Subgroups — C₂²×C₃⋊(C₃⋊D₃) #6339

Order 216.

448 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃⋊(C₃⋊D₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊D₃	—	3	13	no	maximal
4	C₂×C₃³	C₂²	1	3	yes	complemented
5	C₃⋊(C₃⋊D₃)	C₂²	1	4	yes	direct product factor
6	C₂²×C₃²	D₃	1	13	yes	complemented
7	C₂×C₃⋊D₃	—	3	78	no	—
8	C₃³	C₂³	1	1	yes	characteristic, complemented, Sylow
9	C₂²×D₃	—	9	13	no	—
10	C₂×C₃²	C₂×D₃	1	39	yes	complemented
11	C₃⋊D₃	—	3	52	no	—
12	C₂²×C₃	C₃⋊D₃	1	13	yes	complemented
13	C₂×D₃	—	9	78	no	—
14	C₃²	C₂²×D₃	1	13	yes	complemented
15	C₂³	—	27	1	no	Sylow
16	C₂×C₃	C₂×C₃⋊D₃	1	39	yes	complemented
17	D₃	—	9	52	no	—
18	C₂²	—	27	6	no	—
19	C₂²	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
20	C₃	C₂²×C₃⋊D₃	1	13	yes	complemented
21	C₂	—	27	4	no	—
22	C₂	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂²

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.