Subgroups — #153705

Order 512.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13741	C₂	1	2	yes	maximal, complemented
2	Q₂₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₂₅₆	C₂	1	2	yes	maximal, complemented
4	D₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#2262	—	2	2	no	—
7	Q₁₂₈	C₂²	1	2	yes	—
8	SD₁₂₈	—	2	2	no	—
9	D₆₄	C₂²	1	2	yes	—
10	C₂×C₆₄	C₂²	1	1	yes	characteristic
11	C₁₂₈	C₂²	1	1	yes	characteristic
12	C₁₂₈	C₂²	1	1	yes	characteristic
13	#508	—	4	2	no	—
14	Q₆₄	—	2	2	no	—
15	SD₆₄	—	4	2	no	—
16	D₃₂	—	2	2	no	—
17	C₂×C₃₂	D₄	1	1	yes	characteristic
18	C₆₄	D₄	1	1	yes	characteristic
19	C₆₄	C₂³	1	1	yes	characteristic
20	#135	—	8	2	no	—
21	Q₃₂	—	4	2	no	—
22	SD₃₂	—	8	2	no	—
23	D₁₆	—	4	2	no	—
24	C₂×C₁₆	D₈	1	1	yes	characteristic
25	C₃₂	D₈	1	1	yes	characteristic
26	C₃₂	C₂×D₄	1	1	yes	characteristic
27	#41	—	16	2	no	—
28	Q₁₆	—	8	2	no	—
29	SD₁₆	—	16	2	no	—
30	D₈	—	8	2	no	—
31	C₂×C₈	D₁₆	1	1	yes	characteristic
32	C₁₆	D₁₆	1	1	yes	characteristic
33	C₁₆	C₂×D₈	1	1	yes	characteristic
34	Q₈	—	16	2	no	—
35	D₄	—	16	2	no	—
36	D₄	—	32	2	no	—
37	C₂×C₄	—	32	2	no	—
38	C₂×C₄	D₃₂	1	1	yes	characteristic
39	C₈	D₃₂	1	1	yes	characteristic
40	C₈	C₂×D₁₆	1	1	yes	characteristic
41	C₂²	—	32	2	no	—
42	C₂²	D₆₄	1	1	yes	characteristic
43	C₄	—	32	2	no	—
44	C₄	D₆₄	1	1	yes	characteristic
45	C₄	C₂×D₃₂	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	64	2	no	—
48	C₂	C₂×D₆₄	1	1	yes	characteristic

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

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

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