Subgroups — #88786

Order 432.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3748	—	3	1	no	maximal
5	#3711	—	3	4	no	maximal
6	C₂×C₃²×D₃	C₄	1	1	yes	characteristic
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃²×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₃²×C₄	—	3	1	no	—
10	C₃×C₄×D₃	D₃	1	4	yes	—
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	C₃⋊(C₃⋊C₈)	—	3	4	no	—
13	C₃⋊(C₃⋊C₈)	—	6	4	no	—
14	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
15	C₃×C₃⋊C₈	—	3	4	no	—
16	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
17	C₃²×D₃	—	2	1	no	—
18	#208	—	9	4	no	—
19	ΓL(1,𝔽₂₅)	—	9	1	no	—
20	C₂²×C₃²	—	3	1	no	—
21	C₂×C₃×D₃	Dic₃	1	4	yes	—
22	C₃²×C₄	—	2	4	no	—
23	C₃²×C₄	—	3	1	no	—
24	C₃²×C₄	C₂×D₃	1	4	yes	—
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃×Dic₃	Dic₃	1	4	yes	—
27	C₃³	M₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₄	—	3	4	no	—
29	C₄×D₃	C₃⋊D₃	1	1	yes	characteristic
30	C₃×C₈	—	9	1	no	—
31	C₃⋊C₈	—	3	4	no	—
32	C₃⋊C₈	—	9	4	no	—
33	C₃⋊C₈	—	9	1	no	—
34	C₃⋊C₈	—	18	4	no	—
35	C₂×C₃²	—	2	4	no	—
36	C₂×C₃²	—	6	1	no	—
37	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
38	C₂×C₃²	C₂×Dic₃	1	4	yes	—
39	C₃×D₃	—	2	4	no	—
40	M₁₆	—	27	1	no	Sylow
41	C₂²×C₃	—	3	4	no	—
42	C₂×D₃	C₃⋊Dic₃	1	1	yes	characteristic
43	C₃×C₄	—	2	4	no	—
44	C₃×C₄	—	3	4	no	—
45	C₃×C₄	D₃²	1	4	yes	—
46	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
47	Dic₃	C₃⋊Dic₃	1	1	yes	characteristic
48	C₃²	—	2	4	no	—
49	C₃²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
50	C₃²	#208	1	4	yes	complemented
51	C₂×C₄	—	3	1	no	—
52	C₈	—	9	1	no	—
53	C₈	—	27	1	no	—
54	C₂×C₃	—	2	4	no	—
55	C₂×C₃	—	6	4	no	—
56	C₂×C₃	D₃×Dic₃	1	4	yes	—
57	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
58	D₃	—	2	1	no	—
59	C₂²	—	3	1	no	—
60	C₄	—	3	1	no	—
61	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
62	C₃	—	2	4	no	—
63	C₃	#3711	1	4	yes	complemented
64	C₃	#3748	1	1	yes	characteristic, complemented
65	C₂	—	6	1	no	—
66	C₂	D₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃×C₃⋊D₃
commutator, socle	C₂×C₃³	C₂×C₄
Fitting, cosocle	C₃³×C₄	C₂²
prime radical	C₂×C₃²×D₃	C₄
hypocenter	C₃³	M₁₆
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
3. C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
14. C₃⋊(C₃⋊C₈)	D₃	semidirect
27. C₃³	M₁₆	semidirect
49. C₃²	ΓL(1,𝔽₂₅)	semidirect
50. C₃²	#208	semidirect
63. C₃	#3711	semidirect
64. C₃	#3748	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₃	2
C₃⋊(C₃⋊(C₃⋊C₈))	2
C₃×C₃⋊(C₃⋊C₈)	2
#3748	3
#3711	3