Subgroups — C₂²×(C₃⋊D₃)⋊C₃ #6273

Order 216.

122 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, complemented
2	C₂×(C₃⋊D₃)⋊C₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×D₃	—	3	1	no	maximal
5	C₂×H(3,𝔽₃)	C₂²	1	3	yes	complemented
6	(C₃⋊D₃)⋊C₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₃²	—	2	1	no	—
8	C₂²×C₃²	D₃	1	1	yes	characteristic, complemented
9	C₂²×C₃²	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃⋊D₃	C₂×C₃	1	6	yes	complemented
11	C₂×C₃×D₃	—	3	6	no	—
12	H(3,𝔽₃)	C₂³	1	1	yes	characteristic, complemented, Sylow
13	C₂³×C₃	—	9	1	no	—
14	C₂²×D₃	—	3	1	no	—
15	C₂²×D₃	—	9	1	no	—
16	C₂×C₃²	—	2	3	no	—
17	C₂×C₃²	C₂×D₃	1	3	yes	complemented
18	C₂×C₃²	C₂²×C₃	1	3	yes	complemented
19	C₃⋊D₃	C₂²×C₃	1	4	yes	complemented
20	C₃×D₃	—	3	4	no	—
21	C₂²×C₃	—	3	1	no	—
22	C₂²×C₃	—	3	1	no	—
23	C₂²×C₃	—	6	1	no	—
24	C₂²×C₃	—	9	6	no	—
25	C₂²×C₃	C₃×D₃	1	1	yes	characteristic
26	C₂×D₃	—	3	6	no	—
27	C₂×D₃	—	9	6	no	—
28	C₃²	—	2	1	no	—
29	C₃²	C₂²×D₃	1	1	yes	characteristic, complemented
30	C₃²	C₂³×C₃	1	1	yes	characteristic, complemented
31	C₂³	—	9	1	no	Sylow
32	C₂×C₃	—	3	3	no	—
33	C₂×C₃	—	3	3	no	—
34	C₂×C₃	—	6	3	no	—
35	C₂×C₃	—	9	4	no	—
36	C₂×C₃	C₂×C₃×D₃	1	3	yes	—
37	D₃	—	3	4	no	—
38	D₃	—	9	4	no	—
39	C₂²	—	9	6	no	—
40	C₂²	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, direct product factor
41	C₃	—	3	1	no	—
42	C₃	—	3	1	no	—
43	C₃	—	6	1	no	—
44	C₃	C₂²×C₃×D₃	1	1	yes	characteristic
45	C₂	—	9	4	no	—
46	C₂	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃²	C₂³×C₃
Frattini	C₃	C₂²×C₃×D₃
Fitting	C₂²×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₃×D₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×H(3,𝔽₃)	C₂	semidirect
2. C₂×(C₃⋊D₃)⋊C₃	C₂	direct (N × H)
3. C₂²×C₃⋊D₃	C₃	semidirect
5. C₂×H(3,𝔽₃)	C₂²	semidirect
6. (C₃⋊D₃)⋊C₃	C₂²	direct (N × H)
8. C₂²×C₃²	D₃	semidirect
9. C₂²×C₃²	C₂×C₃	semidirect
10. C₂×C₃⋊D₃	C₂×C₃	semidirect
12. H(3,𝔽₃)	C₂³	semidirect
17. C₂×C₃²	C₂×D₃	semidirect
18. C₂×C₃²	C₂²×C₃	semidirect
19. C₃⋊D₃	C₂²×C₃	semidirect
29. C₃²	C₂²×D₃	semidirect
30. C₃²	C₂³×C₃	semidirect
40. C₂²	(C₃⋊D₃)⋊C₃	direct (N × H)
46. C₂	C₂×(C₃⋊D₃)⋊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
C₂×(C₃⋊D₃)⋊C₃	2
C₂²×C₃⋊D₃	3
C₂²×C₃×D₃	3

Subgroups — C22×(C3⋊D3)⋊C3 #6273

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×(C₃⋊D₃)⋊C₃ #6273