Subgroups — #72256437

Order 4096.

At least 13101 conjugacy classes of subgroups; only the 537 normal subgroups are shown.

The subgroup lattice has 266 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251906	C₂	1	2	yes	maximal, complemented
2	#72251872	C₂	1	2	yes	maximal, complemented
3	#72251733	C₂	1	4	yes	maximal, complemented
4	#72251681	C₂²	1	4	yes	complemented
5	#72238605	C₂	1	1	yes	maximal, characteristic, complemented
6	#72238418	C₂	1	2	yes	maximal, complemented
7	#72237547	C₂²	1	1	yes	characteristic, complemented
8	#72237543	C₂²	1	1	yes	characteristic, complemented
9	#72237534	C₂²	1	2	yes	complemented
10	#72237049	C₂²	1	4	yes	complemented
11	#72236862	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236704	C₂²	1	2	yes	complemented
13	#72236557	C₂²	1	2	yes	complemented
14	#72235124	C₂	1	1	yes	maximal, characteristic
15	#72235121	C₄	1	1	yes	characteristic, complemented
16	#72235115	C₄	1	1	yes	characteristic, complemented
17	#72235111	C₂²	1	1	yes	characteristic
18	#72235102	C₂²	1	2	yes	—
19	#72235102	C₂²	1	2	yes	—
20	#72235097	C₄	1	2	yes	complemented
21	#72235097	C₄	1	2	yes	complemented
22	#72235088	C₄	1	1	yes	characteristic, complemented
23	#72235079	C₄	1	1	yes	characteristic, complemented
24	C₂×#7630939	C₂²	1	1	yes	characteristic
25	#72216489	C₂	1	2	yes	maximal, complemented
26	#72214353	C₂²	1	2	yes	—
27	#72213548	C₂²	1	1	yes	characteristic
28	#72213429	C₂²	1	4	yes	complemented
29	#72213428	C₂²	1	4	yes	complemented
30	#72210555	C₂²	1	2	yes	—
31	C₂⁶×D₄	D₄	1	1	yes	characteristic
32	#10586169	C₂×C₄	1	1	yes	characteristic, complemented
33	C₂²×#3484	D₄	1	1	yes	characteristic
34	C₂²×#3484	D₄	1	1	yes	characteristic
35	C₂²×#3484	C₂³	1	1	yes	characteristic
36	#10575547	C₂³	1	2	yes	—
37	#10575142	C₂×C₄	1	2	yes	complemented
38	#10574294	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#36614	C₂×C₄	1	1	yes	characteristic
40	C₂×#36614	D₄	1	1	yes	characteristic
41	C₂×#35222	C₂×C₄	1	1	yes	characteristic
42	C₂×#35222	D₄	1	1	yes	characteristic
43	C₂×#35222	D₄	1	1	yes	characteristic
44	C₂×#35222	D₄	1	1	yes	characteristic
45	C₂×#35222	C₂³	1	1	yes	characteristic
46	C₂×#35222	C₂³	1	1	yes	characteristic
47	C₂×#34171	C₂×C₄	1	1	yes	characteristic
48	C₂×#34171	C₂×C₄	1	1	yes	characteristic
49	C₂×#34171	D₄	1	1	yes	characteristic
50	#6451869	C₂×C₄	1	2	yes	—
51	#6451869	C₂×C₄	1	2	yes	complemented
52	#6451748	C₂×C₄	1	2	yes	—
53	#6451748	C₂×C₄	1	2	yes	complemented
54	#6374136	C₂×C₄	1	1	yes	characteristic, complemented
55	#6374123	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂×#12707	D₄	1	8	yes	complemented
57	#493385	C₂³	1	2	yes	—
58	#493372	C₂³	1	2	yes	—
59	#493370	C₂³	1	4	yes	—
60	#417954	D₄	1	8	yes	—
61	#417774	D₄	1	8	yes	—
62	#392259	C₂×C₄	1	2	yes	complemented
63	#382414	C₂³	1	2	yes	—
64	#359766	C₂×C₄	1	2	yes	—
65	#359766	C₂×C₄	1	2	yes	complemented
66	#359762	C₂×C₄	1	2	yes	—
67	#359762	C₂×C₄	1	2	yes	complemented
68	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
69	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
73	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
74	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
75	C₂⁶×C₄	C₂²⋊C₄	1	1	yes	characteristic
76	#62814	C₂²×C₄	1	2	yes	—
77	C₂×#3484	C₂×D₄	1	4	yes	—
78	C₂×#3484	#41	1	4	yes	—
79	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×#535	C₂×D₄	1	1	yes	characteristic
82	C₂²×#535	C₂×D₄	1	1	yes	characteristic
83	C₂²×#535	C₂×D₄	1	1	yes	characteristic
84	C₂²×#535	C₂⁴	1	1	yes	characteristic
85	C₂²×#525	C₂²⋊C₄	1	1	yes	characteristic
86	C₂²×#525	C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×#525	C₂²×C₄	1	1	yes	characteristic
88	C₂²×#525	C₂×D₄	1	1	yes	characteristic
89	C₂²×#525	C₂×D₄	1	1	yes	characteristic
90	C₂²×#525	C₂×D₄	1	1	yes	characteristic
91	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×#518	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×#518	C₂²×C₄	1	1	yes	characteristic
94	#60644	C₂×D₄	1	4	yes	—
95	#60619	C₂×D₄	1	4	yes	—
96	#60124	C₂²×C₄	1	2	yes	—
97	#35222	C₂×D₄	1	8	yes	—
98	C₂×#2309	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×#2309	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×#2309	C₂²×C₄	1	1	yes	characteristic
101	C₂×#2309	C₂²×C₄	1	1	yes	characteristic
102	C₂×#2309	C₂×D₄	1	1	yes	characteristic
103	C₂×#2309	C₂×D₄	1	1	yes	characteristic
104	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×#2280	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×#2280	C₂²×C₄	1	1	yes	characteristic
109	C₂×#2280	C₂²×C₄	1	1	yes	characteristic
110	C₂×#2127	C₂×D₄	1	8	yes	complemented
111	#23467	C₂²⋊C₄	1	4	yes	complemented
112	#23463	C₂²⋊C₄	1	4	yes	complemented
113	#23443	C₂²⋊C₄	1	4	yes	complemented
114	#23407	C₂²⋊C₄	1	4	yes	complemented
115	#20694	C₂²×C₄	1	2	yes	—
116	#20688	C₂²×C₄	1	2	yes	—
117	#11006	C₂×D₄	1	8	yes	—
118	#10948	C₂²⋊C₄	1	8	yes	complemented
119	#10937	C₂²⋊C₄	1	8	yes	complemented
120	C₂⁷	#99	1	1	yes	characteristic
121	C₂²×H(5,𝔽₂)	#121	1	4	yes	—
122	C₂³×#41	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂³×#41	C₂²×D₄	1	1	yes	characteristic
124	C₂⁴×D₄	#99	1	2	yes	—
125	C₂⁴×D₄	#99	1	2	yes	—
126	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂⁴×D₄	#120	1	4	yes	—
130	C₂⁴×D₄	#120	1	2	yes	—
131	C₂⁴×D₄	#123	1	2	yes	—
132	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
133	C₂⁵×C₄	#99	1	1	yes	characteristic
134	C₂⁵×C₄	#99	1	1	yes	characteristic
135	C₂⁵×C₄	#99	1	1	yes	characteristic
136	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	#3458	C₂×C₂²⋊C₄	1	2	yes	complemented
140	#3449	C₂×C₂²⋊C₄	1	2	yes	complemented
141	C₂×#535	#120	1	4	yes	—
142	C₂×#535	#120	1	2	yes	—
143	C₂×#535	#123	1	2	yes	—
144	C₂×#535	C₂²×D₄	1	4	yes	—
145	C₂×#535	C₂²×D₄	1	2	yes	—
146	C₂×#535	C₂×#41	1	2	yes	—
147	C₂²×#127	#99	1	2	yes	—
148	C₂²×#127	H(5,𝔽₂)	1	2	yes	—
149	C₂²×#124	#99	1	2	yes	—
150	C₂²×#124	H(5,𝔽₂)	1	2	yes	—
151	C₂×#525	#99	1	4	yes	—
152	C₂×#525	C₄×D₄	1	4	yes	—
153	C₂×#525	#120	1	4	yes	—
154	C₂×#525	#121	1	4	yes	—
155	C₂×#518	C₄×D₄	1	4	yes	—
156	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×#117	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×#117	C₂³×C₄	1	1	yes	characteristic
162	#2873	C₂×C₂²⋊C₄	1	4	yes	complemented
163	#2309	C₂×C₂²⋊C₄	1	4	yes	—
164	#2309	#120	1	4	yes	—
165	#2280	C₂×C₂²⋊C₄	1	4	yes	—
166	#1742	C₂×C₂²⋊C₄	1	4	yes	complemented
167	#1740	C₂×C₂²⋊C₄	1	4	yes	complemented
168	C₂⁶	C₂×#99	1	1	yes	characteristic
169	C₂⁶	C₂×#99	1	1	yes	characteristic
170	C₂⁶	C₂×#99	1	1	yes	characteristic
171	C₂²×#41	#386	1	2	yes	—
172	C₂²×#41	C₂×#99	1	2	yes	—
173	C₂²×#41	C₂×#120	1	2	yes	—
174	C₂²×#41	C₂×#121	1	2	yes	—
175	C₂³×Q₈	C₂×#99	1	1	yes	characteristic
176	C₂³×Q₈	#534	1	1	yes	characteristic
177	C₂³×D₄	#379	1	2	yes	—
178	C₂³×D₄	#386	1	2	yes	—
179	C₂³×D₄	C₂×#99	1	1	yes	characteristic
180	C₂³×D₄	C₂×#99	1	2	yes	—
181	C₂³×D₄	C₂×#99	1	2	yes	—
182	C₂³×D₄	C₂×#99	1	1	yes	characteristic
183	C₂³×D₄	C₂×#99	1	1	yes	characteristic
184	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
185	C₂³×D₄	B(4,𝔽₂)	1	4	yes	—
186	C₂³×D₄	C₂×#120	1	2	yes	—
187	C₂³×D₄	C₂×#120	1	1	yes	characteristic
188	C₂³×D₄	C₂×#123	1	1	yes	characteristic
189	C₂³×D₄	#534	1	2	yes	—
190	C₂³×D₄	#534	1	1	yes	characteristic
191	C₂⁴×C₄	#379	1	2	yes	—
192	C₂⁴×C₄	#386	1	2	yes	—
193	C₂⁴×C₄	C₂×#99	1	2	yes	—
194	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
195	C₂⁴×C₄	C₂×#99	1	2	yes	—
196	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
197	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
198	C₂⁴×C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×#127	D₄²	1	4	yes	—
200	C₂×#127	#546	1	4	yes	—
201	C₂×#124	D₄²	1	4	yes	—
202	C₂×#124	#546	1	4	yes	—
203	C₂×#117	#379	1	2	yes	—
204	C₂×#117	#379	1	2	yes	—
205	C₂×#117	#386	1	2	yes	—
206	C₂×#117	C₂×#99	1	2	yes	—
207	C₂×#117	C₂²×C₂²⋊C₄	1	2	yes	—
208	C₂×#117	C₂×C₄×D₄	1	2	yes	—
209	C₂²×C₄²	C₂×#99	1	2	yes	—
210	C₂²×C₄²	#518	1	2	yes	—
211	C₂⁵	#1781	1	2	yes	—
212	C₂⁵	#1898	1	2	yes	—
213	C₂⁵	C₂²×#99	1	1	yes	characteristic
214	C₂²×Q₈	#1898	1	2	yes	—
215	C₂²×Q₈	#2410	1	2	yes	—
216	C₂²×D₄	#1781	1	2	yes	—
217	C₂²×D₄	#1781	1	2	yes	—
218	C₂²×D₄	#1889	1	4	yes	—
219	C₂²×D₄	#1898	1	2	yes	—
220	C₂²×D₄	#2403	1	2	yes	—
221	C₂²×D₄	#2405	1	2	yes	—
222	C₂²×D₄	#2410	1	2	yes	—
223	C₂²×D₄	C₂×B(4,𝔽₂)	1	4	yes	—
224	C₂³×C₄	#1781	1	2	yes	—
225	C₂³×C₄	#1889	1	2	yes	—
226	C₂³×C₄	#1889	1	2	yes	—
227	C₂³×C₄	#1898	1	2	yes	—
228	C₂³×C₄	#1913	1	2	yes	—
229	C₂³×C₄	#1913	1	2	yes	—
230	C₂³×C₄	C₂×#379	1	1	yes	characteristic
231	C₂³×C₄	C₂×#386	1	1	yes	characteristic
232	C₂³×C₄	#2308	1	1	yes	characteristic
233	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
234	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
235	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
236	C₂×C₄²	D₄×C₂²⋊C₄	1	4	yes	—
237	C₂⁴	#9495	1	2	yes	—
238	C₂⁴	#9495	1	2	yes	—
239	C₂⁴	#12707	1	2	yes	—
240	C₂⁴	#12707	1	2	yes	—
241	C₂⁴	C₂×#1781	1	1	yes	characteristic
242	C₂⁴	C₂×#1898	1	1	yes	characteristic
243	C₂⁴	#21746	1	1	yes	characteristic
244	C₂×D₄	#12707	1	4	yes	—
245	C₂×D₄	#23900	1	4	yes	—
246	C₂²×C₄	#9495	1	2	yes	—
247	C₂²×C₄	#9495	1	2	yes	—
248	C₂²×C₄	#13867	1	1	yes	characteristic
249	C₂²×C₄	C₂×#1781	1	1	yes	characteristic
250	C₂²×C₄	C₂×#1889	1	2	yes	—
251	C₂²×C₄	C₂×#1898	1	1	yes	characteristic
252	C₂²×C₄	C₂×#1913	1	2	yes	—
253	C₂²×C₄	#21746	1	1	yes	characteristic
254	C₂³	C₂×#9495	1	2	yes	—
255	C₂³	#359428	1	1	yes	characteristic
256	C₂³	C₂×#12707	1	2	yes	—
257	C₂×C₄	#201664	1	2	yes	—
258	C₂×C₄	#359442	1	1	yes	characteristic
259	C₂×C₄	#359501	1	1	yes	characteristic
260	C₂×C₄	C₂×#12707	1	2	yes	—
261	C₂²	#72218057	1	1	yes	characteristic
262	C₂²	#72218057	1	1	yes	characteristic
263	C₂²	#72218846	1	1	yes	characteristic
264	C₂	#72240489	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72240489
commutator	C₂²×#117	C₂³×C₄
Frattini, cosocle	C₂²×#535	C₂⁴
prime radical	#72235124	C₂
LCS	G ▷ C₂²×#117 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂×#7630939 ◁ G
LECS	G ▷ C₂²×#535 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72251906	C₂	semidirect
2. #72251872	C₂	semidirect
3. #72251733	C₂	semidirect
4. #72251681	C₂²	semidirect
5. #72238605	C₂	semidirect
6. #72238418	C₂	semidirect
7. #72237547	C₂²	semidirect
8. #72237543	C₂²	semidirect
9. #72237534	C₂²	semidirect
10. #72237049	C₂²	semidirect
11. #72236862	C₂	semidirect
12. #72236704	C₂²	semidirect
13. #72236557	C₂²	semidirect
15. #72235121	C₄	semidirect
16. #72235115	C₄	semidirect
20. #72235097	C₄	semidirect
21. #72235097	C₄	semidirect
22. #72235088	C₄	semidirect
23. #72235079	C₄	semidirect
25. #72216489	C₂	semidirect
28. #72213429	C₂²	semidirect
29. #72213428	C₂²	semidirect
32. #10586169	C₂×C₄	semidirect
37. #10575142	C₂×C₄	semidirect
38. #10574294	C₂×C₄	semidirect
51. #6451869	C₂×C₄	semidirect
53. #6451748	C₂×C₄	semidirect
54. #6374136	C₂×C₄	semidirect
55. #6374123	C₂×C₄	semidirect
56. C₂×#12707	D₄	semidirect
62. #392259	C₂×C₄	semidirect
65. #359766	C₂×C₄	semidirect
67. #359762	C₂×C₄	semidirect
110. C₂×#2127	C₂×D₄	semidirect
111. #23467	C₂²⋊C₄	semidirect
112. #23463	C₂²⋊C₄	semidirect
113. #23443	C₂²⋊C₄	semidirect
114. #23407	C₂²⋊C₄	semidirect
118. #10948	C₂²⋊C₄	semidirect
119. #10937	C₂²⋊C₄	semidirect
139. #3458	C₂×C₂²⋊C₄	semidirect
140. #3449	C₂×C₂²⋊C₄	semidirect
162. #2873	C₂×C₂²⋊C₄	semidirect
166. #1742	C₂×C₂²⋊C₄	semidirect
167. #1740	C₂×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72251906	2
#72251872	2
#72251733	2
#72238605	2
#72238418	2
#72236862	2
#72235124	2
#72216489	2