Subgroups — #3799

Order 144.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#626	C₂	1	2	yes	maximal, complemented
5	#624	C₂	1	2	yes	maximal, complemented
6	#239	—	3	1	no	maximal
7	#235	—	3	1	no	maximal
8	C₂×C₃⋊D₃	C₂²	1	2	yes	—
9	C₂×C₃×D₃	C₂²	1	1	yes	characteristic
10	C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₃×Dic₃	C₂²	1	1	yes	characteristic
12	C₃×Dic₃	C₂²	1	2	yes	—
13	C₃×Q₈	—	3	1	no	—
14	C₂×C₃×C₄	—	3	1	no	—
15	#67	—	3	2	no	—
16	D₁₂	—	3	2	no	—
17	D₁₂	—	3	1	no	—
18	D₁₂	—	3	1	no	—
19	D₁₂	—	6	1	no	—
20	C₄×D₃	—	3	2	no	—
21	C₄×D₃	—	3	2	no	—
22	C₄×D₃	D₃	1	1	yes	characteristic, complemented
23	Dic₆	D₃	1	1	yes	characteristic, complemented
24	C₂×C₃²	C₂³	1	1	yes	characteristic
25	C₃⋊D₃	—	2	2	no	—
26	C₃×D₃	—	2	1	no	—
27	#41	—	9	1	no	Sylow
28	C₂²×C₃	—	3	1	no	—
29	C₂×D₃	—	3	2	no	—
30	C₂×D₃	—	3	2	no	—
31	C₂×D₃	—	6	2	no	—
32	C₂×D₃	C₂×D₃	1	1	yes	characteristic
33	C₃×C₄	—	2	1	no	—
34	C₃×C₄	—	3	1	no	—
35	C₃×C₄	—	3	2	no	—
36	C₃×C₄	C₂×D₃	1	1	yes	characteristic
37	C₃×C₄	C₂×D₃	1	1	yes	characteristic
38	Dic₃	C₂×D₃	1	2	yes	—
39	Dic₃	C₂×D₃	1	1	yes	characteristic
40	C₃²	#41	1	1	yes	characteristic, complemented, Sylow
41	Q₈	—	3	1	no	—
42	D₄	—	9	2	no	—
43	D₄	—	9	1	no	—
44	C₂×C₄	—	3	1	no	—
45	C₂×C₄	—	9	2	no	—
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	—	6	1	no	—
48	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
49	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
50	D₃	—	2	1	no	—
51	D₃	—	6	2	no	—
52	D₃	—	6	2	no	—
53	D₃	—	12	2	no	—
54	C₂²	—	3	1	no	—
55	C₂²	—	9	2	no	—
56	C₄	—	3	1	no	—
57	C₄	—	3	2	no	—
58	C₄	D₃²	1	1	yes	characteristic
59	C₃	—	2	1	no	—
60	C₃	#235	1	1	yes	characteristic, complemented
61	C₃	#239	1	1	yes	characteristic, complemented
62	C₂	—	6	1	no	—
63	C₂	—	18	2	no	—
64	C₂	C₂×D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃²
commutator, socle, cosocle	C₂×C₃²	C₂³
Fitting	C₃²×C₄	C₂²
hypercenter	C₄	D₃²
hypocenter	C₃²	#41
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊(C₃⋊D₃)	C₂	semidirect
2. C₃×C₄×D₃	C₂	semidirect
3. C₃×Dic₆	C₂	semidirect
4. #626	C₂	semidirect
5. #624	C₂	semidirect
22. C₄×D₃	D₃	semidirect
23. Dic₆	D₃	semidirect
40. C₃²	#41	semidirect
60. C₃	#235	semidirect
61. C₃	#239	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄⋊(C₃⋊D₃)	2
C₃×C₄×D₃	2
C₃×Dic₆	2
#626	2
#624	2
#239	3
#235	3