Subgroups — C₂³×C₃²×D₃ #89125

Order 432.

932 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃²×D₃	C₂	1	14	yes	maximal, direct product factor
3	C₂⁴×C₃²	—	3	1	no	maximal
4	C₂³×C₃×D₃	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃³	C₂²	1	7	yes	complemented
6	C₂×C₃²×D₃	C₂²	1	28	yes	direct product factor
7	C₂³×C₃²	—	2	4	no	—
8	C₂³×C₃²	—	3	14	no	—
9	C₂³×C₃²	D₃	1	1	yes	characteristic, direct product factor
10	C₂³×C₃²	C₂×C₃	1	4	yes	complemented
11	C₂²×C₃×D₃	C₂×C₃	1	56	yes	direct product factor
12	C₂×C₃³	C₂³	1	7	yes	complemented
13	C₃²×D₃	C₂³	1	8	yes	direct product factor
14	C₂⁴×C₃	—	3	4	no	—
15	C₂³×D₃	C₃²	1	1	yes	characteristic, direct product factor
16	C₂²×C₃²	—	2	28	no	—
17	C₂²×C₃²	—	3	28	no	—
18	C₂²×C₃²	C₂×D₃	1	7	yes	direct product factor
19	C₂²×C₃²	C₂²×C₃	1	28	yes	complemented
20	C₂×C₃×D₃	C₂²×C₃	1	112	yes	direct product factor
21	C₃³	C₂⁴	1	1	yes	characteristic, complemented, Sylow
22	C₂³×C₃	—	2	4	no	—
23	C₂³×C₃	—	3	56	no	—
24	C₂³×C₃	C₃×D₃	1	4	yes	direct product factor
25	C₂³×C₃	C₂×C₃²	1	1	yes	characteristic, complemented
26	C₂²×D₃	C₂×C₃²	1	14	yes	direct product factor
27	C₂×C₃²	—	2	28	no	—
28	C₂×C₃²	—	3	8	no	—
29	C₂×C₃²	C₂²×D₃	1	7	yes	direct product factor
30	C₂×C₃²	C₂³×C₃	1	28	yes	complemented
31	C₃×D₃	C₂³×C₃	1	32	yes	direct product factor
32	C₂⁴	—	3	1	no	Sylow
33	C₂²×C₃	—	2	28	no	—
34	C₂²×C₃	—	3	112	no	—
35	C₂²×C₃	C₂×C₃×D₃	1	28	yes	direct product factor
36	C₂²×C₃	C₂²×C₃²	1	7	yes	complemented
37	C₂×D₃	C₂²×C₃²	1	28	yes	direct product factor
38	C₃²	—	2	4	no	—
39	C₃²	C₂³×D₃	1	1	yes	characteristic, direct product factor
40	C₃²	C₂⁴×C₃	1	4	yes	complemented
41	C₂³	—	3	14	no	—
42	C₂³	C₃²×D₃	1	1	yes	characteristic, direct product factor
43	C₂×C₃	—	2	28	no	—
44	C₂×C₃	—	3	32	no	—
45	C₂×C₃	C₂²×C₃×D₃	1	28	yes	direct product factor
46	C₂×C₃	C₂³×C₃²	1	7	yes	complemented
47	D₃	C₂³×C₃²	1	8	yes	direct product factor
48	C₂²	—	3	28	no	—
49	C₂²	C₂×C₃²×D₃	1	7	yes	direct product factor
50	C₃	—	2	4	no	—
51	C₃	C₂³×C₃×D₃	1	4	yes	direct product factor
52	C₃	C₂⁴×C₃²	1	1	yes	characteristic, complemented
53	C₂	—	3	8	no	—
54	C₂	C₂²×C₃²×D₃	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³×C₃²	D₃
commutator, cosocle, hypocenter	C₃	C₂⁴×C₃²
Frattini	C₁	G
Fitting, socle	C₂³×C₃³	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₃²

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃³	C₂	semidirect
2. C₂²×C₃²×D₃	C₂	direct (N × H)
4. C₂³×C₃×D₃	C₃	direct (N × H)
5. C₂²×C₃³	C₂²	semidirect
6. C₂×C₃²×D₃	C₂²	direct (N × H)
9. C₂³×C₃²	D₃	direct (N × H)
10. C₂³×C₃²	C₂×C₃	semidirect
11. C₂²×C₃×D₃	C₂×C₃	direct (N × H)
12. C₂×C₃³	C₂³	semidirect
13. C₃²×D₃	C₂³	direct (N × H)
15. C₂³×D₃	C₃²	direct (N × H)
18. C₂²×C₃²	C₂×D₃	direct (N × H)
19. C₂²×C₃²	C₂²×C₃	semidirect
20. C₂×C₃×D₃	C₂²×C₃	direct (N × H)
21. C₃³	C₂⁴	semidirect
24. C₂³×C₃	C₃×D₃	direct (N × H)
25. C₂³×C₃	C₂×C₃²	semidirect
26. C₂²×D₃	C₂×C₃²	direct (N × H)
29. C₂×C₃²	C₂²×D₃	direct (N × H)
30. C₂×C₃²	C₂³×C₃	semidirect
31. C₃×D₃	C₂³×C₃	direct (N × H)
35. C₂²×C₃	C₂×C₃×D₃	direct (N × H)
36. C₂²×C₃	C₂²×C₃²	semidirect
37. C₂×D₃	C₂²×C₃²	direct (N × H)
39. C₃²	C₂³×D₃	direct (N × H)
40. C₃²	C₂⁴×C₃	semidirect
42. C₂³	C₃²×D₃	direct (N × H)
45. C₂×C₃	C₂²×C₃×D₃	direct (N × H)
46. C₂×C₃	C₂³×C₃²	semidirect
47. D₃	C₂³×C₃²	direct (N × H)
49. C₂²	C₂×C₃²×D₃	direct (N × H)
51. C₃	C₂³×C₃×D₃	direct (N × H)
52. C₃	C₂⁴×C₃²	semidirect
54. C₂	C₂²×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
C₂³×C₃³	2
C₂²×C₃²×D₃	2
C₂⁴×C₃²	3
C₂³×C₃×D₃	3

Subgroups — C23×C32×D3 #89125

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×D₃ #89125