Subgroups — D₃² #158

Order 36.

22 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, complemented
2	C₃×D₃	C₂	1	2	yes	maximal, complemented
3	C₂×D₃	—	3	2	no	maximal
4	C₃²	C₂²	1	1	yes	characteristic, complemented, Sylow
5	C₂×C₃	—	3	2	no	—
6	D₃	—	3	2	no	—
7	D₃	—	6	1	no	—
8	D₃	D₃	1	2	yes	direct product factor
9	C₂²	—	9	1	no	Sylow
10	C₃	—	2	1	no	—
11	C₃	C₂×D₃	1	2	yes	complemented
12	C₂	—	3	2	no	—
13	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²	C₂²
LCS	G ▷ C₃²
UCS	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.