Subgroups — C₃⋊((C₃⋊D₃)⋊C₃) #4204

Order 162.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(C₃⋊D₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊D₃	—	3	1	no	maximal
4	(C₃⋊D₃)⋊C₃	—	3	3	no	maximal
5	C₃³	—	2	1	no	—
6	C₃³	D₃	1	1	yes	characteristic, complemented
7	C₃³	C₂×C₃	1	1	yes	characteristic, complemented
8	H(3,𝔽₃)	—	2	3	no	—
9	H(3,𝔽₃)	D₃	1	3	yes	complemented
10	C₃⋊D₃	—	3	1	no	—
11	C₃⋊D₃	—	3	3	no	—
12	C₃⋊D₃	—	9	3	no	—
13	C₃×D₃	—	9	3	no	—
14	C₃×D₃	—	9	1	no	—
15	C₃²	—	2	1	no	—
16	C₃²	—	2	3	no	—
17	C₃²	—	3	3	no	—
18	C₃²	—	3	3	no	—
19	C₃²	—	6	3	no	—
20	C₃²	C₃×D₃	1	1	yes	characteristic
21	C₃²	C₃×D₃	1	3	yes	complemented
22	C₃²	C₃⋊D₃	1	1	yes	characteristic, complemented
23	C₂×C₃	—	27	1	no	—
24	D₃	—	9	3	no	—
25	D₃	—	9	1	no	—
26	D₃	—	27	3	no	—
27	C₃	—	3	3	no	—
28	C₃	—	3	1	no	—
29	C₃	—	6	1	no	—
30	C₃	—	6	3	no	—
31	C₃	(C₃⋊D₃)⋊C₃	1	3	yes	complemented
32	C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
33	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³	C₂×C₃
Frattini	C₃	C₃×C₃⋊D₃
Fitting	C₃×H(3,𝔽₃)	C₂
socle	C₃²	C₃×D₃
LCS	G ▷ C₃³
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.