Subgroups — C₃⋊((C₃⋊Dic₃)⋊C₃) #66406

Order 324.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊Dic₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊Dic₃	—	3	1	no	maximal
4	(C₃⋊Dic₃)⋊C₃	—	3	3	no	maximal
5	C₃×H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	—	2	1	no	—
7	C₂×C₃³	D₃	1	1	yes	characteristic
8	C₂×C₃³	C₂×C₃	1	1	yes	characteristic
9	C₂×H(3,𝔽₃)	—	2	3	no	—
10	C₂×H(3,𝔽₃)	D₃	1	3	yes	—
11	C₃⋊Dic₃	—	3	1	no	—
12	C₃⋊Dic₃	—	3	3	no	—
13	C₃⋊Dic₃	—	9	3	no	—
14	C₃×Dic₃	—	9	3	no	—
15	C₃×Dic₃	—	9	1	no	—
16	C₃³	—	2	1	no	—
17	C₃³	Dic₃	1	1	yes	characteristic, complemented
18	C₃³	C₃×C₄	1	1	yes	characteristic, complemented
19	H(3,𝔽₃)	—	2	3	no	—
20	H(3,𝔽₃)	Dic₃	1	3	yes	complemented
21	C₂×C₃²	—	2	1	no	—
22	C₂×C₃²	—	2	3	no	—
23	C₂×C₃²	—	3	3	no	—
24	C₂×C₃²	—	3	3	no	—
25	C₂×C₃²	—	6	3	no	—
26	C₂×C₃²	C₃×D₃	1	3	yes	—
27	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
28	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic
29	C₃×C₄	—	27	1	no	—
30	Dic₃	—	9	3	no	—
31	Dic₃	—	9	1	no	—
32	Dic₃	—	27	3	no	—
33	C₃²	—	2	1	no	—
34	C₃²	—	2	3	no	—
35	C₃²	—	3	3	no	—
36	C₃²	—	3	3	no	—
37	C₃²	—	6	3	no	—
38	C₃²	C₃×Dic₃	1	1	yes	characteristic
39	C₃²	C₃×Dic₃	1	3	yes	complemented
40	C₃²	C₃⋊Dic₃	1	1	yes	characteristic, complemented
41	C₂×C₃	—	3	3	no	—
42	C₂×C₃	—	3	1	no	—
43	C₂×C₃	—	6	1	no	—
44	C₂×C₃	—	6	3	no	—
45	C₂×C₃	(C₃⋊D₃)⋊C₃	1	3	yes	—
46	C₂×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
47	C₄	—	27	1	no	Sylow
48	C₃	—	3	3	no	—
49	C₃	—	3	1	no	—
50	C₃	—	6	1	no	—
51	C₃	—	6	3	no	—
52	C₃	(C₃⋊Dic₃)⋊C₃	1	3	yes	complemented
53	C₃	C₃×C₃⋊Dic₃	1	1	yes	characteristic
54	C₂	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	C₃⋊((C₃⋊D₃)⋊C₃)
commutator, hypocenter	C₃³	C₃×C₄
Frattini	C₂×C₃	C₃×C₃⋊D₃
Fitting, prime radical	C₂×C₃×H(3,𝔽₃)	C₂
socle	C₂×C₃²	C₃×D₃
cosocle	C₂×C₃³	C₂×C₃
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊Dic₃)	C₃	semidirect
5. C₃×H(3,𝔽₃)	C₄	semidirect
17. C₃³	Dic₃	semidirect
18. C₃³	C₃×C₄	semidirect
20. H(3,𝔽₃)	Dic₃	semidirect
39. C₃²	C₃×Dic₃	semidirect
40. C₃²	C₃⋊Dic₃	semidirect
52. C₃	(C₃⋊Dic₃)⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×H(3,𝔽₃)	2
C₃⋊(C₃⋊Dic₃)	3
C₃×C₃⋊Dic₃	3
(C₃⋊Dic₃)⋊C₃	3

Subgroups — C3⋊((C3⋊Dic3)⋊C3) #66406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊((C₃⋊Dic₃)⋊C₃) #66406