Subgroups — #3811

Order 144.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#67	C₂	1	2	yes	maximal, complemented
4	#626	C₂	1	2	yes	maximal, complemented
5	#624	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×D₄	—	3	2	no	maximal
7	C₂²×C₃²	C₂²	1	1	yes	characteristic, complemented
8	C₂×C₃⋊D₃	—	2	2	no	—
9	C₂×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊D₃	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₃×D₃	C₂²	1	2	yes	complemented
12	D₃²	—	2	2	no	—
13	C₃×Dic₃	C₂²	1	2	yes	complemented
14	C₂²×D₃	—	3	2	no	—
15	C₂²×D₃	—	3	2	no	—
16	C₂²×D₃	—	6	1	no	—
17	C₃×D₄	—	3	2	no	—
18	#67	—	3	2	no	—
19	#67	D₃	1	2	yes	complemented
20	D₁₂	—	3	2	no	—
21	C₄×D₃	—	3	2	no	—
22	C₂×C₃²	—	2	1	no	—
23	C₂×C₃²	C₂³	1	1	yes	characteristic
24	C₃⋊D₃	—	2	1	no	—
25	C₃⋊D₃	D₄	1	2	yes	complemented
26	C₃×D₃	—	2	2	no	—
27	C₂×D₄	—	9	1	no	Sylow
28	C₂²×C₃	—	2	1	no	—
29	C₂²×C₃	—	3	2	no	—
30	C₂²×C₃	C₂×D₃	1	2	yes	complemented
31	C₂×D₃	—	3	2	no	—
32	C₂×D₃	—	3	2	no	—
33	C₂×D₃	—	6	1	no	—
34	C₂×D₃	—	6	1	no	—
35	C₂×D₃	—	6	4	no	—
36	C₂×D₃	—	6	4	no	—
37	C₂×D₃	—	6	4	no	—
38	C₂×D₃	C₂×D₃	1	2	yes	complemented
39	C₃×C₄	—	3	2	no	—
40	Dic₃	C₂×D₃	1	2	yes	complemented
41	C₃²	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
42	C₂³	—	9	1	no	—
43	C₂³	—	9	1	no	—
44	D₄	—	3	2	no	—
45	D₄	—	9	2	no	—
46	C₂×C₄	—	9	1	no	—
47	C₂×C₃	—	2	2	no	—
48	C₂×C₃	—	2	2	no	—
49	C₂×C₃	—	2	1	no	—
50	C₂×C₃	—	6	2	no	—
51	C₂×C₃	C₂²×D₃	1	2	yes	—
52	D₃	—	2	2	no	—
53	D₃	—	3	4	no	—
54	D₃	—	6	2	no	—
55	D₃	—	6	2	no	—
56	D₃	—	6	2	no	—
57	C₂²	—	3	2	no	—
58	C₂²	—	9	1	no	—
59	C₂²	—	9	1	no	—
60	C₂²	—	18	2	no	—
61	C₂²	—	18	2	no	—
62	C₂²	D₃²	1	1	yes	characteristic, complemented
63	C₄	—	3	2	no	—
64	C₃	—	2	1	no	—
65	C₃	D₃×D₄	1	2	yes	complemented
66	C₂	—	2	1	no	—
67	C₂	—	6	2	no	—
68	C₂	—	9	2	no	—
69	C₂	—	18	1	no	—
70	C₂	C₂×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂³
Fitting	C₂²×C₃²	C₂²
hypercenter	C₂²	D₃²
hypocenter	C₃²	C₂×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃⋊D₃	C₂	semidirect
2. C₂×D₃²	C₂	semidirect
3. C₃×#67	C₂	semidirect
4. #626	C₂	semidirect
5. #624	C₂	semidirect
7. C₂²×C₃²	C₂²	semidirect
10. C₂×C₃⋊D₃	C₂²	semidirect
11. C₂×C₃×D₃	C₂²	semidirect
13. C₃×Dic₃	C₂²	semidirect
19. #67	D₃	semidirect
25. C₃⋊D₃	D₄	semidirect
30. C₂²×C₃	C₂×D₃	semidirect
38. C₂×D₃	C₂×D₃	semidirect
40. Dic₃	C₂×D₃	semidirect
41. C₃²	C₂×D₄	semidirect
62. C₂²	D₃²	semidirect
65. C₃	D₃×D₄	semidirect

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×D₃²	2
C₃×#67	2
#626	2
#624	2
D₃×D₄	3