Subgroups — #13645

Order 256.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	C₈×D₄	C₄	1	1	yes	characteristic
11	M₆₄	—	2	2	no	—
12	C₂×C₃₂	—	2	2	no	—
13	C₂×C₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₂	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆	C₄	1	1	yes	characteristic
18	C₂²⋊C₁₆	C₄	1	1	yes	characteristic
19	C₂²⋊C₁₆	C₄	1	1	yes	characteristic
20	C₄×C₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×D₄	C₈	1	1	yes	characteristic
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆	C₂³	1	1	yes	characteristic
32	C₄⋊C₈	C₈	1	1	yes	characteristic
33	C₂²⋊C₈	C₈	1	1	yes	characteristic
34	C₂²⋊C₈	C₈	1	1	yes	characteristic
35	C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃₂	—	2	1	no	—
37	C₃₂	—	2	1	no	—
38	C₃₂	—	2	1	no	—
39	C₃₂	D₄	1	2	yes	complemented
40	C₂×D₄	C₁₆	1	1	yes	characteristic
41	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	1	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₄⋊C₄	C₁₆	1	1	yes	characteristic
52	C₂²⋊C₄	C₁₆	1	1	yes	characteristic
53	C₂²⋊C₄	C₁₆	1	1	yes	characteristic
54	C₄²	C₂×C₈	1	1	yes	characteristic
55	C₁₆	—	2	1	no	—
56	C₁₆	—	2	1	no	—
57	C₁₆	—	2	1	no	—
58	C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₁₆	#41	1	1	yes	characteristic
60	C₂³	C₂×C₁₆	1	1	yes	characteristic
61	C₂³	C₂×C₁₆	1	1	yes	characteristic
62	D₄	—	2	2	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
68	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
69	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
70	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
71	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	—	2	1	no	—
75	C₈	C₄×D₄	1	1	yes	characteristic
76	C₈	#131	1	1	yes	characteristic
77	C₂²	—	2	1	no	—
78	C₂²	—	2	1	no	—
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	M₆₄	1	2	yes	complemented
83	C₂²	C₂²×C₁₆	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	2	1	no	—
86	C₄	—	2	1	no	—
87	C₄	—	2	1	no	—
88	C₄	C₈×D₄	1	1	yes	characteristic
89	C₄	#504	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	—	4	1	no	—
92	C₂	D₄×C₁₆	1	1	yes	characteristic
93	C₂	C₂×M₆₄	1	1	yes	characteristic
94	C₂	#2258	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂²×C₁₆
prime radical	C₂×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×M₆₄	C₂	semidirect
2. C₂²×C₃₂	C₂	semidirect
4. C₄⋊C₃₂	C₂	semidirect
5. C₂²⋊C₃₂	C₂	semidirect
6. C₂²⋊C₃₂	C₂	semidirect
7. #1397	C₂	semidirect
39. C₃₂	D₄	semidirect
82. C₂²	M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄	2
C₂²×C₃₂	2
D₄×C₁₆	2
C₄⋊C₃₂	2
C₂²⋊C₃₂	2
#1397	2