Subgroups — #153706

Order 512.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13741	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₂₅₆	C₂	1	2	yes	maximal, complemented
4	D₁₂₈	C₂	1	2	yes	maximal, complemented
5	M₂₅₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#2262	—	2	1	no	—
7	C₂×D₃₂	—	2	1	no	—
8	Q₁₂₈	C₂²	1	1	yes	characteristic, complemented
9	SD₁₂₈	—	2	1	no	—
10	D₆₄	—	2	1	no	—
11	D₆₄	C₂²	1	2	yes	—
12	D₆₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₆₄	C₂²	1	1	yes	characteristic
14	C₁₂₈	C₂²	1	2	yes	complemented
15	#508	—	4	1	no	—
16	C₂×D₁₆	—	4	1	no	—
17	Q₆₄	—	2	1	no	—
18	SD₆₄	—	4	1	no	—
19	D₃₂	—	2	2	no	—
20	D₃₂	—	2	1	no	—
21	D₃₂	—	4	1	no	—
22	C₂×C₃₂	D₄	1	1	yes	characteristic
23	C₆₄	D₄	1	1	yes	characteristic
24	C₆₄	C₂³	1	1	yes	characteristic
25	#135	—	8	1	no	—
26	C₂×D₈	—	8	1	no	—
27	Q₃₂	—	4	1	no	—
28	SD₃₂	—	8	1	no	—
29	D₁₆	—	4	2	no	—
30	D₁₆	—	4	1	no	—
31	D₁₆	—	8	1	no	—
32	C₂×C₁₆	D₈	1	1	yes	characteristic
33	C₃₂	D₈	1	1	yes	characteristic
34	C₃₂	C₂×D₄	1	1	yes	characteristic
35	#41	—	16	1	no	—
36	C₂×D₄	—	16	1	no	—
37	Q₁₆	—	8	1	no	—
38	SD₁₆	—	16	1	no	—
39	D₈	—	8	2	no	—
40	D₈	—	8	1	no	—
41	D₈	—	16	1	no	—
42	C₂×C₈	D₁₆	1	1	yes	characteristic
43	C₁₆	D₁₆	1	1	yes	characteristic
44	C₁₆	C₂×D₈	1	1	yes	characteristic
45	C₂³	—	32	1	no	—
46	Q₈	—	16	1	no	—
47	D₄	—	16	2	no	—
48	D₄	—	16	1	no	—
49	D₄	—	32	1	no	—
50	D₄	—	32	1	no	—
51	C₂×C₄	—	32	1	no	—
52	C₂×C₄	D₃₂	1	1	yes	characteristic
53	C₈	D₃₂	1	1	yes	characteristic
54	C₈	C₂×D₁₆	1	1	yes	characteristic
55	C₂²	—	32	1	no	—
56	C₂²	—	32	2	no	—
57	C₂²	—	64	2	no	—
58	C₂²	D₆₄	1	1	yes	characteristic
59	C₄	—	32	1	no	—
60	C₄	D₆₄	1	1	yes	characteristic
61	C₄	C₂×D₃₂	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	64	1	no	—
64	C₂	—	64	2	no	—
65	C₂	C₂×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×D₆₄
commutator, Frattini, cosocle	C₆₄	C₂³
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #13741	C₂	semidirect
2. C₂×D₆₄	C₂	semidirect
3. SD₂₅₆	C₂	semidirect
4. D₁₂₈	C₂	semidirect
5. M₂₅₆	C₂	semidirect
8. Q₁₂₈	C₂²	semidirect
12. D₆₄	C₂²	semidirect
14. C₁₂₈	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#13741	2
C₂×D₆₄	2
SD₂₅₆	2
D₁₂₈	2
M₂₅₆	2