Subgroups — #509

Order 64.

45 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₃₂	C₂	1	2	yes	maximal, complemented
4	D₁₆	C₂	1	2	yes	maximal, complemented
5	M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#41	—	2	1	no	—
7	C₂×D₄	—	2	1	no	—
8	Q₁₆	C₂²	1	1	yes	characteristic, complemented
9	SD₁₆	—	2	1	no	—
10	D₈	—	2	1	no	—
11	D₈	C₂²	1	2	yes	—
12	D₈	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₈	C₂²	1	1	yes	characteristic
14	C₁₆	C₂²	1	2	yes	complemented
15	C₂³	—	4	1	no	—
16	Q₈	—	2	1	no	—
17	D₄	—	2	2	no	—
18	D₄	—	2	1	no	—
19	D₄	—	4	1	no	—
20	D₄	—	4	1	no	—
21	C₂×C₄	—	4	1	no	—
22	C₂×C₄	D₄	1	1	yes	characteristic
23	C₈	D₄	1	1	yes	characteristic
24	C₈	C₂³	1	1	yes	characteristic
25	C₂²	—	4	1	no	—
26	C₂²	—	4	2	no	—
27	C₂²	—	8	2	no	—
28	C₂²	D₈	1	1	yes	characteristic
29	C₄	—	4	1	no	—
30	C₄	D₈	1	1	yes	characteristic
31	C₄	C₂×D₄	1	1	yes	characteristic
32	C₂	—	2	1	no	—
33	C₂	—	8	1	no	—
34	C₂	—	8	2	no	—
35	C₂	C₂×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂³
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ G
LECS	G ▷ 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.