Subgroups — #135

Order 32.

31 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41	C₂	1	2	yes	maximal, complemented
2	Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₁₆	C₂	1	2	yes	maximal, complemented
4	D₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈	C₂²	1	2	yes	—
7	D₄	—	2	2	no	—
8	D₄	C₂²	1	2	yes	—
9	C₂×C₄	—	2	2	no	—
10	C₂×C₄	C₂²	1	1	yes	characteristic
11	C₈	C₂²	1	1	yes	characteristic
12	C₈	C₂²	1	1	yes	characteristic
13	C₂²	—	2	2	no	—
14	C₂²	D₄	1	1	yes	characteristic
15	C₄	—	2	2	no	—
16	C₄	D₄	1	1	yes	characteristic
17	C₄	C₂³	1	1	yes	characteristic
18	C₂	—	2	1	no	—
19	C₂	—	4	2	no	—
20	C₂	C₂×D₄	1	1	yes	characteristic

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