Subgroups — #446

Order 64.

47 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⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈	C₂	1	2	yes	maximal
4	#97	C₂	1	2	yes	maximal
5	C₄×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈	C₂²	1	2	yes	—
9	C₂×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊C₄	C₄	1	2	yes	—
11	C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₄²	C₂²	1	1	yes	characteristic
13	C₄²	C₂²	1	2	yes	—
14	Q₈	—	2	2	no	—
15	C₂×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄	C₂×C₄	1	2	yes	—
17	C₂×C₄	C₂×C₄	1	2	yes	—
18	C₂×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄	C₂³	1	1	yes	characteristic
20	C₈	—	2	1	no	—
21	C₈	—	2	2	no	—
22	C₈	Q₈	1	2	yes	complemented
23	C₂²	C₂²×C₄	1	1	yes	characteristic
24	C₄	—	2	2	no	—
25	C₄	—	2	1	no	—
26	C₄	—	2	2	no	—
27	C₄	M₁₆	1	2	yes	—
28	C₄	C₂×Q₈	1	1	yes	characteristic
29	C₄	#41	1	1	yes	characteristic
30	C₂	C₄×Q₈	1	1	yes	characteristic
31	C₂	C₂×M₁₆	1	1	yes	characteristic
32	C₂	#131	1	1	yes	characteristic

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

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
22. C₈	Q₈	semidirect

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