Finite Group Databasebeta

← C₃×C₃⋊(C₃⋊D₃) #4217

Subgroups — C₃×C₃⋊(C₃⋊D₃) #4217

Order 162.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(C₃⋊D₃)	C₃	1	1	yes	maximal, characteristic, direct product factor
3	C₃×C₃⋊D₃	—	3	13	no	maximal
4	C₃³	—	2	13	no	—
5	C₃³	D₃	1	13	yes	complemented
6	C₃³	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃⋊D₃	—	3	13	no	—
8	C₃×D₃	—	9	13	no	—
9	C₃²	—	2	52	no	—
10	C₃²	C₃×D₃	1	13	yes	complemented
11	C₃²	C₃⋊D₃	1	13	yes	complemented
12	C₂×C₃	—	27	1	no	—
13	D₃	—	9	13	no	—
14	C₃	—	2	13	no	—
15	C₃	C₃×C₃⋊D₃	1	13	yes	complemented
16	C₃	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
17	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃⁴	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃⁴	C₂	semidirect
2. C₃⋊(C₃⋊D₃)	C₃	direct (N × H)
5. C₃³	D₃	semidirect
6. C₃³	C₂×C₃	semidirect
10. C₃²	C₃×D₃	semidirect
11. C₃²	C₃⋊D₃	semidirect
15. C₃	C₃×C₃⋊D₃	semidirect
16. C₃	C₃⋊(C₃⋊D₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴	2
C₃⋊(C₃⋊D₃)	3
C₃×C₃⋊D₃	3