Subgroups — C₂²×C₃²×D₃ #6337

Order 216.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×D₃	C₂	1	6	yes	maximal, direct product factor
3	C₂³×C₃²	—	3	1	no	maximal
4	C₂²×C₃×D₃	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃³	C₂²	1	3	yes	complemented
6	C₃²×D₃	C₂²	1	4	yes	direct product factor
7	C₂²×C₃²	—	2	4	no	—
8	C₂²×C₃²	—	3	6	no	—
9	C₂²×C₃²	D₃	1	1	yes	characteristic, direct product factor
10	C₂²×C₃²	C₂×C₃	1	4	yes	complemented
11	C₂×C₃×D₃	C₂×C₃	1	24	yes	direct product factor
12	C₃³	C₂³	1	1	yes	characteristic, complemented, Sylow
13	C₂³×C₃	—	3	4	no	—
14	C₂²×D₃	C₃²	1	1	yes	characteristic, direct product factor
15	C₂×C₃²	—	2	12	no	—
16	C₂×C₃²	—	3	4	no	—
17	C₂×C₃²	C₂×D₃	1	3	yes	direct product factor
18	C₂×C₃²	C₂²×C₃	1	12	yes	complemented
19	C₃×D₃	C₂²×C₃	1	16	yes	direct product factor
20	C₂²×C₃	—	2	4	no	—
21	C₂²×C₃	—	3	24	no	—
22	C₂²×C₃	C₃×D₃	1	4	yes	direct product factor
23	C₂²×C₃	C₂×C₃²	1	1	yes	characteristic, complemented
24	C₂×D₃	C₂×C₃²	1	6	yes	direct product factor
25	C₃²	—	2	4	no	—
26	C₃²	C₂²×D₃	1	1	yes	characteristic, direct product factor
27	C₃²	C₂³×C₃	1	4	yes	complemented
28	C₂³	—	3	1	no	Sylow
29	C₂×C₃	—	2	12	no	—
30	C₂×C₃	—	3	16	no	—
31	C₂×C₃	C₂×C₃×D₃	1	12	yes	direct product factor
32	C₂×C₃	C₂²×C₃²	1	3	yes	complemented
33	D₃	C₂²×C₃²	1	4	yes	direct product factor
34	C₂²	—	3	6	no	—
35	C₂²	C₃²×D₃	1	1	yes	characteristic, direct product factor
36	C₃	—	2	4	no	—
37	C₃	C₂²×C₃×D₃	1	4	yes	direct product factor
38	C₃	C₂³×C₃²	1	1	yes	characteristic, complemented
39	C₂	—	3	4	no	—
40	C₂	C₂×C₃²×D₃	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃²	D₃
commutator, cosocle, hypocenter	C₃	C₂³×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃³	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂²×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³	C₂	semidirect
2. C₂×C₃²×D₃	C₂	direct (N × H)
4. C₂²×C₃×D₃	C₃	direct (N × H)
5. C₂×C₃³	C₂²	semidirect
6. C₃²×D₃	C₂²	direct (N × H)
9. C₂²×C₃²	D₃	direct (N × H)
10. C₂²×C₃²	C₂×C₃	semidirect
11. C₂×C₃×D₃	C₂×C₃	direct (N × H)
12. C₃³	C₂³	semidirect
14. C₂²×D₃	C₃²	direct (N × H)
17. C₂×C₃²	C₂×D₃	direct (N × H)
18. C₂×C₃²	C₂²×C₃	semidirect
19. C₃×D₃	C₂²×C₃	direct (N × H)
22. C₂²×C₃	C₃×D₃	direct (N × H)
23. C₂²×C₃	C₂×C₃²	semidirect
24. C₂×D₃	C₂×C₃²	direct (N × H)
26. C₃²	C₂²×D₃	direct (N × H)
27. C₃²	C₂³×C₃	semidirect
31. C₂×C₃	C₂×C₃×D₃	direct (N × H)
32. C₂×C₃	C₂²×C₃²	semidirect
33. D₃	C₂²×C₃²	direct (N × H)
35. C₂²	C₃²×D₃	direct (N × H)
37. C₃	C₂²×C₃×D₃	direct (N × H)
38. C₃	C₂³×C₃²	semidirect
40. C₂	C₂×C₃²×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
C₂²×C₃³	2
C₂×C₃²×D₃	2
C₂³×C₃²	3
C₂²×C₃×D₃	3

Subgroups — C22×C32×D3 #6337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×D₃ #6337