Subgroups — #382

Order 64.

77 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄²	C₂	1	1	yes	maximal, characteristic
3	#95	C₂	1	4	yes	maximal
4	C₂²×C₄	C₂²	1	4	yes	—
5	C₂²×C₄	C₂²	1	2	yes	—
6	C₂²×C₄	C₂²	1	1	yes	characteristic
7	C₄⋊C₄	—	2	4	no	—
8	C₄²	C₄	1	4	yes	complemented
9	C₂³	C₂³	1	1	yes	characteristic
10	C₂×C₄	—	2	4	no	—
11	C₂×C₄	—	2	4	no	—
12	C₂×C₄	—	2	8	no	—
13	C₂×C₄	C₂×C₄	1	4	yes	—
14	C₂×C₄	C₂×C₄	1	2	yes	—
15	C₂×C₄	D₄	1	2	yes	—
16	C₂×C₄	Q₈	1	2	yes	—
17	C₂²	C₂²×C₄	1	1	yes	characteristic
18	C₂²	C₂×D₄	1	1	yes	characteristic
19	C₂²	C₂×Q₈	1	1	yes	characteristic
20	C₂²	#41	1	4	yes	—
21	C₄	—	2	4	no	—
22	C₄	—	4	4	no	—
23	C₄	C₄⋊C₄	1	4	yes	—
24	C₂	C₂×C₄⋊C₄	1	1	yes	characteristic
25	C₂	#117	1	2	yes	—
26	C₂	#124	1	2	yes	—
27	C₂	#125	1	2	yes	—

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

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

N	H ≅ G/N	type
8. C₄²	C₄	semidirect

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