	1	2	yes	—
249	C₂⁵	#2433	1	2	yes	—
250	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
251	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
252	C₂⁵	#3027	1	2	yes	—
253	C₂⁵	#3027	1	2	yes	—
254	C₂⁵	C₂²×#120	1	1	yes	characteristic
255	C₂⁵	C₂²×#127	1	1	yes	characteristic
256	C₂⁵	C₂×#535	1	1	yes	characteristic
257	C₂⁵	#3484	1	1	yes	characteristic
258	C₂⁵	#3484	1	1	yes	characteristic
259	C₂⁵	#3484	1	2	yes	—
260	C₂⁵	#3484	1	1	yes	characteristic
261	C₂⁵	#3484	1	1	yes	characteristic
262	C₂⁵	#3484	1	2	yes	—
263	C₂⁵	#3537	1	2	yes	—
264	C₂²×D₄	C₂×D₄²	1	4	yes	—
265	C₂²×D₄	#3484	1	4	yes	—
266	C₂²×D₄	#3498	1	4	yes	—
267	C₂²×D₄	#3537	1	4	yes	—
268	C₂³×C₄	#3484	1	2	yes	—
269	C₂³×C₄	#3484	1	2	yes	—
270	C₂³×C₄	#3537	1	2	yes	—
271	C₂×C₂²⋊C₄	#2433	1	4	yes	—
272	C₂×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	8	yes	—
273	C₂⁴	#25771	1	2	yes	—
274	C₂⁴	#25771	1	2	yes	—
275	C₂⁴	#25771	1	2	yes	—
276	C₂⁴	#25771	1	2	yes	—
277	C₂⁴	C₂×#2433	1	1	yes	characteristic
278	C₂⁴	#35346	1	1	yes	characteristic
279	C₂⁴	#35390	1	1	yes	characteristic
280	C₂⁴	#35390	1	1	yes	characteristic
281	C₂⁴	#35527	1	1	yes	characteristic
282	C₂⁴	#36717	1	1	yes	characteristic
283	C₂⁴	#36717	1	1	yes	characteristic
284	C₂⁴	#43001	1	4	yes	—
285	C₂⁴	C₂²×B(4,𝔽₂)	1	2	yes	—
286	C₂⁴	C₂×#3027	1	2	yes	—
287	C₂²×C₄	#43001	1	4	yes	—
288	C₂³	#763814	1	1	yes	characteristic
289	C₂³	C₂×#25771	1	2	yes	—
290	C₂³	#6499867	1	1	yes	characteristic
291	C₂³	#6499871	1	1	yes	characteristic
292	C₂³	#6499874	1	4	yes	—
293	C₂³	#6516837	1	2	yes	—
294	C₂²	#72210439	1	2	yes	—
295	C₂²	#72225644	1	2	yes	—
296	C₂²	#72240637	1	2	yes	—
297	C₂²	#72251139	1	1	yes	characteristic
298	C₂	#72251646	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72251646
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁵
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72262720	C₂	semidirect
2. #72262718	C₂	semidirect
3. #72262702	C₂	semidirect
4. #72262701	C₂²	semidirect
5. #72262683	C₂	semidirect
6. #72262641	C₂	semidirect
7. #72262637	C₂²	semidirect
8. #72256281	C₂	semidirect
9. #72256233	C₂	semidirect
10. #72256222	C₂²	semidirect
11. #72256024	C₂	semidirect
12. #72256001	C₂²	semidirect
13. #72255986	C₂²	semidirect
14. #72255893	C₂²	semidirect
15. #72255857	C₂	semidirect
16. #72255853	C₂²	semidirect
17. #72255849	C₂²	semidirect
18. #72255803	C₂²	semidirect
19. #72255785	C₂²	semidirect
20. C₂×#144380	C₂²	semidirect
21. #72251552	C₂²	semidirect
22. #72251475	C₂²	semidirect
23. #72249634	C₂	semidirect
24. #72249626	C₂²	semidirect
25. #72249624	C₂²	semidirect
26. #72249612	C₂²	semidirect
27. #72249593	C₂²	semidirect
28. #72240083	C₂²	semidirect
29. #72238974	C₂²	semidirect
30. #72238598	C₂	semidirect
31. #72237900	C₂²	semidirect
32. #72237381	C₂	semidirect
33. #72237374	C₂²	semidirect
34. #72237039	C₂	semidirect
35. #72237037	C₂²	semidirect
36. #72237034	C₂²	semidirect
37. #72237033	C₂²	semidirect
38. #72236928	C₂²	semidirect
39. #72236631	C₂²	semidirect
40. #72236521	C₂²	semidirect
41. #72236508	C₂²	semidirect
42. #72235238	C₂²	semidirect
43. #72235234	C₂²	semidirect
44. #72235130	C₂²	semidirect
45. #72234953	C₂²	semidirect
46. #72234675	C₂²	semidirect
49. C₂×#511771	C₂²	semidirect
50. C₂×#188035	C₂²	semidirect
52. #72225712	C₂²	semidirect
54. #72225644	C₂²	semidirect
56. #72225610	C₂²	semidirect
58. #72219649	C₂²	semidirect
59. #72219424	C₂²	semidirect
61. #72219158	C₂²	semidirect
62. #72218838	C₂²	semidirect
63. #72218114	C₂²	semidirect
64. C₂×#144378	C₂²	semidirect
65. C₂×#139731	C₂²	semidirect
66. C₂×#511790	C₂²	semidirect
74. #6442728	D₄	semidirect
87. C₂²×#1781	D₄	semidirect
102. #364532	D₄	semidirect
103. #364173	D₄	semidirect
104. #359705	D₄	semidirect
107. #355913	D₄	semidirect
118. #133826	D₄	semidirect
119. #133825	D₄	semidirect
120. #133824	D₄	semidirect
121. #133822	D₄	semidirect
131. #62842	C₂×D₄	semidirect
139. C₂×#2883	C₂×D₄	semidirect
154. #20633	C₂×D₄	semidirect
155. #20609	C₂×D₄	semidirect
162. #20354	C₂×D₄	semidirect
164. #12688	C₂×D₄	semidirect
167. #12687	C₂×D₄	semidirect
168. #10768	C₂×D₄	semidirect
169. #10754	C₂×D₄	semidirect
171. #8548	C₂×D₄	semidirect
172. #8547	C₂×D₄	semidirect
173. #8543	C₂×D₄	semidirect
174. #8541	C₂×D₄	semidirect
175. C₂⁷	#127	semidirect
180. C₂⁴×D₄	#127	semidirect
184. C₂⁵×C₄	#127	semidirect
205. C₂×#381	#127	semidirect
206. C₂×#381	#127	semidirect
209. C₂×#379	#127	semidirect
210. C₂×#379	#127	semidirect
211. C₂×#379	#127	semidirect
212. C₂×#379	#127	semidirect
213. C₂×#379	#127	semidirect
246. #379	B(4,𝔽₂)	semidirect
247. #379	B(4,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#72262720	2
#72262718	2
#72262702	2
#72262683	2
#72262641	2
#72256281	2
#72256233	2
#72256024	2
#72255857	2
#72249634	2
#72238598	2
#72237381	2
#72237039	2