Subgroups — C₂²×C₃×D₁₂ #64342

Order 288.

499 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×D₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₁₂	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×D₄	—	3	1	no	maximal
5	C₂²×D₁₂	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂³×C₃²	C₂²	1	1	yes	characteristic
7	C₂²×C₃×D₃	—	2	8	no	—
8	C₂²×C₃×D₃	C₂²	1	12	yes	complemented
9	C₂×C₃²×C₄	C₂²	1	6	yes	complemented
10	C₃×D₁₂	C₂²	1	16	yes	direct product factor
11	C₂⁴×C₃	—	3	2	no	—
12	C₂³×D₃	C₂×C₃	1	2	yes	complemented
13	C₂×C₃×D₄	—	3	12	no	—
14	C₂²×C₃×C₄	—	2	1	no	—
15	C₂²×C₃×C₄	D₃	1	1	yes	characteristic, complemented
16	C₂²×C₃×C₄	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×D₁₂	C₂×C₃	1	12	yes	direct product factor
18	C₂²×C₃²	D₄	1	4	yes	complemented
19	C₂²×C₃²	C₂³	1	3	yes	—
20	C₂×C₃×D₃	—	2	24	no	—
21	C₂×C₃×D₃	C₂³	1	8	yes	complemented
22	C₃²×C₄	C₂³	1	4	yes	complemented
23	C₂²×D₄	—	3	1	no	Sylow
24	C₂³×C₃	—	2	1	no	—
25	C₂³×C₃	—	3	12	no	—
26	C₂³×C₃	—	6	8	no	—
27	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
28	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
29	C₂²×D₃	—	2	8	no	—
30	C₂²×D₃	C₂²×C₃	1	12	yes	complemented
31	C₃×D₄	—	3	16	no	—
32	C₂×C₃×C₄	—	2	6	no	—
33	C₂×C₃×C₄	C₂×D₃	1	6	yes	complemented
34	C₂×C₃×C₄	C₂²×C₃	1	6	yes	complemented
35	D₁₂	C₂²×C₃	1	16	yes	direct product factor
36	C₂×C₃²	C₂×D₄	1	6	yes	complemented
37	C₂×C₃²	C₂⁴	1	1	yes	characteristic
38	C₃×D₃	—	2	8	no	—
39	C₂⁴	—	3	2	no	—
40	C₂×D₄	—	3	12	no	—
41	C₂²×C₄	C₃×D₃	1	1	yes	characteristic, complemented
42	C₂²×C₃	—	2	4	no	—
43	C₂²×C₃	—	2	3	no	—
44	C₂²×C₃	—	3	8	no	—
45	C₂²×C₃	—	6	24	no	—
46	C₂²×C₃	D₁₂	1	4	yes	direct product factor
47	C₂²×C₃	C₃×D₄	1	4	yes	complemented
48	C₂²×C₃	C₂²×D₃	1	3	yes	—
49	C₂²×C₃	C₂³×C₃	1	3	yes	—
50	C₂×D₃	—	2	24	no	—
51	C₂×D₃	C₂³×C₃	1	8	yes	complemented
52	C₃×C₄	—	2	4	no	—
53	C₃×C₄	C₂²×D₃	1	4	yes	complemented
54	C₃×C₄	C₂³×C₃	1	4	yes	complemented
55	C₃²	C₂²×D₄	1	1	yes	characteristic, complemented, Sylow
56	C₂³	—	3	12	no	—
57	C₂³	—	6	8	no	—
58	C₂³	C₂×C₃×D₃	1	1	yes	characteristic
59	D₄	—	3	16	no	—
60	C₂×C₄	C₂×C₃×D₃	1	6	yes	complemented
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	—	2	6	no	—
63	C₂×C₃	—	6	8	no	—
64	C₂×C₃	C₂×D₁₂	1	6	yes	direct product factor
65	C₂×C₃	C₂×C₃×D₄	1	6	yes	complemented
66	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
67	C₂×C₃	C₂⁴×C₃	1	1	yes	characteristic
68	D₃	—	2	8	no	—
69	C₂²	—	3	8	no	—
70	C₂²	—	6	24	no	—
71	C₂²	C₃×D₁₂	1	4	yes	direct product factor
72	C₂²	C₂²×C₃×D₃	1	3	yes	—
73	C₄	C₂²×C₃×D₃	1	4	yes	complemented
74	C₃	—	2	1	no	—
75	C₃	C₂²×D₁₂	1	1	yes	characteristic, direct product factor
76	C₃	C₂²×C₃×D₄	1	1	yes	characteristic, complemented
77	C₂	—	6	8	no	—
78	C₂	C₂×C₃×D₁₂	1	6	yes	direct product factor
79	C₂	C₂³×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂³×C₃²	C₂²
hypercenter	C₂²×C₃×C₄	D₃
hypocenter	C₃	C₂²×C₃×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×D₃	C₂	semidirect
2. C₂²×C₃²×C₄	C₂	semidirect
3. C₂×C₃×D₁₂	C₂	direct (N × H)
5. C₂²×D₁₂	C₃	direct (N × H)
8. C₂²×C₃×D₃	C₂²	semidirect
9. C₂×C₃²×C₄	C₂²	semidirect
10. C₃×D₁₂	C₂²	direct (N × H)
12. C₂³×D₃	C₂×C₃	semidirect
15. C₂²×C₃×C₄	D₃	semidirect
16. C₂²×C₃×C₄	C₂×C₃	semidirect
17. C₂×D₁₂	C₂×C₃	direct (N × H)
18. C₂²×C₃²	D₄	semidirect
21. C₂×C₃×D₃	C₂³	semidirect
22. C₃²×C₄	C₂³	semidirect
30. C₂²×D₃	C₂²×C₃	semidirect
33. C₂×C₃×C₄	C₂×D₃	semidirect
34. C₂×C₃×C₄	C₂²×C₃	semidirect
35. D₁₂	C₂²×C₃	direct (N × H)
36. C₂×C₃²	C₂×D₄	semidirect
41. C₂²×C₄	C₃×D₃	semidirect
46. C₂²×C₃	D₁₂	direct (N × H)
47. C₂²×C₃	C₃×D₄	semidirect
51. C₂×D₃	C₂³×C₃	semidirect
53. C₃×C₄	C₂²×D₃	semidirect
54. C₃×C₄	C₂³×C₃	semidirect
55. C₃²	C₂²×D₄	semidirect
60. C₂×C₄	C₂×C₃×D₃	semidirect
64. C₂×C₃	C₂×D₁₂	direct (N × H)
65. C₂×C₃	C₂×C₃×D₄	semidirect
71. C₂²	C₃×D₁₂	direct (N × H)
73. C₄	C₂²×C₃×D₃	semidirect
75. C₃	C₂²×D₁₂	direct (N × H)
76. C₃	C₂²×C₃×D₄	semidirect
78. C₂	C₂×C₃×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
C₂³×C₃×D₃	2
C₂²×C₃²×C₄	2
C₂×C₃×D₁₂	2
C₂²×C₃×D₄	3
C₂²×D₁₂	3

Subgroups — C22×C3×D12 #64342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×D₁₂ #64342