Subgroups — #72311021

Order 4096.

At least 10625 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

The subgroup lattice has 232 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72310991	C₂	1	4	yes	maximal, complemented
2	#72310468	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310189	C₂	1	1	yes	maximal, characteristic
4	#72309421	C₂	1	1	yes	maximal, characteristic, complemented
5	#72302137	C₂	1	1	yes	maximal, characteristic, complemented
6	#72259157	C₄	1	2	yes	complemented
7	#72259147	C₄	1	2	yes	complemented
8	#72259136	C₄	1	1	yes	characteristic, complemented
9	#72259117	C₄	1	1	yes	characteristic, complemented
10	#72258997	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237589	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237033	C₄	1	1	yes	characteristic, complemented
14	#72235847	C₂²	1	1	yes	characteristic
15	#72235718	C₂²	1	1	yes	characteristic
16	#72226093	C₂²	1	2	yes	complemented
17	#72225712	C₄	1	1	yes	characteristic, complemented
18	#72224136	C₂²	1	1	yes	characteristic
19	#72219314	C₂²	1	1	yes	characteristic
20	#72219311	C₂²	1	1	yes	characteristic
21	#72219311	C₂²	1	1	yes	characteristic
22	#72219264	C₂²	1	1	yes	characteristic
23	#72216490	C₂	1	4	yes	maximal, complemented
24	#72216484	C₂²	1	4	yes	complemented
25	#72216481	C₂²	1	2	yes	complemented
26	#72216427	C₂²	1	4	yes	complemented
27	#72215671	C₂²	1	2	yes	complemented
28	#72211163	C₂²	1	2	yes	complemented
29	#72210587	C₂²	1	2	yes	—
30	#72210489	C₂²	1	4	yes	complemented
31	#72210482	C₂²	1	4	yes	complemented
32	#72209835	C₂²	1	2	yes	—
33	#10586404	C₂×C₄	1	1	yes	characteristic, complemented
34	C₂²×#3532	D₄	1	1	yes	characteristic
35	C₂×#34980	C₂³	1	1	yes	characteristic
36	#6499870	C₂×C₄	1	2	yes	complemented
37	#6499867	C₂³	1	2	yes	—
38	#6374128	C₂×C₄	1	1	yes	characteristic, complemented
39	#5025789	C₂×C₄	1	1	yes	characteristic
40	#5025782	D₄	1	1	yes	characteristic
41	#5021140	D₄	1	1	yes	characteristic
42	#5021138	C₂³	1	1	yes	characteristic
43	#4422949	C₂×C₄	1	1	yes	characteristic
44	#4420586	D₄	1	1	yes	characteristic
45	#763333	C₂×C₄	1	1	yes	characteristic
46	#686187	D₄	1	1	yes	characteristic
47	#647656	C₂³	1	1	yes	characteristic
48	#616961	D₄	1	1	yes	characteristic
49	#616960	C₂×C₄	1	1	yes	characteristic
50	C₂×#15947	D₄	1	1	yes	characteristic
51	#552670	D₄	1	1	yes	characteristic
52	#392109	D₄	1	2	yes	complemented
53	#347042	C₂×C₄	1	2	yes	complemented
54	#347042	C₂×C₄	1	2	yes	complemented
55	#346456	C₂×C₄	1	2	yes	complemented
56	#346446	C₂×C₄	1	2	yes	complemented
57	#190116	D₄	1	2	yes	complemented
58	#188109	C₂×C₄	1	1	yes	characteristic, complemented
59	#188049	C₂×C₄	1	1	yes	characteristic, complemented
60	#139720	C₂×C₄	1	2	yes	complemented
61	#139718	C₂×C₄	1	2	yes	complemented
62	#139716	C₂×C₄	1	2	yes	complemented
63	#139716	C₂×C₄	1	2	yes	complemented
64	#134033	C₂³	1	2	yes	—
65	#133825	C₂×C₄	1	2	yes	complemented
66	#133822	C₂³	1	2	yes	—
67	#115690	D₄	1	2	yes	complemented
68	#115690	C₂³	1	2	yes	—
69	#115690	C₂³	1	2	yes	—
70	#115690	C₂³	1	2	yes	—
71	#115631	D₄	1	2	yes	complemented
72	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
73	#62817	C₂²×C₄	1	2	yes	—
74	C₂²×#551	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×D₄²	C₂×D₄	1	1	yes	characteristic
76	C₂²×H(3,𝔽₄)	C₂×D₄	1	1	yes	characteristic
77	C₂²×#560	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×#534	C₂²×C₄	1	1	yes	characteristic
80	C₂²×#534	C₂×D₄	1	1	yes	characteristic
81	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×#120	C₂×D₄	1	1	yes	characteristic
83	C₂×#2849	C₂×D₄	1	1	yes	characteristic
84	C₂×#2849	#41	1	1	yes	characteristic
85	C₂×#2846	C₂×D₄	1	1	yes	characteristic
86	C₂×#2846	C₂⁴	1	1	yes	characteristic
87	C₂×#2613	C₂²×C₄	1	1	yes	characteristic
88	C₂×#2613	C₂×D₄	1	1	yes	characteristic
89	C₂×#2613	C₂×D₄	1	1	yes	characteristic
90	C₂×#2613	#41	1	1	yes	characteristic
91	C₂×#2308	C₂×D₄	1	1	yes	characteristic
92	C₂×#2308	#41	1	1	yes	characteristic
93	C₂²×#379	#41	1	1	yes	characteristic
94	#21746	C₂²×C₄	1	2	yes	—
95	#15948	C₂²⋊C₄	1	1	yes	characteristic
96	#15948	C₂²⋊C₄	1	1	yes	characteristic
97	#15947	C₂²⋊C₄	1	1	yes	characteristic
98	#15947	C₂²⋊C₄	1	1	yes	characteristic
99	#15947	C₂²×C₄	1	1	yes	characteristic
100	#15947	C₂²×C₄	1	1	yes	characteristic
101	#15947	C₂×D₄	1	1	yes	characteristic
102	#15947	C₂×D₄	1	1	yes	characteristic
103	#15262	C₂²⋊C₄	1	1	yes	characteristic
104	#15261	C₂²⋊C₄	1	1	yes	characteristic
105	#15259	C₂²⋊C₄	1	1	yes	characteristic
106	#15259	C₂²×C₄	1	1	yes	characteristic
107	#15259	C₂×D₄	1	1	yes	characteristic
108	#15258	C₂²⋊C₄	1	1	yes	characteristic
109	#15258	C₂²×C₄	1	1	yes	characteristic
110	#15258	C₂×D₄	1	1	yes	characteristic
111	#14679	C₂×D₄	1	4	yes	complemented
112	#14658	C₂²⋊C₄	1	1	yes	characteristic
113	#14657	C₂×D₄	1	1	yes	characteristic
114	#14335	C₂²⋊C₄	1	1	yes	characteristic
115	#14167	C₂²⋊C₄	1	1	yes	characteristic
116	#14134	C₂×D₄	1	1	yes	characteristic
117	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic
118	#8547	C₂²×C₄	1	2	yes	—
119	#8541	C₂²×C₄	1	2	yes	—
120	#8101	C₂×D₄	1	4	yes	complemented
121	C₂⁷	#99	1	1	yes	characteristic, complemented
122	C₂⁷	#121	1	1	yes	characteristic
123	C₂⁷	C₂²×D₄	1	1	yes	characteristic
124	C₂³×#41	#99	1	1	yes	characteristic, complemented
125	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
127	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
128	C₂⁴×D₄	#120	1	1	yes	characteristic
129	C₂⁴×D₄	#120	1	1	yes	characteristic
130	C₂⁵×C₄	#121	1	1	yes	characteristic
131	C₂²×#127	#99	1	1	yes	characteristic, complemented
132	C₂²×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×#124	#99	1	1	yes	characteristic, complemented
134	C₂²×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×#123	C₄×D₄	1	1	yes	characteristic
138	C₂²×#123	#123	1	1	yes	characteristic
139	C₂²×#120	#99	1	1	yes	characteristic, complemented
140	C₂²×#120	#99	1	1	yes	characteristic, complemented
141	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×#120	C₄×D₄	1	1	yes	characteristic
145	C₂²×#120	#120	1	1	yes	characteristic
146	C₂²×#120	#121	1	1	yes	characteristic
147	C₂²×#120	#121	1	1	yes	characteristic
148	C₂²×#120	C₂³×C₄	1	1	yes	characteristic
149	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
150	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
151	C₂²×#117	#99	1	1	yes	characteristic, complemented
152	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic, complemented
153	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
154	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
155	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
156	C₂×#381	C₄×D₄	1	1	yes	characteristic
157	C₂×#381	#121	1	1	yes	characteristic
158	C₂×#381	#121	1	1	yes	characteristic
159	C₂×#381	#123	1	1	yes	characteristic
160	C₂×#379	C₄×D₄	1	1	yes	characteristic
161	C₂×#379	C₄×D₄	1	1	yes	characteristic
162	C₂×#379	#120	1	1	yes	characteristic
163	C₂×#379	#121	1	1	yes	characteristic
164	C₂×#379	#123	1	1	yes	characteristic
165	C₂×#379	C₂²×D₄	1	1	yes	characteristic
166	C₂×#379	C₂×#41	1	1	yes	characteristic
167	C₂×#379	C₂×#41	1	1	yes	characteristic
168	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂⁶	#386	1	1	yes	characteristic
173	C₂⁶	#386	1	1	yes	characteristic
174	C₂⁶	C₂×#99	1	1	yes	characteristic
175	C₂⁶	C₂×#99	1	1	yes	characteristic
176	C₂⁶	C₂×#99	1	1	yes	characteristic
177	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂⁶	C₂×#120	1	1	yes	characteristic
179	C₂⁶	C₂×#121	1	1	yes	characteristic
180	C₂³×Q₈	C₂×#99	1	1	yes	characteristic
181	C₂³×D₄	#386	1	1	yes	characteristic
182	C₂³×D₄	C₂×#99	1	1	yes	characteristic
183	C₂³×D₄	C₂×#99	1	1	yes	characteristic
184	C₂³×D₄	C₂×#99	1	1	yes	characteristic
185	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
186	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
187	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
188	C₂⁴×C₄	#386	1	1	yes	characteristic
189	C₂⁴×C₄	#386	1	1	yes	characteristic
190	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
193	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
194	C₂²×C₂²⋊C₄	C₂×#99	1	1	yes	characteristic
195	C₂²×C₂²⋊C₄	C₂×#99	1	1	yes	characteristic
196	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
198	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
199	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
200	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
201	C₂⁵	#1889	1	1	yes	characteristic
202	C₂⁵	#1889	1	1	yes	characteristic
203	C₂⁵	#1898	1	1	yes	characteristic
204	C₂⁵	#1898	1	1	yes	characteristic
205	C₂⁵	#1898	1	1	yes	characteristic
206	C₂⁵	#1913	1	1	yes	characteristic
207	C₂⁵	#1913	1	1	yes	characteristic
208	C₂⁵	C₂×#386	1	1	yes	characteristic
209	C₂⁵	C₂²×#99	1	1	yes	characteristic
210	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
211	C₂³×C₄	#1889	1	2	yes	—
212	C₂³×C₄	#1898	1	1	yes	characteristic
213	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
214	C₂⁴	#9495	1	1	yes	characteristic
215	C₂⁴	#9495	1	1	yes	characteristic
216	C₂⁴	#9495	1	1	yes	characteristic
217	C₂⁴	#9495	1	1	yes	characteristic
218	C₂⁴	C₂×#1889	1	1	yes	characteristic
219	C₂⁴	C₂×#1898	1	1	yes	characteristic
220	C₂⁴	C₂×#1913	1	1	yes	characteristic
221	C₂⁴	#33551	1	4	yes	—
222	C₂³	#139743	1	1	yes	characteristic
223	C₂³	#139743	1	1	yes	characteristic
224	C₂³	#139750	1	2	yes	—
225	C₂³	C₂×#9495	1	1	yes	characteristic
226	C₂³	#494575	1	2	yes	—
227	C₂²	C₂×#139743	1	1	yes	characteristic
228	C₂²	C₂×#139750	1	1	yes	characteristic
229	C₂²	#72237037	1	1	yes	characteristic
230	C₂	#72266177	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266177
commutator	C₂²×#120	C₂³×C₄
Frattini, cosocle	C₂×#2846	C₂⁴
prime radical	#72310189	C₂
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2846 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72310991	C₂	semidirect
2. #72310468	C₂	semidirect
4. #72309421	C₂	semidirect
5. #72302137	C₂	semidirect
6. #72259157	C₄	semidirect
7. #72259147	C₄	semidirect
8. #72259136	C₄	semidirect
9. #72259117	C₄	semidirect
10. #72258997	C₂	semidirect
11. #72237596	C₂	semidirect
12. #72237589	C₂	semidirect
13. #72237033	C₄	semidirect
16. #72226093	C₂²	semidirect
17. #72225712	C₄	semidirect
23. #72216490	C₂	semidirect
24. #72216484	C₂²	semidirect
25. #72216481	C₂²	semidirect
26. #72216427	C₂²	semidirect
27. #72215671	C₂²	semidirect
28. #72211163	C₂²	semidirect
30. #72210489	C₂²	semidirect
31. #72210482	C₂²	semidirect
33. #10586404	C₂×C₄	semidirect
36. #6499870	C₂×C₄	semidirect
38. #6374128	C₂×C₄	semidirect
52. #392109	D₄	semidirect
53. #347042	C₂×C₄	semidirect
54. #347042	C₂×C₄	semidirect
55. #346456	C₂×C₄	semidirect
56. #346446	C₂×C₄	semidirect
57. #190116	D₄	semidirect
58. #188109	C₂×C₄	semidirect
59. #188049	C₂×C₄	semidirect
60. #139720	C₂×C₄	semidirect
61. #139718	C₂×C₄	semidirect
62. #139716	C₂×C₄	semidirect
63. #139716	C₂×C₄	semidirect
65. #133825	C₂×C₄	semidirect
67. #115690	D₄	semidirect
71. #115631	D₄	semidirect
111. #14679	C₂×D₄	semidirect
120. #8101	C₂×D₄	semidirect
121. C₂⁷	#99	semidirect
124. C₂³×#41	#99	semidirect
131. C₂²×#127	#99	semidirect
133. C₂²×#124	#99	semidirect
139. C₂²×#120	#99	semidirect
140. C₂²×#120	#99	semidirect
151. C₂²×#117	#99	semidirect
152. C₂³×C₂²⋊C₄	#99	semidirect

Maximal subgroups

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

subgroup	index
#72310991	2
#72310468	2
#72310189	2
#72309421	2
#72302137	2
#72258997	2
#72237596	2
#72237589	2
#72216490	2