Subgroups — #493

Order 64.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈	C₂	1	4	yes	maximal, complemented
2	#127	C₂	1	2	yes	maximal, complemented
3	C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄	—	2	4	no	—
5	C₂×D₄	C₂²	1	4	yes	—
6	D₈	—	2	8	no	—
7	C₂×C₈	C₂²	1	2	yes	—
8	C₄²	C₂²	1	1	yes	characteristic
9	C₂³	—	2	4	no	—
10	D₄	—	2	8	no	—
11	D₄	—	4	8	no	—
12	C₂×C₄	D₄	1	2	yes	—
13	C₂×C₄	C₂³	1	1	yes	characteristic
14	C₈	D₄	1	4	yes	complemented
15	C₂²	—	4	4	no	—
16	C₂²	—	4	8	no	—
17	C₂²	C₂×D₄	1	1	yes	characteristic
18	C₄	D₈	1	4	yes	complemented
19	C₄	C₂×D₄	1	2	yes	—
20	C₂	—	8	4	no	—
21	C₂	#127	1	1	yes	characteristic
22	C₂	C₂×D₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₂×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.