Subgroups — C₂×(C₃⋊Dic₃)⋊C₃ #6222

Order 216.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
2. (C₃⋊Dic₃)⋊C₃	C₂	direct (N × H)
3. C₂×C₃⋊Dic₃	C₃	semidirect
5. C₂×H(3,𝔽₃)	C₄	semidirect
10. C₃⋊Dic₃	C₂×C₃	semidirect
12. H(3,𝔽₃)	C₂×C₄	semidirect
18. C₂×C₃²	Dic₃	semidirect
19. C₂×C₃²	C₃×C₄	semidirect
30. C₃²	C₂×Dic₃	semidirect
31. C₃²	C₂×C₃×C₄	semidirect
47. C₂	(C₃⋊Dic₃)⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×H(3,𝔽₃)	2
(C₃⋊Dic₃)⋊C₃	2
C₂×C₃⋊Dic₃	3
C₂×C₃×Dic₃	3

Subgroups — C2×(C3⋊Dic3)⋊C3 #6222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₃⋊Dic₃)⋊C₃ #6222