Subgroups — C₂³×C₃×C₃⋊D₃ #89126

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

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃³	C₂	semidirect
2. C₂²×C₃×C₃⋊D₃	C₂	direct (N × H)
3. C₂³×C₃⋊D₃	C₃	direct (N × H)
5. C₂²×C₃³	C₂²	semidirect
6. C₂×C₃×C₃⋊D₃	C₂²	direct (N × H)
8. C₂³×C₃²	D₃	semidirect
9. C₂³×C₃²	C₂×C₃	semidirect
10. C₂²×C₃⋊D₃	C₂×C₃	direct (N × H)
12. C₂×C₃³	C₂³	semidirect
13. C₃×C₃⋊D₃	C₂³	direct (N × H)
17. C₂²×C₃²	C₂×D₃	semidirect
18. C₂²×C₃²	C₂²×C₃	semidirect
19. C₂×C₃⋊D₃	C₂²×C₃	direct (N × H)
21. C₃³	C₂⁴	semidirect
24. C₂³×C₃	C₃×D₃	semidirect
25. C₂³×C₃	C₃⋊D₃	direct (N × H)
28. C₂×C₃²	C₂²×D₃	semidirect
29. C₂×C₃²	C₂³×C₃	semidirect
30. C₃⋊D₃	C₂³×C₃	direct (N × H)
35. C₂²×C₃	C₂×C₃×D₃	semidirect
36. C₂²×C₃	C₂×C₃⋊D₃	direct (N × H)
39. C₃²	C₂³×D₃	semidirect
40. C₃²	C₂⁴×C₃	semidirect
42. C₂³	C₃×C₃⋊D₃	direct (N × H)
45. C₂×C₃	C₂²×C₃×D₃	semidirect
46. C₂×C₃	C₂²×C₃⋊D₃	direct (N × H)
49. C₂²	C₂×C₃×C₃⋊D₃	direct (N × H)
51. C₃	C₂³×C₃×D₃	semidirect
52. C₃	C₂³×C₃⋊D₃	direct (N × H)
54. C₂	C₂²×C₃×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
C₂³×C₃³	2
C₂²×C₃×C₃⋊D₃	2
C₂³×C₃⋊D₃	3
C₂³×C₃×D₃	3

Subgroups — C23×C3×C3⋊D3 #89126

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₃⋊D₃ #89126