Subgroups — #500

Order 64.

49 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈	C₂	1	2	yes	maximal
2	#107	C₂	1	4	yes	maximal
3	C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈	—	2	2	no	—
5	C₂×C₈	C₂²	1	2	yes	—
6	C₄⋊C₄	—	2	2	no	—
7	C₄⋊C₄	C₂²	1	4	yes	—
8	C₄²	C₂²	1	1	yes	characteristic
9	Q₈	—	4	4	no	—
10	C₂×C₄	—	2	4	no	—
11	C₂×C₄	D₄	1	1	yes	characteristic
12	C₂×C₄	D₄	1	1	yes	characteristic
13	C₂×C₄	C₂³	1	1	yes	characteristic
14	C₈	Q₈	1	4	yes	complemented
15	C₂²	C₂×D₄	1	1	yes	characteristic
16	C₄	—	4	4	no	—
17	C₄	D₈	1	2	yes	—
18	C₄	Q₁₆	1	2	yes	—
19	C₄	C₂×Q₈	1	2	yes	—
20	C₂	C₄⋊Q₈	1	1	yes	characteristic
21	C₂	C₂×D₈	1	1	yes	characteristic
22	C₂	C₂×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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.

N	H ≅ G/N	type
14. C₈	Q₈	semidirect

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