Subgroups — C₂×C₃²×D₁₂ #89055

Order 432.

388 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₃³×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₁₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×D₄	—	3	1	no	maximal
5	C₂×C₃×D₁₂	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³	C₂²	1	1	yes	characteristic
7	C₂×C₃²×D₃	—	2	4	no	—
8	C₂×C₃²×D₃	C₂²	1	4	yes	complemented
9	C₃³×C₄	C₂²	1	2	yes	complemented
10	C₂³×C₃²	—	3	2	no	—
11	C₂²×C₃×D₃	C₂×C₃	1	8	yes	complemented
12	C₃²×D₄	—	3	4	no	—
13	C₂×C₃²×C₄	—	2	4	no	—
14	C₂×C₃²×C₄	D₃	1	1	yes	characteristic, complemented
15	C₂×C₃²×C₄	C₂×C₃	1	4	yes	complemented
16	C₃×D₁₂	C₂×C₃	1	16	yes	direct product factor
17	C₂×C₃³	D₄	1	2	yes	complemented
18	C₂×C₃³	C₂³	1	1	yes	characteristic
19	C₃²×D₃	—	2	4	no	—
20	C₂×C₃×D₄	—	3	4	no	—
21	C₂×D₁₂	C₃²	1	1	yes	characteristic, direct product factor
22	C₂²×C₃²	—	2	4	no	—
23	C₂²×C₃²	—	3	4	no	—
24	C₂²×C₃²	—	6	4	no	—
25	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
26	C₂²×C₃²	C₂²×C₃	1	4	yes	—
27	C₂×C₃×D₃	—	2	16	no	—
28	C₂×C₃×D₃	C₂²×C₃	1	16	yes	complemented
29	C₃²×C₄	—	2	8	no	—
30	C₃²×C₄	C₂×D₃	1	2	yes	complemented
31	C₃²×C₄	C₂²×C₃	1	8	yes	complemented
32	C₃³	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
33	C₂³×C₃	—	3	8	no	—
34	C₂²×D₃	C₂×C₃²	1	2	yes	complemented
35	C₃×D₄	—	3	16	no	—
36	C₂×C₃×C₄	—	2	4	no	—
37	C₂×C₃×C₄	C₃×D₃	1	4	yes	complemented
38	C₂×C₃×C₄	C₂×C₃²	1	1	yes	characteristic, complemented
39	D₁₂	C₂×C₃²	1	4	yes	direct product factor
40	C₂×C₃²	—	2	8	no	—
41	C₂×C₃²	—	2	4	no	—
42	C₂×C₃²	—	6	4	no	—
43	C₂×C₃²	D₁₂	1	2	yes	direct product factor
44	C₂×C₃²	C₃×D₄	1	8	yes	complemented
45	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
46	C₂×C₃²	C₂³×C₃	1	4	yes	—
47	C₃×D₃	—	2	16	no	—
48	C₂×D₄	—	3	1	no	Sylow
49	C₂²×C₃	—	2	4	no	—
50	C₂²×C₃	—	3	16	no	—
51	C₂²×C₃	—	6	16	no	—
52	C₂²×C₃	C₂×C₃×D₃	1	4	yes	—
53	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
54	C₂×D₃	—	2	4	no	—
55	C₂×D₃	C₂²×C₃²	1	4	yes	complemented
56	C₃×C₄	—	2	8	no	—
57	C₃×C₄	C₂×C₃×D₃	1	8	yes	complemented
58	C₃×C₄	C₂²×C₃²	1	2	yes	complemented
59	C₃²	—	2	4	no	—
60	C₃²	C₂×D₁₂	1	1	yes	characteristic, direct product factor
61	C₃²	C₂×C₃×D₄	1	4	yes	complemented
62	C₂³	—	3	2	no	—
63	D₄	—	3	4	no	—
64	C₂×C₄	C₃²×D₃	1	1	yes	characteristic, complemented
65	C₂×C₃	—	2	8	no	—
66	C₂×C₃	—	2	4	no	—
67	C₂×C₃	—	6	16	no	—
68	C₂×C₃	C₃×D₁₂	1	8	yes	direct product factor
69	C₂×C₃	C₃²×D₄	1	2	yes	complemented
70	C₂×C₃	C₂²×C₃×D₃	1	4	yes	—
71	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
72	D₃	—	2	4	no	—
73	C₂²	—	3	4	no	—
74	C₂²	—	6	4	no	—
75	C₂²	C₂×C₃²×D₃	1	1	yes	characteristic
76	C₄	C₂×C₃²×D₃	1	2	yes	complemented
77	C₃	—	2	4	no	—
78	C₃	C₂×C₃×D₁₂	1	4	yes	direct product factor
79	C₃	C₂×C₃²×D₄	1	1	yes	characteristic, complemented
80	C₂	—	6	4	no	—
81	C₂	C₃²×D₁₂	1	2	yes	direct product factor
82	C₂	C₂²×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₃²
Frattini	C₂	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₄	C₂
socle	C₂²×C₃³	C₂²
hypercenter	C₂×C₃²×C₄	D₃
hypocenter	C₃	C₂×C₃²×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×D₃	C₂	semidirect
2. C₂×C₃³×C₄	C₂	semidirect
3. C₃²×D₁₂	C₂	direct (N × H)
5. C₂×C₃×D₁₂	C₃	direct (N × H)
8. C₂×C₃²×D₃	C₂²	semidirect
9. C₃³×C₄	C₂²	semidirect
11. C₂²×C₃×D₃	C₂×C₃	semidirect
14. C₂×C₃²×C₄	D₃	semidirect
15. C₂×C₃²×C₄	C₂×C₃	semidirect
16. C₃×D₁₂	C₂×C₃	direct (N × H)
17. C₂×C₃³	D₄	semidirect
21. C₂×D₁₂	C₃²	direct (N × H)
28. C₂×C₃×D₃	C₂²×C₃	semidirect
30. C₃²×C₄	C₂×D₃	semidirect
31. C₃²×C₄	C₂²×C₃	semidirect
32. C₃³	C₂×D₄	semidirect
34. C₂²×D₃	C₂×C₃²	semidirect
37. C₂×C₃×C₄	C₃×D₃	semidirect
38. C₂×C₃×C₄	C₂×C₃²	semidirect
39. D₁₂	C₂×C₃²	direct (N × H)
43. C₂×C₃²	D₁₂	direct (N × H)
44. C₂×C₃²	C₃×D₄	semidirect
55. C₂×D₃	C₂²×C₃²	semidirect
57. C₃×C₄	C₂×C₃×D₃	semidirect
58. C₃×C₄	C₂²×C₃²	semidirect
60. C₃²	C₂×D₁₂	direct (N × H)
61. C₃²	C₂×C₃×D₄	semidirect
64. C₂×C₄	C₃²×D₃	semidirect
68. C₂×C₃	C₃×D₁₂	direct (N × H)
69. C₂×C₃	C₃²×D₄	semidirect
76. C₄	C₂×C₃²×D₃	semidirect
78. C₃	C₂×C₃×D₁₂	direct (N × H)
79. C₃	C₂×C₃²×D₄	semidirect
81. C₂	C₃²×D₁₂	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×C₃²×D₃	2
C₂×C₃³×C₄	2
C₃²×D₁₂	2
C₂×C₃²×D₄	3
C₂×C₃×D₁₂	3

Subgroups — C2×C32×D12 #89055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×D₁₂ #89055