Subgroups — #13742

Order 256.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2262	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₁₂₈	C₂	1	2	yes	maximal, complemented
4	D₆₄	C₂	1	2	yes	maximal, complemented
5	M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#508	—	2	1	no	—
7	C₂×D₁₆	—	2	1	no	—
8	Q₆₄	C₂²	1	1	yes	characteristic, complemented
9	SD₆₄	—	2	1	no	—
10	D₃₂	—	2	1	no	—
11	D₃₂	C₂²	1	2	yes	—
12	D₃₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃₂	C₂²	1	1	yes	characteristic
14	C₆₄	C₂²	1	2	yes	complemented
15	#135	—	4	1	no	—
16	C₂×D₈	—	4	1	no	—
17	Q₃₂	—	2	1	no	—
18	SD₃₂	—	4	1	no	—
19	D₁₆	—	2	2	no	—
20	D₁₆	—	2	1	no	—
21	D₁₆	—	4	1	no	—
22	C₂×C₁₆	D₄	1	1	yes	characteristic
23	C₃₂	D₄	1	1	yes	characteristic
24	C₃₂	C₂³	1	1	yes	characteristic
25	#41	—	8	1	no	—
26	C₂×D₄	—	8	1	no	—
27	Q₁₆	—	4	1	no	—
28	SD₁₆	—	8	1	no	—
29	D₈	—	4	2	no	—
30	D₈	—	4	1	no	—
31	D₈	—	8	1	no	—
32	C₂×C₈	D₈	1	1	yes	characteristic
33	C₁₆	D₈	1	1	yes	characteristic
34	C₁₆	C₂×D₄	1	1	yes	characteristic
35	C₂³	—	16	1	no	—
36	Q₈	—	8	1	no	—
37	D₄	—	8	2	no	—
38	D₄	—	8	1	no	—
39	D₄	—	16	1	no	—
40	D₄	—	16	1	no	—
41	C₂×C₄	—	16	1	no	—
42	C₂×C₄	D₁₆	1	1	yes	characteristic
43	C₈	D₁₆	1	1	yes	characteristic
44	C₈	C₂×D₈	1	1	yes	characteristic
45	C₂²	—	16	1	no	—
46	C₂²	—	16	2	no	—
47	C₂²	—	32	2	no	—
48	C₂²	D₃₂	1	1	yes	characteristic
49	C₄	—	16	1	no	—
50	C₄	D₃₂	1	1	yes	characteristic
51	C₄	C₂×D₁₆	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	32	1	no	—
54	C₂	—	32	2	no	—
55	C₂	C₂×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #2262	C₂	semidirect
2. C₂×D₃₂	C₂	semidirect
3. SD₁₂₈	C₂	semidirect
4. D₆₄	C₂	semidirect
5. M₁₂₈	C₂	semidirect
8. Q₆₄	C₂²	semidirect
12. D₃₂	C₂²	semidirect
14. C₆₄	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#2262	2
C₂×D₃₂	2
SD₁₂₈	2
D₆₄	2
M₁₂₈	2