Subgroups — C₂×C₃²×C₃⋊D₃ #66481

Order 324.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×C₃⋊D₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₃⋊D₃	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×D₃	—	3	4	no	maximal
5	C₃⁴	C₂²	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	—	2	16	no	—
7	C₂×C₃³	D₃	1	4	yes	complemented
8	C₂×C₃³	C₂×C₃	1	4	yes	complemented
9	C₃×C₃⋊D₃	C₂×C₃	1	8	yes	direct product factor
10	C₃²×D₃	—	3	8	no	—
11	C₂²×C₃²	—	9	1	no	—
12	C₂×C₃⋊D₃	C₃²	1	1	yes	characteristic, direct product factor
13	C₂×C₃×D₃	—	3	16	no	—
14	C₃³	—	2	16	no	—
15	C₃³	C₂×D₃	1	4	yes	complemented
16	C₃³	C₂²×C₃	1	4	yes	complemented
17	C₂×C₃²	—	2	24	no	—
18	C₂×C₃²	—	2	16	no	—
19	C₂×C₃²	—	2	16	no	—
20	C₂×C₃²	—	9	2	no	—
21	C₂×C₃²	C₃×D₃	1	16	yes	complemented
22	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic, direct product factor
23	C₂×C₃²	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₃⋊D₃	C₂×C₃²	1	2	yes	direct product factor
25	C₃×D₃	—	3	32	no	—
26	C₂²×C₃	—	9	4	no	—
27	C₂×D₃	—	3	4	no	—
28	C₃²	—	2	24	no	—
29	C₃²	—	2	16	no	—
30	C₃²	—	2	16	no	—
31	C₃²	C₂×C₃×D₃	1	16	yes	complemented
32	C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic, direct product factor
33	C₃²	C₂²×C₃²	1	1	yes	characteristic, complemented
34	C₂×C₃	—	2	16	no	—
35	C₂×C₃	—	9	8	no	—
36	C₂×C₃	C₃²×D₃	1	4	yes	complemented
37	C₂×C₃	C₃×C₃⋊D₃	1	4	yes	direct product factor
38	D₃	—	3	8	no	—
39	C₂²	—	9	1	no	Sylow
40	C₃	—	2	16	no	—
41	C₃	C₂×C₃²×D₃	1	4	yes	complemented
42	C₃	C₂×C₃×C₃⋊D₃	1	4	yes	direct product factor
43	C₂	—	9	2	no	—
44	C₂	C₃²×C₃⋊D₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃²	C₃⋊D₃
commutator, cosocle, hypocenter	C₃²	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃⁴	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃⁴	C₂	semidirect
2. C₃²×C₃⋊D₃	C₂	direct (N × H)
3. C₂×C₃×C₃⋊D₃	C₃	direct (N × H)
5. C₃⁴	C₂²	semidirect
7. C₂×C₃³	D₃	semidirect
8. C₂×C₃³	C₂×C₃	semidirect
9. C₃×C₃⋊D₃	C₂×C₃	direct (N × H)
12. C₂×C₃⋊D₃	C₃²	direct (N × H)
15. C₃³	C₂×D₃	semidirect
16. C₃³	C₂²×C₃	semidirect
21. C₂×C₃²	C₃×D₃	semidirect
22. C₂×C₃²	C₃⋊D₃	direct (N × H)
23. C₂×C₃²	C₂×C₃²	semidirect
24. C₃⋊D₃	C₂×C₃²	direct (N × H)
31. C₃²	C₂×C₃×D₃	semidirect
32. C₃²	C₂×C₃⋊D₃	direct (N × H)
33. C₃²	C₂²×C₃²	semidirect
36. C₂×C₃	C₃²×D₃	semidirect
37. C₂×C₃	C₃×C₃⋊D₃	direct (N × H)
41. C₃	C₂×C₃²×D₃	semidirect
42. C₃	C₂×C₃×C₃⋊D₃	direct (N × H)
44. C₂	C₃²×C₃⋊D₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃⁴	2
C₃²×C₃⋊D₃	2
C₂×C₃×C₃⋊D₃	3
C₂×C₃²×D₃	3

Subgroups — C2×C32×C3⋊D3 #66481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₃⋊D₃ #66481