Subgroups — C₂³×D₄ #580

Order 64.

681 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵	C₂	1	2	yes	maximal, complemented
2	C₂²×D₄	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴	—	2	16	no	—
5	C₂⁴	C₂²	1	1	yes	characteristic
6	C₂⁴	C₂²	1	28	yes	complemented
7	C₂×D₄	C₂²	1	112	yes	direct product factor
8	C₂²×C₄	C₂²	1	14	yes	complemented
9	C₂³	—	2	112	no	—
10	C₂³	D₄	1	8	yes	direct product factor
11	C₂³	C₂³	1	7	yes	—
12	C₂³	C₂³	1	56	yes	complemented
13	D₄	C₂³	1	64	yes	direct product factor
14	C₂×C₄	C₂³	1	28	yes	complemented
15	C₂²	—	2	112	no	—
16	C₂²	C₂×D₄	1	28	yes	direct product factor
17	C₂²	C₂⁴	1	7	yes	—
18	C₂²	C₂⁴	1	16	yes	complemented
19	C₄	C₂⁴	1	8	yes	complemented
20	C₂	—	2	16	no	—
21	C₂	C₂²×D₄	1	14	yes	direct product factor
22	C₂	C₂⁵	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴	C₂²
commutator, Frattini, cosocle	C₂	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ 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.