Subgroups — C₃²×C₄×D₃ #6299

Order 216.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×C₄	—	3	1	no	maximal
5	C₃×C₄×D₃	C₃	1	4	yes	maximal, direct product factor
6	C₂×C₃³	C₂²	1	1	yes	characteristic
7	C₃²×D₃	C₄	1	2	yes	direct product factor
8	C₂²×C₃²	—	3	1	no	—
9	C₂×C₃×D₃	C₂×C₃	1	4	yes	—
10	C₃²×C₄	—	2	4	no	—
11	C₃²×C₄	—	3	1	no	—
12	C₃²×C₄	D₃	1	1	yes	characteristic, direct product factor
13	C₃²×C₄	C₂×C₃	1	4	yes	complemented
14	C₃×Dic₃	C₂×C₃	1	4	yes	complemented
15	C₃³	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃×C₄	—	3	4	no	—
17	C₄×D₃	C₃²	1	1	yes	characteristic, direct product factor
18	C₂×C₃²	—	2	4	no	—
19	C₂×C₃²	—	3	2	no	—
20	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
21	C₂×C₃²	C₂²×C₃	1	4	yes	—
22	C₃×D₃	C₃×C₄	1	8	yes	direct product factor
23	C₂²×C₃	—	3	4	no	—
24	C₂×D₃	C₂×C₃²	1	1	yes	characteristic
25	C₃×C₄	—	2	4	no	—
26	C₃×C₄	—	3	4	no	—
27	C₃×C₄	C₃×D₃	1	4	yes	direct product factor
28	C₃×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
29	Dic₃	C₂×C₃²	1	1	yes	characteristic, complemented
30	C₃²	—	2	4	no	—
31	C₃²	C₄×D₃	1	1	yes	characteristic, direct product factor
32	C₃²	C₂×C₃×C₄	1	4	yes	complemented
33	C₂×C₄	—	3	1	no	Sylow
34	C₂×C₃	—	2	4	no	—
35	C₂×C₃	—	3	8	no	—
36	C₂×C₃	C₂×C₃×D₃	1	4	yes	—
37	C₂×C₃	C₂²×C₃²	1	1	yes	characteristic
38	D₃	C₃²×C₄	1	2	yes	direct product factor
39	C₂²	—	3	1	no	—
40	C₄	—	3	1	no	—
41	C₄	C₃²×D₃	1	1	yes	characteristic, direct product factor
42	C₃	—	2	4	no	—
43	C₃	C₃×C₄×D₃	1	4	yes	direct product factor
44	C₃	C₂×C₃²×C₄	1	1	yes	characteristic, complemented
45	C₂	—	3	2	no	—
46	C₂	C₂×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₃²×C₄	D₃
commutator, hypocenter	C₃	C₂×C₃²×C₄
Frattini	C₂	C₂×C₃²×D₃
Fitting	C₃³×C₄	C₂
socle	C₂×C₃³	C₂²
prime radical	C₂×C₃²×D₃	C₂
cosocle	C₂×C₃	C₂²×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃³×C₄	C₂	semidirect
3. C₃²×Dic₃	C₂	semidirect
5. C₃×C₄×D₃	C₃	direct (N × H)
7. C₃²×D₃	C₄	direct (N × H)
12. C₃²×C₄	D₃	direct (N × H)
13. C₃²×C₄	C₂×C₃	semidirect
14. C₃×Dic₃	C₂×C₃	semidirect
15. C₃³	C₂×C₄	semidirect
17. C₄×D₃	C₃²	direct (N × H)
22. C₃×D₃	C₃×C₄	direct (N × H)
27. C₃×C₄	C₃×D₃	direct (N × H)
28. C₃×C₄	C₂×C₃²	semidirect
29. Dic₃	C₂×C₃²	semidirect
31. C₃²	C₄×D₃	direct (N × H)
32. C₃²	C₂×C₃×C₄	semidirect
38. D₃	C₃²×C₄	direct (N × H)
41. C₄	C₃²×D₃	direct (N × H)
43. C₃	C₃×C₄×D₃	direct (N × H)
44. C₃	C₂×C₃²×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₃	2
C₃³×C₄	2
C₃²×Dic₃	2
C₂×C₃²×C₄	3
C₃×C₄×D₃	3

Subgroups — C32×C4×D3 #6299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×D₃ #6299