Subgroups — #88435

Order 432.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#272	C₂	1	1	yes	maximal, characteristic
2	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	2	yes	maximal, complemented
3	#3711	—	3	2	no	maximal
4	C₂×#272	C₄	1	1	yes	characteristic
5	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	#1084	C₄	1	1	yes	characteristic
7	C₃×C₄×D₃	—	3	2	no	—
8	C₃×C₄×D₃	—	6	1	no	—
9	C₃⋊(C₃⋊C₈)	D₃	1	2	yes	complemented
10	C₃×C₃⋊C₈	—	3	2	no	—
11	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	#272	—	2	1	no	—
13	#208	—	9	1	no	—
14	ΓL(1,𝔽₂₅)	—	9	2	no	—
15	C₂×C₃×D₃	—	3	2	no	—
16	C₂×C₃×D₃	—	6	1	no	—
17	C₃²×C₄	—	2	1	no	—
18	C₃²×C₄	C₂×D₃	1	2	yes	—
19	C₃×Dic₃	—	3	2	no	—
20	C₃×Dic₃	—	6	1	no	—
21	H(3,𝔽₃)	M₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₄	—	9	1	no	—
23	C₄×D₃	—	3	2	no	—
24	C₄×D₃	—	6	1	no	—
25	C₃×C₈	—	9	2	no	—
26	C₃⋊C₈	—	3	2	no	—
27	C₃⋊C₈	—	18	2	no	—
28	C₂×C₃²	—	2	1	no	—
29	C₂×C₃²	C₄×D₃	1	2	yes	—
30	C₃×D₃	—	6	2	no	—
31	C₃×D₃	—	6	2	no	—
32	M₁₆	—	27	1	no	Sylow
33	C₂²×C₃	—	9	1	no	—
34	C₂×D₃	—	3	2	no	—
35	C₂×D₃	—	6	1	no	—
36	C₃×C₄	—	6	2	no	—
37	C₃×C₄	—	9	1	no	—
38	C₃×C₄	—	12	1	no	—
39	C₃×C₄	D₃²	1	1	yes	characteristic
40	Dic₃	—	3	2	no	—
41	Dic₃	—	6	1	no	—
42	C₃²	—	2	1	no	—
43	C₃²	ΓL(1,𝔽₂₅)	1	2	yes	complemented
44	C₂×C₄	—	9	1	no	—
45	C₈	—	9	2	no	—
46	C₂×C₃	—	6	2	no	—
47	C₂×C₃	—	12	1	no	—
48	C₂×C₃	—	18	1	no	—
49	C₂×C₃	#624	1	1	yes	characteristic
50	D₃	—	6	2	no	—
51	D₃	—	6	2	no	—
52	C₂²	—	9	1	no	—
53	C₄	—	9	1	no	—
54	C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
55	C₃	—	6	2	no	—
56	C₃	—	12	1	no	—
57	C₃	#3712	1	1	yes	characteristic
58	C₂	—	18	1	no	—
59	C₂	#6199	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	((C₃⋊D₃)⋊C₃)⋊C₂
commutator	C₂×H(3,𝔽₃)	C₂×C₄
Frattini	C₃×C₄	D₃²
Fitting, cosocle	C₄×H(3,𝔽₃)	C₂²
socle	C₂×C₃	#624
prime radical	C₂×#272	C₄
hypocenter	H(3,𝔽₃)	M₁₆
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
9. C₃⋊(C₃⋊C₈)	D₃	semidirect
21. H(3,𝔽₃)	M₁₆	semidirect
43. C₃²	ΓL(1,𝔽₂₅)	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
(C₃⋊(C₃⋊C₈))⋊C₃	2
#3711	3