Subgroups — D₁₆ #111

Order 32.

14 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈	C₂	1	2	yes	maximal, complemented
2	C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄	—	2	2	no	—
4	C₈	C₂²	1	1	yes	characteristic
5	C₂²	—	4	2	no	—
6	C₄	D₄	1	1	yes	characteristic
7	C₂	—	8	2	no	—
8	C₂	D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	D₈
commutator, Frattini, cosocle	C₈	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
1. D₈	C₂	semidirect
2. C₁₆	C₂	semidirect

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

subgroup	index
D₈	2
C₁₆	2