Subgroups — C₂×C₃×D₁₂ #3817

Order 144.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₃	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₁₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×D₄	—	3	1	no	maximal
5	C₂×D₁₂	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃²	C₂²	1	1	yes	characteristic
7	C₂×C₃×D₃	—	2	4	no	—
8	C₂×C₃×D₃	C₂²	1	4	yes	complemented
9	C₃²×C₄	C₂²	1	2	yes	complemented
10	C₂³×C₃	—	3	2	no	—
11	C₂²×D₃	C₂×C₃	1	2	yes	complemented
12	C₃×D₄	—	3	4	no	—
13	C₂×C₃×C₄	—	2	1	no	—
14	C₂×C₃×C₄	D₃	1	1	yes	characteristic, complemented
15	C₂×C₃×C₄	C₂×C₃	1	1	yes	characteristic, complemented
16	D₁₂	C₂×C₃	1	4	yes	direct product factor
17	C₂×C₃²	D₄	1	2	yes	complemented
18	C₂×C₃²	C₂³	1	1	yes	characteristic
19	C₃×D₃	—	2	4	no	—
20	C₂×D₄	—	3	1	no	Sylow
21	C₂²×C₃	—	2	1	no	—
22	C₂²×C₃	—	3	4	no	—
23	C₂²×C₃	—	6	4	no	—
24	C₂²×C₃	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₃	C₂²×C₃	1	1	yes	characteristic
26	C₂×D₃	—	2	4	no	—
27	C₂×D₃	C₂²×C₃	1	4	yes	complemented
28	C₃×C₄	—	2	2	no	—
29	C₃×C₄	C₂×D₃	1	2	yes	complemented
30	C₃×C₄	C₂²×C₃	1	2	yes	complemented
31	C₃²	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
32	C₂³	—	3	2	no	—
33	D₄	—	3	4	no	—
34	C₂×C₄	C₃×D₃	1	1	yes	characteristic, complemented
35	C₂×C₃	—	2	2	no	—
36	C₂×C₃	—	2	1	no	—
37	C₂×C₃	—	6	4	no	—
38	C₂×C₃	D₁₂	1	2	yes	direct product factor
39	C₂×C₃	C₃×D₄	1	2	yes	complemented
40	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
41	C₂×C₃	C₂³×C₃	1	1	yes	characteristic
42	D₃	—	2	4	no	—
43	C₂²	—	3	4	no	—
44	C₂²	—	6	4	no	—
45	C₂²	C₂×C₃×D₃	1	1	yes	characteristic
46	C₄	C₂×C₃×D₃	1	2	yes	complemented
47	C₃	—	2	1	no	—
48	C₃	C₂×D₁₂	1	1	yes	characteristic, direct product factor
49	C₃	C₂×C₃×D₄	1	1	yes	characteristic, complemented
50	C₂	—	6	4	no	—
51	C₂	C₃×D₁₂	1	2	yes	direct product factor
52	C₂	C₂²×C₃×D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₃	C₂	semidirect
2. C₂×C₃²×C₄	C₂	semidirect
3. C₃×D₁₂	C₂	direct (N × H)
5. C₂×D₁₂	C₃	direct (N × H)
8. C₂×C₃×D₃	C₂²	semidirect
9. C₃²×C₄	C₂²	semidirect
11. C₂²×D₃	C₂×C₃	semidirect
14. C₂×C₃×C₄	D₃	semidirect
15. C₂×C₃×C₄	C₂×C₃	semidirect
16. D₁₂	C₂×C₃	direct (N × H)
17. C₂×C₃²	D₄	semidirect
27. C₂×D₃	C₂²×C₃	semidirect
29. C₃×C₄	C₂×D₃	semidirect
30. C₃×C₄	C₂²×C₃	semidirect
31. C₃²	C₂×D₄	semidirect
34. C₂×C₄	C₃×D₃	semidirect
38. C₂×C₃	D₁₂	direct (N × H)
39. C₂×C₃	C₃×D₄	semidirect
46. C₄	C₂×C₃×D₃	semidirect
48. C₃	C₂×D₁₂	direct (N × H)
49. C₃	C₂×C₃×D₄	semidirect
51. 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
C₂²×C₃×D₃	2
C₂×C₃²×C₄	2
C₃×D₁₂	2
C₂×C₃×D₄	3
C₂×D₁₂	3

Subgroups — C2×C3×D12 #3817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₁₂ #3817