Subgroups — #72310543

Order 4096.

At least 16741 conjugacy classes of subgroups; only the 543 normal subgroups are shown.

The subgroup lattice has 286 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310542	C₂	1	4	yes	maximal, complemented
2	#72310517	C₂	1	4	yes	maximal, complemented
3	#72309397	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#97551	C₂²	1	2	yes	complemented
5	#72237628	C₂	1	1	yes	maximal, characteristic
6	#72237624	C₄	1	2	yes	complemented
7	#72237620	C₄	1	2	yes	complemented
8	#72237618	C₄	1	1	yes	characteristic, complemented
9	#72237612	C₂²	1	2	yes	—
10	#72237038	C₂	1	2	yes	maximal, complemented
11	#72237035	C₂²	1	8	yes	complemented
12	#72237032	C₂	1	2	yes	maximal, complemented
13	C₂×#97550	C₂²	1	2	yes	complemented
14	#72234688	C₂	1	1	yes	maximal, characteristic, complemented
15	#72234668	C₄	1	1	yes	characteristic, complemented
16	#72234625	C₄	1	1	yes	characteristic, complemented
17	#72234620	C₂²	1	2	yes	complemented
18	#72232592	C₄	1	1	yes	characteristic, complemented
19	#72225776	C₂²	1	2	yes	complemented
20	#72225733	C₂²	1	2	yes	—
21	C₂×#7670203	C₂²	1	1	yes	characteristic
22	#72219022	C₂²	1	8	yes	complemented
23	#72218518	C₂²	1	2	yes	—
24	#72218375	C₂²	1	2	yes	—
25	#72217789	C₂²	1	1	yes	characteristic
26	#72217511	C₂²	1	1	yes	characteristic
27	C₂×#62817	C₂×C₄	1	1	yes	characteristic, complemented
28	C₂²×#3484	D₄	1	1	yes	characteristic
29	C₂²×#3484	D₄	1	1	yes	characteristic
30	C₂⁴×#120	D₄	1	1	yes	characteristic
31	C₂⁴×#120	C₂³	1	1	yes	characteristic
32	#7642273	C₂³	1	2	yes	—
33	C₂²×#2846	D₄	1	1	yes	characteristic
34	#7630944	C₂×C₄	1	2	yes	complemented
35	C₂×#35390	D₄	1	1	yes	characteristic
36	C₂×#35348	D₄	1	1	yes	characteristic
37	C₂×#35348	C₂³	1	1	yes	characteristic
38	C₂×#35346	C₂³	1	1	yes	characteristic
39	C₂×#34981	C₂×C₄	1	1	yes	characteristic
40	C₂×#34981	C₂×C₄	1	1	yes	characteristic
41	C₂×#34980	C₂×C₄	1	1	yes	characteristic
42	C₂×#34980	D₄	1	1	yes	characteristic
43	#7626202	C₂×C₄	1	1	yes	characteristic, complemented
44	C₂×#34198	C₂×C₄	1	1	yes	characteristic
45	C₂×#34198	D₄	1	1	yes	characteristic
46	#6453039	C₂×C₄	1	2	yes	complemented
47	#6451874	C₂×C₄	1	2	yes	—
48	#6451874	C₂×C₄	1	2	yes	—
49	#6451874	C₂×C₄	1	2	yes	complemented
50	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
52	#417896	D₄	1	4	yes	—
53	#417896	D₄	1	4	yes	—
54	#417824	D₄	1	4	yes	—
55	#417824	D₄	1	4	yes	—
56	#392207	C₂×C₄	1	2	yes	complemented
57	#392109	D₄	1	4	yes	complemented
58	#382422	C₂³	1	2	yes	—
59	#359921	C₂×C₄	1	2	yes	complemented
60	#359705	C₂×C₄	1	2	yes	—
61	#359705	C₂×C₄	1	2	yes	—
62	#359705	C₂×C₄	1	2	yes	complemented
63	C₂×#10965	C₂³	1	4	yes	—
64	C₂×#8541	D₄	1	4	yes	complemented
65	C₂×#8541	C₂³	1	4	yes	—
66	C₂⁸	C₂×D₄	1	1	yes	characteristic
67	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
68	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂⁵×D₄	C₂×D₄	1	2	yes	—
70	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
71	C₂×#3490	C₂²×C₄	1	2	yes	—
72	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×#534	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×#534	C₂×D₄	1	1	yes	characteristic
77	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
79	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
80	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
81	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
82	C₂³×#120	C₂×D₄	1	1	yes	characteristic
83	C₂³×#120	C₂×D₄	1	1	yes	characteristic
84	C₂³×#120	C₂×D₄	1	1	yes	characteristic
85	C₂³×#120	C₂×D₄	1	1	yes	characteristic
86	C₂³×#120	C₂×D₄	1	1	yes	characteristic
87	C₂³×#120	C₂×D₄	1	2	yes	—
88	C₂³×#120	#41	1	2	yes	—
89	C₂³×#120	C₂⁴	1	1	yes	characteristic
90	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×#518	C₂×D₄	1	1	yes	characteristic
92	C₂⁴×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
93	C₂⁴×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂⁴×C₂²⋊C₄	#41	1	2	yes	—
95	#35390	C₂×D₄	1	4	yes	—
96	#35367	C₂×D₄	1	4	yes	—
97	#35346	C₂×D₄	1	4	yes	—
98	#35346	C₂×D₄	1	4	yes	—
99	#34176	C₂²×C₄	1	2	yes	—
100	C₂×#2315	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×#2315	C₂²×C₄	1	1	yes	characteristic
102	C₂×#2308	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
104	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
105	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
106	C₂×#2308	C₂×D₄	1	1	yes	characteristic
107	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
108	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
109	C₂²×#379	C₂×D₄	1	1	yes	characteristic
110	C₂×#2273	C₂²⋊C₄	1	1	yes	characteristic
111	#23468	C₂²⋊C₄	1	4	yes	complemented
112	#23468	C₂²⋊C₄	1	2	yes	complemented
113	#23468	C₂²⋊C₄	1	2	yes	complemented
114	#23444	C₂²⋊C₄	1	4	yes	complemented
115	#23444	C₂²⋊C₄	1	2	yes	complemented
116	#23444	C₂²⋊C₄	1	2	yes	complemented
117	C₂×#1819	C₂²×C₄	1	2	yes	—
118	C₂×#1819	C₂²×C₄	1	2	yes	—
119	C₂×#1781	C₂×D₄	1	8	yes	complemented
120	#10965	C₂×D₄	1	4	yes	—
121	#10965	C₂×D₄	1	4	yes	—
122	#10938	C₂²⋊C₄	1	4	yes	complemented
123	#10938	C₂²⋊C₄	1	4	yes	complemented
124	#10938	C₂²⋊C₄	1	4	yes	complemented
125	#10938	C₂²⋊C₄	1	4	yes	complemented
126	C₂⁷	#99	1	1	yes	characteristic
127	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂⁷	#120	1	1	yes	characteristic
130	C₂⁷	#121	1	2	yes	—
131	C₂⁷	#121	1	2	yes	—
132	C₂⁷	C₂²×D₄	1	1	yes	characteristic
133	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
134	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
135	C₂⁴×D₄	#99	1	1	yes	characteristic
136	C₂⁴×D₄	#99	1	1	yes	characteristic
137	C₂⁴×D₄	#99	1	2	yes	—
138	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂⁴×D₄	C₄×D₄	1	2	yes	—
143	C₂⁴×D₄	#120	1	1	yes	characteristic
144	C₂⁴×D₄	#120	1	2	yes	—
145	C₂⁴×D₄	#121	1	2	yes	—
146	C₂⁴×D₄	#123	1	1	yes	characteristic
147	C₂⁴×D₄	C₂²×D₄	1	2	yes	—
148	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
149	C₂⁵×C₄	#99	1	1	yes	characteristic
150	C₂⁵×C₄	C₄×D₄	1	2	yes	—
151	C₂⁵×C₄	#123	1	1	yes	characteristic
152	C₂×#543	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×#534	#99	1	2	yes	—
154	C₂×#534	#99	1	2	yes	—
155	C₂×#534	#99	1	4	yes	—
156	C₂×#534	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂²×#120	#120	1	2	yes	—
158	C₂²×#120	#120	1	2	yes	—
159	C₂²×#120	#120	1	4	yes	—
160	C₂²×#120	C₂²×D₄	1	4	yes	—
161	C₂²×#120	H(5,𝔽₂)	1	4	yes	—
162	C₂³×C₂²⋊C₄	#99	1	2	yes	—
163	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
170	C₂³×C₂²⋊C₄	C₄×D₄	1	2	yes	—
171	C₂³×C₂²⋊C₄	#121	1	2	yes	—
172	C₂³×C₂²⋊C₄	#123	1	2	yes	—
173	C₂³×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
174	C₂³×C₂²⋊C₄	C₂×#41	1	2	yes	—
175	#2308	C₂×C₂²⋊C₄	1	2	yes	—
176	#2308	C₂×C₂²⋊C₄	1	2	yes	—
177	#2308	C₂×C₂²⋊C₄	1	4	yes	—
178	#2308	#120	1	4	yes	—
179	#2273	C₂×C₂²⋊C₄	1	4	yes	—
180	C₂×#342	C₂×C₂²⋊C₄	1	4	yes	—
181	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
182	C₂×#342	C₂×C₂²⋊C₄	1	2	yes	—
183	C₂⁶	#386	1	1	yes	characteristic
184	C₂⁶	#386	1	1	yes	characteristic
185	C₂⁶	#386	1	1	yes	characteristic
186	C₂⁶	#386	1	2	yes	—
187	C₂⁶	C₂×#99	1	1	yes	characteristic
188	C₂⁶	C₂×#99	1	2	yes	—
189	C₂⁶	C₂×#99	1	1	yes	characteristic
190	C₂⁶	C₂×#99	1	1	yes	characteristic
191	C₂⁶	C₂×#99	1	1	yes	characteristic
192	C₂⁶	C₂×#99	1	1	yes	characteristic
193	C₂⁶	C₂²×C₂²⋊C₄	1	1	yes	characteristic
194	C₂⁶	C₂×#120	1	1	yes	characteristic
195	C₂⁶	C₂×#120	1	2	yes	—
196	C₂⁶	C₂×#121	1	2	yes	—
197	C₂⁶	#534	1	2	yes	—
198	C₂³×D₄	#379	1	1	yes	characteristic
199	C₂³×D₄	#379	1	1	yes	characteristic
200	C₂³×D₄	#379	1	2	yes	—
201	C₂³×D₄	C₂×#99	1	2	yes	—
202	C₂³×D₄	C₂×#99	1	2	yes	—
203	C₂³×D₄	C₂×#99	1	1	yes	characteristic
204	C₂³×D₄	C₂×#99	1	1	yes	characteristic
205	C₂³×D₄	C₂×#99	1	2	yes	—
206	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
207	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
208	C₂³×D₄	C₂×#120	1	2	yes	—
209	C₂³×D₄	#534	1	2	yes	—
210	C₂³×D₄	D₄²	1	8	yes	—
211	C₂⁴×C₄	#386	1	2	yes	—
212	C₂⁴×C₄	#386	1	1	yes	characteristic
213	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
214	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
215	C₂⁴×C₄	C₂×#99	1	2	yes	—
216	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
217	C₂⁴×C₄	C₂×C₄×D₄	1	2	yes	—
218	C₂⁴×C₄	C₂×#123	1	1	yes	characteristic
219	C₂²×C₂²⋊C₄	#379	1	2	yes	—
220	C₂²×C₂²⋊C₄	#379	1	2	yes	—
221	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
222	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
223	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₂²⋊C₄	#518	1	2	yes	—
225	C₂²×C₂²⋊C₄	#546	1	8	yes	—
226	C₂⁵	#1781	1	1	yes	characteristic
227	C₂⁵	#1781	1	2	yes	—
228	C₂⁵	#1781	1	1	yes	characteristic
229	C₂⁵	#1889	1	2	yes	—
230	C₂⁵	#1889	1	2	yes	—
231	C₂⁵	#1898	1	2	yes	—
232	C₂⁵	#1898	1	2	yes	—
233	C₂⁵	#1898	1	2	yes	—
234	C₂⁵	#1913	1	2	yes	—
235	C₂⁵	#1913	1	2	yes	—
236	C₂⁵	C₂×#379	1	1	yes	characteristic
237	C₂⁵	C₂×#386	1	1	yes	characteristic
238	C₂⁵	#2308	1	1	yes	characteristic
239	C₂⁵	#2403	1	2	yes	—
240	C₂⁵	#2410	1	4	yes	—
241	C₂⁵	C₂²×#99	1	1	yes	characteristic
242	C₂⁵	C₂²×#99	1	1	yes	characteristic
243	C₂⁵	C₂²×#99	1	1	yes	characteristic
244	C₂⁵	C₂²×#99	1	1	yes	characteristic
245	C₂⁵	C₂×B(4,𝔽₂)	1	4	yes	—
246	C₂³×C₄	#1781	1	2	yes	—
247	C₂³×C₄	#1781	1	1	yes	characteristic
248	C₂³×C₄	#1781	1	1	yes	characteristic
249	C₂³×C₄	#1889	1	4	yes	—
250	C₂³×C₄	#1898	1	2	yes	—
251	C₂³×C₄	D₄×C₂²⋊C₄	1	4	yes	—
252	C₂³×C₄	#2405	1	2	yes	—
253	C₂⁴	#9495	1	2	yes	—
254	C₂⁴	#9495	1	2	yes	—
255	C₂⁴	#9495	1	2	yes	—
256	C₂⁴	#9495	1	2	yes	—
257	C₂⁴	#12707	1	2	yes	complemented
258	C₂⁴	#13867	1	1	yes	characteristic
259	C₂⁴	C₂×#1781	1	1	yes	characteristic
260	C₂⁴	C₂×#1781	1	1	yes	characteristic
261	C₂⁴	C₂×#1889	1	2	yes	—
262	C₂⁴	C₂×#1898	1	1	yes	characteristic
263	C₂⁴	C₂×#1898	1	1	yes	characteristic
264	C₂⁴	C₂×#1913	1	2	yes	—
265	C₂⁴	#21746	1	1	yes	characteristic
266	C₂⁴	#21746	1	1	yes	characteristic
267	C₂⁴	#23900	1	4	yes	—
268	C₂²×C₄	#12707	1	2	yes	complemented
269	C₂³	#139743	1	4	yes	—
270	C₂³	C₂×#9495	1	2	yes	—
271	C₂³	#201664	1	2	yes	—
272	C₂³	#359428	1	1	yes	characteristic
273	C₂³	#359442	1	1	yes	characteristic
274	C₂³	#359501	1	1	yes	characteristic
275	C₂³	C₂×#12707	1	1	yes	characteristic
276	C₂³	C₂×#12707	1	1	yes	characteristic
277	C₂³	#493397	1	2	yes	—
278	C₂²	#72218057	1	1	yes	characteristic
279	C₂²	C₂×#139743	1	2	yes	—
280	C₂²	#72218846	1	1	yes	characteristic
281	C₂²	#72235239	1	1	yes	characteristic
282	C₂²	#72255841	1	2	yes	—
283	C₂	#72258080	1	2	yes	—
284	C₂	#72266167	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72218846
commutator	C₂³×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂³×#120	C₂⁴
prime radical	#72237628	C₂
LCS	G ▷ C₂³×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×#120 ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72310542	C₂	semidirect
2. #72310517	C₂	semidirect
3. #72309397	C₂	semidirect
4. C₂×#97551	C₂²	semidirect
6. #72237624	C₄	semidirect
7. #72237620	C₄	semidirect
8. #72237618	C₄	semidirect
10. #72237038	C₂	semidirect
11. #72237035	C₂²	semidirect
12. #72237032	C₂	semidirect
13. C₂×#97550	C₂²	semidirect
14. #72234688	C₂	semidirect
15. #72234668	C₄	semidirect
16. #72234625	C₄	semidirect
17. #72234620	C₂²	semidirect
18. #72232592	C₄	semidirect
19. #72225776	C₂²	semidirect
22. #72219022	C₂²	semidirect
27. C₂×#62817	C₂×C₄	semidirect
34. #7630944	C₂×C₄	semidirect
43. #7626202	C₂×C₄	semidirect
46. #6453039	C₂×C₄	semidirect
49. #6451874	C₂×C₄	semidirect
50. C₂×#21746	C₂×C₄	semidirect
51. C₂×#21746	C₂×C₄	semidirect
56. #392207	C₂×C₄	semidirect
57. #392109	D₄	semidirect
59. #359921	C₂×C₄	semidirect
62. #359705	C₂×C₄	semidirect
64. C₂×#8541	D₄	semidirect
111. #23468	C₂²⋊C₄	semidirect
112. #23468	C₂²⋊C₄	semidirect
113. #23468	C₂²⋊C₄	semidirect
114. #23444	C₂²⋊C₄	semidirect
115. #23444	C₂²⋊C₄	semidirect
116. #23444	C₂²⋊C₄	semidirect
119. C₂×#1781	C₂×D₄	semidirect
122. #10938	C₂²⋊C₄	semidirect
123. #10938	C₂²⋊C₄	semidirect
124. #10938	C₂²⋊C₄	semidirect
125. #10938	C₂²⋊C₄	semidirect
257. C₂⁴	#12707	semidirect
268. C₂²×C₄	#12707	semidirect

Maximal subgroups

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

subgroup	index
#72310542	2
#72310517	2
#72309397	2
#72237628	2
#72237038	2
#72237032	2
#72234688	2