Subgroups — #3745⋊C₃ #88471

Order 432.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊((C₃⋊D₃)⋊C₃)	C₂	1	2	yes	maximal, complemented
2	C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	#3745	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×D₂₄	—	3	1	no	maximal
5	C₂×(C₃⋊D₃)⋊C₃	—	2	2	no	—
6	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄⋊(C₃⋊D₃)	C₂×C₃	1	2	yes	complemented
8	C₃×D₁₂	—	3	2	no	—
9	C₃²×C₈	—	2	1	no	—
10	C₃²×C₈	D₃	1	1	yes	characteristic, complemented
11	C₃²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
13	(C₃⋊D₃)⋊C₃	—	4	2	no	—
14	C₃×D₈	—	9	1	no	—
15	D₂₄	—	3	1	no	—
16	D₂₄	—	9	1	no	—
17	C₂×C₃⋊D₃	—	2	2	no	—
18	C₂×C₃×D₃	—	6	2	no	—
19	C₃²×C₄	—	2	1	no	—
20	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
21	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
22	H(3,𝔽₃)	D₈	1	1	yes	characteristic, complemented, Sylow
23	C₃×D₄	—	9	2	no	—
24	D₁₂	—	3	2	no	—
25	D₁₂	—	9	2	no	—
26	C₃×C₈	—	3	1	no	—
27	C₃×C₈	—	3	1	no	—
28	C₃×C₈	—	6	1	no	—
29	C₃×C₈	C₃×D₃	1	1	yes	characteristic
30	C₂×C₃²	—	2	1	no	—
31	C₂×C₃²	D₁₂	1	1	yes	characteristic
32	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
33	C₃⋊D₃	—	4	2	no	—
34	C₃×D₃	—	12	2	no	—
35	D₈	—	9	1	no	Sylow
36	C₂²×C₃	—	18	2	no	—
37	C₂×D₃	—	6	2	no	—
38	C₂×D₃	—	18	2	no	—
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	—	6	1	no	—
42	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
43	C₃²	—	2	1	no	—
44	C₃²	D₂₄	1	1	yes	characteristic, complemented
45	C₃²	C₃×D₈	1	1	yes	characteristic, complemented
46	D₄	—	9	2	no	—
47	C₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
48	C₂×C₃	—	3	1	no	—
49	C₂×C₃	—	3	1	no	—
50	C₂×C₃	—	6	1	no	—
51	C₂×C₃	—	36	2	no	—
52	C₂×C₃	C₃×D₁₂	1	1	yes	characteristic
53	D₃	—	12	2	no	—
54	D₃	—	36	2	no	—
55	C₂²	—	18	2	no	—
56	C₄	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
57	C₃	—	3	1	no	—
58	C₃	—	3	1	no	—
59	C₃	—	6	1	no	—
60	C₃	C₃×D₂₄	1	1	yes	characteristic
61	C₂	—	36	2	no	—
62	C₂	C₄⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle	C₃²×C₄	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₈×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₃×D₁₂
hypercenter	C₈	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
1. C₄⋊((C₃⋊D₃)⋊C₃)	C₂	semidirect
2. C₈×H(3,𝔽₃)	C₂	semidirect
3. #3745	C₃	semidirect
7. C₄⋊(C₃⋊D₃)	C₂×C₃	semidirect
10. C₃²×C₈	D₃	semidirect
11. C₃²×C₈	C₂×C₃	semidirect
22. H(3,𝔽₃)	D₈	semidirect
44. C₃²	D₂₄	semidirect
45. C₃²	C₃×D₈	semidirect
47. C₈	(C₃⋊D₃)⋊C₃	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)⋊C₃)	2
C₈×H(3,𝔽₃)	2
#3745	3
C₃×D₂₄	3

Subgroups — #3745⋊C3 #88471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3745⋊C₃ #88471