Subgroups — #88529

Order 432.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#272	C₂	1	2	yes	maximal, complemented
2	C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₂₄	—	3	1	no	maximal
4	C₃×D₂₄	—	3	3	no	maximal
5	C₂×#272	—	2	2	no	—
6	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₃×D₁₂	—	3	2	no	—
8	C₃×D₁₂	—	3	6	no	—
9	C₃²×C₈	D₃	1	1	yes	characteristic, complemented
10	C₃²×C₈	D₃	1	3	yes	complemented
11	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
12	#272	—	4	2	no	—
13	C₃×D₈	—	9	1	no	—
14	D₂₄	—	3	1	no	—
15	D₂₄	—	3	3	no	—
16	C₂×C₃×D₃	—	6	2	no	—
17	C₂×C₃×D₃	—	6	6	no	—
18	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
19	C₃²×C₄	C₂×D₃	1	3	yes	—
20	H(3,𝔽₃)	D₈	1	1	yes	characteristic, complemented, Sylow
21	C₃×D₄	—	9	2	no	—
22	D₁₂	—	3	2	no	—
23	D₁₂	—	3	6	no	—
24	C₃×C₈	—	3	1	no	—
25	C₃×C₈	—	6	3	no	—
26	C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
27	C₂×C₃²	D₁₂	1	1	yes	characteristic
28	C₂×C₃²	D₁₂	1	3	yes	—
29	C₃×D₃	—	12	2	no	—
30	C₃×D₃	—	12	6	no	—
31	D₈	—	9	1	no	Sylow
32	C₂²×C₃	—	18	2	no	—
33	C₂×D₃	—	6	2	no	—
34	C₂×D₃	—	6	6	no	—
35	C₃×C₄	—	3	1	no	—
36	C₃×C₄	—	6	3	no	—
37	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
38	C₃²	D₂₄	1	1	yes	characteristic, complemented
39	C₃²	D₂₄	1	3	yes	complemented
40	D₄	—	9	2	no	—
41	C₈	#272	1	1	yes	characteristic, complemented
42	C₂×C₃	—	3	1	no	—
43	C₂×C₃	—	6	3	no	—
44	C₂×C₃	—	36	2	no	—
45	C₂×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
46	D₃	—	12	2	no	—
47	D₃	—	12	6	no	—
48	C₂²	—	18	2	no	—
49	C₄	C₂×#272	1	1	yes	characteristic
50	C₃	—	3	1	no	—
51	C₃	—	6	3	no	—
52	C₃	#3745	1	1	yes	characteristic
53	C₂	—	36	2	no	—
54	C₂	C₄⋊#272	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₄⋊#272	C₂	semidirect
2. C₈×H(3,𝔽₃)	C₂	semidirect
9. C₃²×C₈	D₃	semidirect
10. C₃²×C₈	D₃	semidirect
20. H(3,𝔽₃)	D₈	semidirect
38. C₃²	D₂₄	semidirect
39. C₃²	D₂₄	semidirect
41. C₈	#272	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#272	2
C₈×H(3,𝔽₃)	2
C₃×D₂₄	3