Subgroups — ((C₃⋊D₃)⋊C₃)⋊C₄ #6197

Order 216.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#1084	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊Dic₃)⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#624	—	3	1	no	maximal
5	D₃×Dic₃	—	3	1	no	maximal
6	C₂×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	(C₃⋊D₃)⋊C₃	C₄	1	2	yes	complemented
8	C₂×C₃⋊D₃	D₃	1	1	yes	characteristic
9	C₂×C₃×D₃	—	3	1	no	—
10	C₃⋊Dic₃	D₃	1	1	yes	characteristic, complemented
11	C₃×Dic₃	—	3	1	no	—
12	C₃×Dic₃	—	3	1	no	—
13	C₃×Dic₃	—	3	1	no	—
14	C₃×Dic₃	—	6	1	no	—
15	H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
16	C₂×Dic₃	—	9	1	no	—
17	C₄×D₃	—	9	1	no	—
18	C₄×D₃	—	9	1	no	—
19	C₂×C₃²	—	2	1	no	—
20	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
21	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
22	C₃⋊D₃	Dic₃	1	2	yes	complemented
23	C₃×D₃	—	3	2	no	—
24	C₂²×C₃	—	9	1	no	—
25	C₂×D₃	—	3	1	no	—
26	C₂×D₃	—	9	1	no	—
27	C₃×C₄	—	9	1	no	—
28	C₃×C₄	—	9	1	no	—
29	Dic₃	—	3	1	no	—
30	Dic₃	—	3	1	no	—
31	Dic₃	—	3	1	no	—
32	Dic₃	—	6	1	no	—
33	Dic₃	—	9	1	no	—
34	C₃²	—	2	1	no	—
35	C₃²	C₄×D₃	1	1	yes	characteristic, complemented
36	C₃²	C₂×Dic₃	1	1	yes	characteristic, complemented
37	C₂×C₄	—	27	1	no	Sylow
38	C₂×C₃	—	3	1	no	—
39	C₂×C₃	—	3	1	no	—
40	C₂×C₃	—	6	1	no	—
41	C₂×C₃	—	9	2	no	—
42	C₂×C₃	D₃²	1	1	yes	characteristic
43	D₃	—	3	2	no	—
44	D₃	—	9	2	no	—
45	C₂²	—	9	1	no	—
46	C₄	—	9	1	no	—
47	C₄	—	9	1	no	—
48	C₃	—	3	1	no	—
49	C₃	—	3	1	no	—
50	C₃	—	6	1	no	—
51	C₃	D₃×Dic₃	1	1	yes	characteristic
52	C₂	—	9	2	no	—
53	C₂	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, hypocenter	H(3,𝔽₃)	C₂×C₄
Frattini, socle	C₂×C₃	D₃²
Fitting, cosocle	C₂×H(3,𝔽₃)	C₂²
prime radical	C₂×(C₃⋊D₃)⋊C₃	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. #1084	C₂	semidirect
3. (C₃⋊Dic₃)⋊C₃	C₂	semidirect
7. (C₃⋊D₃)⋊C₃	C₄	semidirect
10. C₃⋊Dic₃	D₃	semidirect
15. H(3,𝔽₃)	C₂×C₄	semidirect
22. C₃⋊D₃	Dic₃	semidirect
35. C₃²	C₄×D₃	semidirect
36. C₃²	C₂×Dic₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₃⋊D₃)⋊C₃	2
#1084	2
(C₃⋊Dic₃)⋊C₃	2
#624	3
D₃×Dic₃	3

Subgroups — ((C3⋊D3)⋊C3)⋊C4 #6197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ((C₃⋊D₃)⋊C₃)⋊C₄ #6197