Subgroups — #508

Order 64.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#135	C₂	1	2	yes	maximal, complemented
2	Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₃₂	C₂	1	2	yes	maximal, complemented
4	D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#41	—	2	2	no	—
7	Q₁₆	C₂²	1	2	yes	—
8	SD₁₆	—	2	2	no	—
9	D₈	C₂²	1	2	yes	—
10	C₂×C₈	C₂²	1	1	yes	characteristic
11	C₁₆	C₂²	1	1	yes	characteristic
12	C₁₆	C₂²	1	1	yes	characteristic
13	Q₈	—	2	2	no	—
14	D₄	—	2	2	no	—
15	D₄	—	4	2	no	—
16	C₂×C₄	—	4	2	no	—
17	C₂×C₄	D₄	1	1	yes	characteristic
18	C₈	D₄	1	1	yes	characteristic
19	C₈	C₂³	1	1	yes	characteristic
20	C₂²	—	4	2	no	—
21	C₂²	D₈	1	1	yes	characteristic
22	C₄	—	4	2	no	—
23	C₄	D₈	1	1	yes	characteristic
24	C₄	C₂×D₄	1	1	yes	characteristic
25	C₂	—	2	1	no	—
26	C₂	—	8	2	no	—
27	C₂	C₂×D₈	1	1	yes	characteristic

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