Subgroups — #157

Order 36.

10 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃²	C₄	1	1	yes	characteristic, complemented, Sylow
3	D₃	—	6	2	no	—
4	C₄	—	9	1	no	maximal, Sylow
5	C₃	—	2	2	no	—
6	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²	C₄
prime radical, cosocle	C₃⋊D₃	C₂
LCS	G ▷ C₃²
UCS	C₁

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

N	H ≅ G/N	type
2. C₃²	C₄	semidirect

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

subgroup	index
C₃⋊D₃	2
C₄	9