Subgroups — C₂²×D₈ #569

Order 64.

169 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₈	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴	—	2	2	no	—
5	C₂×D₄	—	2	6	no	—
6	C₂×D₄	C₂²	1	12	yes	complemented
7	C₂²×C₄	C₂²	1	1	yes	characteristic
8	D₈	C₂²	1	16	yes	direct product factor
9	C₂×C₈	C₂²	1	6	yes	complemented
10	C₂³	—	2	12	no	—
11	C₂³	—	4	8	no	—
12	C₂³	D₄	1	1	yes	characteristic
13	D₄	—	2	12	no	—
14	D₄	C₂³	1	8	yes	complemented
15	C₂×C₄	D₄	1	3	yes	—
16	C₂×C₄	C₂³	1	3	yes	—
17	C₈	C₂³	1	4	yes	complemented
18	C₂²	—	2	8	no	—
19	C₂²	—	4	24	no	—
20	C₂²	D₈	1	4	yes	direct product factor
21	C₂²	C₂×D₄	1	3	yes	—
22	C₄	C₂×D₄	1	3	yes	—
23	C₄	C₂⁴	1	1	yes	characteristic
24	C₂	—	4	8	no	—
25	C₂	C₂×D₈	1	6	yes	direct product factor
26	C₂	C₂²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	D₄
commutator, Frattini, cosocle	C₄	C₂⁴
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ G
LECS	G ▷ 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.