Subgroups — Dic₁₈ #607

Order 72.

18 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	Dic₉	C₂	1	2	yes	maximal
3	Dic₆	—	3	1	no	maximal
4	C₂×C₉	C₂²	1	1	yes	characteristic
5	C₃×C₄	D₃	1	1	yes	characteristic
6	Dic₃	—	3	2	no	—
7	C₉	Q₈	1	1	yes	characteristic, complemented, Sylow
8	Q₈	—	9	1	no	Sylow
9	C₂×C₃	C₂×D₃	1	1	yes	characteristic
10	C₄	—	9	2	no	—
11	C₄	D₉	1	1	yes	characteristic
12	C₃	Dic₆	1	1	yes	characteristic
13	C₂	C₂×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₉
commutator, cosocle	C₂×C₉	C₂²
Frattini, socle, prime radical	C₂×C₃	C₂×D₃
Fitting	C₄×C₉	C₂
hypercenter	C₄	D₉
hypocenter	C₉	Q₈
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₄

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

N	H ≅ G/N	type
7. C₉	Q₈	semidirect

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