Subgroups — #477

Order 64.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈	C₂	1	1	yes	maximal, characteristic
3	#107	C₂	1	2	yes	maximal
4	C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#103	C₂	1	2	yes	maximal
6	C₂×Q₈	—	2	1	no	—
7	C₂×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₈	C₂²	1	2	yes	—
9	C₄⋊C₄	—	2	1	no	—
10	C₄⋊C₄	—	2	1	no	—
11	C₄⋊C₄	C₂²	1	2	yes	—
12	C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₄²	—	2	1	no	—
14	C₄²	C₂²	1	1	yes	characteristic
15	Q₈	—	2	1	no	—
16	Q₈	—	4	2	no	—
17	Q₈	Q₈	1	2	yes	complemented
18	C₂×C₄	—	2	1	no	—
19	C₂×C₄	—	2	1	no	—
20	C₂×C₄	—	2	2	no	—
21	C₂×C₄	D₄	1	1	yes	characteristic
22	C₂×C₄	D₄	1	1	yes	characteristic
23	C₂×C₄	C₂³	1	1	yes	characteristic
24	C₈	—	2	2	no	—
25	C₂²	C₂×D₄	1	1	yes	characteristic
26	C₄	—	2	2	no	—
27	C₄	—	2	1	no	—
28	C₄	—	4	1	no	—
29	C₄	—	4	2	no	—
30	C₄	Q₁₆	1	2	yes	—
31	C₄	C₂×Q₈	1	1	yes	characteristic
32	C₄	#41	1	1	yes	characteristic
33	C₂	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂	C₂×Q₁₆	1	1	yes	characteristic
35	C₂	AGL(1,ℤ/8)	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
17. Q₈	Q₈	semidirect

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