Subgroups — C₄⋊((C₃⋊D₃)⋊C₃) #6214

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	2	yes	maximal, complemented
2	C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊D₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×D₁₂	—	3	1	no	maximal
5	C₂×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	(C₃⋊D₃)⋊C₃	—	2	2	no	—
7	C₂×C₃⋊D₃	C₂×C₃	1	2	yes	complemented
8	C₂×C₃×D₃	—	3	2	no	—
9	C₃²×C₄	—	2	1	no	—
10	C₃²×C₄	D₃	1	1	yes	characteristic, complemented
11	C₃²×C₄	C₂×C₃	1	1	yes	characteristic, complemented
12	H(3,𝔽₃)	D₄	1	1	yes	characteristic, complemented, Sylow
13	C₃×D₄	—	9	1	no	—
14	D₁₂	—	3	1	no	—
15	D₁₂	—	9	1	no	—
16	C₂×C₃²	—	2	1	no	—
17	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
18	C₂×C₃²	C₂²×C₃	1	1	yes	characteristic
19	C₃⋊D₃	—	2	2	no	—
20	C₃×D₃	—	6	2	no	—
21	C₂²×C₃	—	9	2	no	—
22	C₂×D₃	—	3	2	no	—
23	C₂×D₃	—	9	2	no	—
24	C₃×C₄	—	3	1	no	—
25	C₃×C₄	—	3	1	no	—
26	C₃×C₄	—	6	1	no	—
27	C₃×C₄	C₃×D₃	1	1	yes	characteristic
28	C₃²	—	2	1	no	—
29	C₃²	D₁₂	1	1	yes	characteristic, complemented
30	C₃²	C₃×D₄	1	1	yes	characteristic, complemented
31	D₄	—	9	1	no	Sylow
32	C₂×C₃	—	3	1	no	—
33	C₂×C₃	—	3	1	no	—
34	C₂×C₃	—	6	1	no	—
35	C₂×C₃	—	18	2	no	—
36	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
37	D₃	—	6	2	no	—
38	D₃	—	18	2	no	—
39	C₂²	—	9	2	no	—
40	C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
41	C₃	—	3	1	no	—
42	C₃	—	3	1	no	—
43	C₃	—	6	1	no	—
44	C₃	C₃×D₁₂	1	1	yes	characteristic
45	C₂	—	18	2	no	—
46	C₂	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×C₃²	C₂²×C₃
Frattini, socle	C₂×C₃	C₂×C₃×D₃
Fitting	C₄×H(3,𝔽₃)	C₂
hypercenter	C₄	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×(C₃⋊D₃)⋊C₃	C₂	semidirect
2. C₄×H(3,𝔽₃)	C₂	semidirect
3. C₄⋊(C₃⋊D₃)	C₃	semidirect
7. C₂×C₃⋊D₃	C₂×C₃	semidirect
10. C₃²×C₄	D₃	semidirect
11. C₃²×C₄	C₂×C₃	semidirect
12. H(3,𝔽₃)	D₄	semidirect
29. C₃²	D₁₂	semidirect
30. C₃²	C₃×D₄	semidirect
40. C₄	(C₃⋊D₃)⋊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⋊D₃)⋊C₃	2
C₄×H(3,𝔽₃)	2
C₄⋊(C₃⋊D₃)	3
C₃×D₁₂	3

Subgroups — C4⋊((C3⋊D3)⋊C3) #6214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊((C₃⋊D₃)⋊C₃) #6214