Subgroups — #72323798

Order 4096.

At least 11163 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

The subgroup lattice has 182 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323795	C₂	1	4	yes	maximal, complemented
2	#72323228	C₂	1	1	yes	maximal, characteristic, complemented
3	#72310468	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266038	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266037	C₂²	1	2	yes	complemented
6	#72258968	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238595	C₂²	1	2	yes	complemented
8	#72237864	C₂	1	1	yes	maximal, characteristic, complemented
9	#72237381	C₂	1	1	yes	maximal, characteristic, complemented
10	#72235847	C₂²	1	1	yes	characteristic
11	#72234625	C₂²	1	1	yes	characteristic
12	#72225997	C₂	1	1	yes	maximal, characteristic, complemented
13	#72225991	C₂²	1	2	yes	complemented
14	#72225870	C₂²	1	1	yes	characteristic
15	#72224136	C₂²	1	1	yes	characteristic
16	#72219314	C₂²	1	1	yes	characteristic
17	#72219311	C₂²	1	1	yes	characteristic
18	#72219254	C₂²	1	1	yes	characteristic
19	#72210568	C₂	1	4	yes	maximal, complemented
20	#72210547	C₂²	1	2	yes	complemented
21	#72210489	C₂²	1	4	yes	complemented
22	#72210482	C₂²	1	4	yes	complemented
23	#72210482	C₂²	1	4	yes	complemented
24	#72209844	C₂²	1	4	yes	—
25	#72209716	C₂²	1	2	yes	complemented
26	#72209716	C₂²	1	2	yes	complemented
27	#10586404	D₄	1	1	yes	characteristic, complemented
28	C₂×#62817	D₄	1	1	yes	characteristic, complemented
29	C₂⁴×#120	D₄	1	1	yes	characteristic, complemented
30	C₂×#34980	C₂³	1	1	yes	characteristic
31	#6499870	D₄	1	2	yes	complemented
32	#6499868	D₄	1	2	yes	complemented
33	#6499867	C₂³	1	2	yes	—
34	#6499867	C₂³	1	2	yes	—
35	#6374128	D₄	1	1	yes	characteristic, complemented
36	C₂×#21746	D₄	1	1	yes	characteristic, complemented
37	#6342371	D₄	1	1	yes	characteristic, complemented
38	#6342371	C₂³	1	1	yes	characteristic
39	#4308128	D₄	1	1	yes	characteristic, complemented
40	#4308128	C₂³	1	1	yes	characteristic
41	#517411	D₄	1	1	yes	characteristic, complemented
42	#190116	D₄	1	2	yes	complemented
43	#188109	D₄	1	1	yes	characteristic, complemented
44	#188049	D₄	1	1	yes	characteristic, complemented
45	C₂×#8101	D₄	1	2	yes	complemented
46	#134033	C₂³	1	2	yes	—
47	#133825	D₄	1	2	yes	complemented
48	#133822	C₂³	1	2	yes	—
49	#115692	D₄	1	2	yes	complemented
50	#115690	C₂³	1	2	yes	—
51	#115690	C₂³	1	2	yes	—
52	#115631	D₄	1	2	yes	complemented
53	C₂⁸	C₂×D₄	1	1	yes	characteristic, complemented
54	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
55	#62817	C₂×D₄	1	2	yes	—
56	#62817	C₂×D₄	1	2	yes	—
57	C₂²×#534	C₂×D₄	1	1	yes	characteristic
58	C₂²×#534	C₂×D₄	1	1	yes	characteristic
59	C₂³×#120	C₂×D₄	1	1	yes	characteristic
60	C₂⁴×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic, complemented
61	C₂×#2849	C₂×D₄	1	1	yes	characteristic
62	C₂×#2849	C₂×D₄	1	1	yes	characteristic
63	C₂×#2846	C₂×D₄	1	1	yes	characteristic
64	C₂×#2846	C₂⁴	1	1	yes	characteristic
65	C₂×#2613	C₂×D₄	1	1	yes	characteristic
66	C₂×#2613	C₂×D₄	1	1	yes	characteristic
67	C₂×#2613	C₂×D₄	1	1	yes	characteristic
68	C₂×#2613	#41	1	1	yes	characteristic
69	C₂×#2308	C₂×D₄	1	1	yes	characteristic
70	C₂×#2308	#41	1	1	yes	characteristic
71	C₂²×#379	C₂×D₄	1	1	yes	characteristic
72	#21746	C₂×D₄	1	2	yes	—
73	#21746	C₂×D₄	1	2	yes	—
74	#14679	C₂×D₄	1	1	yes	characteristic, complemented
75	#14679	C₂×D₄	1	1	yes	characteristic, complemented
76	#8547	C₂×D₄	1	2	yes	—
77	#8541	C₂×D₄	1	2	yes	—
78	#8101	C₂×D₄	1	4	yes	complemented
79	#8101	C₂×D₄	1	4	yes	complemented
80	C₂⁷	#120	1	1	yes	characteristic
81	C₂⁷	C₂²×D₄	1	1	yes	characteristic
82	C₂⁴×D₄	#120	1	1	yes	characteristic
83	C₂⁴×D₄	#120	1	1	yes	characteristic
84	C₂⁴×D₄	#120	1	1	yes	characteristic
85	C₂⁴×D₄	#121	1	1	yes	characteristic
86	C₂⁴×D₄	#121	1	1	yes	characteristic
87	C₂⁵×C₄	#120	1	1	yes	characteristic
88	C₂²×#123	#120	1	1	yes	characteristic
89	C₂²×#123	#120	1	1	yes	characteristic
90	C₂²×#123	#121	1	1	yes	characteristic
91	C₂²×#123	#127	1	1	yes	characteristic
92	C₂²×#120	#120	1	1	yes	characteristic
93	C₂²×#120	#120	1	1	yes	characteristic
94	C₂²×#120	#120	1	1	yes	characteristic
95	C₂²×#120	#121	1	1	yes	characteristic
96	C₂²×#120	#121	1	1	yes	characteristic
97	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
98	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
103	C₂×#381	#121	1	1	yes	characteristic
104	C₂×#381	#121	1	1	yes	characteristic
105	C₂×#381	#121	1	1	yes	characteristic
106	C₂×#381	#127	1	1	yes	characteristic
107	C₂×#379	#120	1	1	yes	characteristic
108	C₂×#379	#120	1	1	yes	characteristic
109	C₂×#379	#121	1	1	yes	characteristic
110	C₂×#379	#121	1	1	yes	characteristic
111	C₂×#379	#127	1	1	yes	characteristic
112	C₂×#379	C₂²×D₄	1	1	yes	characteristic
113	C₂×#379	C₂²×D₄	1	1	yes	characteristic
114	C₂×#379	C₂×#41	1	1	yes	characteristic
115	C₂⁶	#392	1	1	yes	characteristic
116	C₂⁶	#392	1	1	yes	characteristic
117	C₂⁶	B(4,𝔽₂)	1	2	yes	—
118	C₂⁶	B(4,𝔽₂)	1	2	yes	—
119	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
120	C₂⁶	C₂×#120	1	1	yes	characteristic
121	C₂⁶	C₂×#120	1	1	yes	characteristic
122	C₂⁶	C₂×#121	1	1	yes	characteristic
123	C₂³×D₄	#392	1	1	yes	characteristic
124	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
125	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
126	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
127	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
128	C₂³×D₄	C₂×#120	1	1	yes	characteristic
129	C₂⁴×C₄	#392	1	1	yes	characteristic
130	C₂⁴×C₄	#392	1	1	yes	characteristic
131	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
134	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
135	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
136	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
137	C₂²×C₂²⋊C₄	C₂×#127	1	1	yes	characteristic
138	C₂⁵	#2021	1	1	yes	characteristic
139	C₂⁵	#2021	1	1	yes	characteristic
140	C₂⁵	#2021	1	1	yes	characteristic
141	C₂⁵	#2021	1	1	yes	characteristic
142	C₂⁵	#2021	1	2	yes	—
143	C₂⁵	#2021	1	1	yes	characteristic
144	C₂⁵	#2021	1	1	yes	characteristic
145	C₂⁵	#2021	1	1	yes	characteristic
146	C₂⁵	#2196	1	2	yes	—
147	C₂⁵	#2196	1	2	yes	—
148	C₂⁵	#2199	1	2	yes	—
149	C₂⁵	#2199	1	2	yes	—
150	C₂⁵	C₂×#392	1	1	yes	characteristic
151	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
152	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
153	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
154	C₂³×C₄	#2021	1	2	yes	—
155	C₂³×C₄	#2021	1	1	yes	characteristic
156	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
157	C₂⁴	#10336	1	1	yes	characteristic
158	C₂⁴	#10336	1	1	yes	characteristic
159	C₂⁴	#10336	1	1	yes	characteristic
160	C₂⁴	#10336	1	1	yes	characteristic
161	C₂⁴	#13343	1	2	yes	—
162	C₂⁴	#13343	1	2	yes	—
163	C₂⁴	#13343	1	2	yes	—
164	C₂⁴	#13343	1	2	yes	—
165	C₂⁴	C₂×#2021	1	1	yes	characteristic
166	C₂⁴	C₂×#2021	1	1	yes	characteristic
167	C₂⁴	C₂×#2021	1	1	yes	characteristic
168	C₂⁴	C₂×#2196	1	2	yes	—
169	C₂⁴	C₂×#2199	1	2	yes	—
170	C₂⁴	#33551	1	4	yes	—
171	C₂³	#144366	1	2	yes	—
172	C₂³	#144366	1	1	yes	characteristic
173	C₂³	#144366	1	1	yes	characteristic
174	C₂³	C₂×#10336	1	1	yes	characteristic
175	C₂³	C₂×#13343	1	2	yes	—
176	C₂³	#498886	1	2	yes	—
177	C₂²	C₂×#144366	1	1	yes	characteristic
178	C₂²	#72218838	1	1	yes	characteristic
179	C₂²	#72236508	1	1	yes	characteristic
180	C₂	#72266315	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266315
commutator, Frattini, cosocle	C₂×#2846	C₂⁴
LCS	G ▷ C₂×#2846 ▷ 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. #72323795	C₂	semidirect
2. #72323228	C₂	semidirect
3. #72310468	C₂	semidirect
4. #72266038	C₂	semidirect
5. #72266037	C₂²	semidirect
6. #72258968	C₂	semidirect
7. #72238595	C₂²	semidirect
8. #72237864	C₂	semidirect
9. #72237381	C₂	semidirect
12. #72225997	C₂	semidirect
13. #72225991	C₂²	semidirect
19. #72210568	C₂	semidirect
20. #72210547	C₂²	semidirect
21. #72210489	C₂²	semidirect
22. #72210482	C₂²	semidirect
23. #72210482	C₂²	semidirect
25. #72209716	C₂²	semidirect
26. #72209716	C₂²	semidirect
27. #10586404	D₄	semidirect
28. C₂×#62817	D₄	semidirect
29. C₂⁴×#120	D₄	semidirect
31. #6499870	D₄	semidirect
32. #6499868	D₄	semidirect
35. #6374128	D₄	semidirect
36. C₂×#21746	D₄	semidirect
37. #6342371	D₄	semidirect
39. #4308128	D₄	semidirect
41. #517411	D₄	semidirect
42. #190116	D₄	semidirect
43. #188109	D₄	semidirect
44. #188049	D₄	semidirect
45. C₂×#8101	D₄	semidirect
47. #133825	D₄	semidirect
49. #115692	D₄	semidirect
52. #115631	D₄	semidirect
53. C₂⁸	C₂×D₄	semidirect
60. C₂⁴×C₂²⋊C₄	C₂×D₄	semidirect
74. #14679	C₂×D₄	semidirect
75. #14679	C₂×D₄	semidirect
78. #8101	C₂×D₄	semidirect
79. #8101	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72323795	2
#72323228	2
#72310468	2
#72266038	2
#72258968	2
#72237864	2
#72237381	2
#72225997	2
#72210568	2