Subgroups — #88788

Order 432.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3743	—	3	1	no	maximal
5	#3712	—	3	4	no	maximal
6	C₂×C₃⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
7	C₃³×C₄	C₂²	1	1	yes	characteristic
8	C₃⋊(C₃⋊Dic₃)	C₄	1	1	yes	characteristic
9	C₄×C₃⋊D₃	—	3	1	no	—
10	C₄×C₃⋊D₃	—	3	4	no	—
11	C₄×C₃⋊D₃	—	6	4	no	—
12	C₃²×C₈	—	3	1	no	—
13	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic, complemented
14	C₃×C₃⋊C₈	—	3	4	no	—
15	C₃×C₃⋊C₈	D₃	1	4	yes	complemented
16	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
17	C₃⋊(C₃⋊D₃)	—	2	1	no	—
18	ΓL(1,𝔽₂₅)	—	9	4	no	—
19	ΓL(1,𝔽₂₅)	—	9	1	no	—
20	C₂×C₃⋊D₃	—	3	1	no	—
21	C₂×C₃⋊D₃	—	3	4	no	—
22	C₂×C₃⋊D₃	—	6	4	no	—
23	C₃²×C₄	—	2	4	no	—
24	C₃²×C₄	C₂×D₃	1	4	yes	—
25	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃⋊Dic₃	—	3	1	no	—
27	C₃⋊Dic₃	—	3	4	no	—
28	C₃⋊Dic₃	—	6	4	no	—
29	C₃³	M₁₆	1	1	yes	characteristic, complemented, Sylow
30	C₄×D₃	—	9	4	no	—
31	C₄×D₃	—	9	1	no	—
32	C₄×D₃	—	18	4	no	—
33	C₃×C₈	—	3	4	no	—
34	C₃×C₈	—	9	1	no	—
35	C₃⋊C₈	—	3	4	no	—
36	C₃⋊C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
37	C₂×C₃²	—	2	4	no	—
38	C₂×C₃²	C₄×D₃	1	4	yes	—
39	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
40	C₃⋊D₃	—	6	8	no	—
41	C₃⋊D₃	—	6	4	no	—
42	C₃⋊D₃	—	6	1	no	—
43	M₁₆	—	27	1	no	Sylow
44	C₂×D₃	—	9	4	no	—
45	C₂×D₃	—	9	1	no	—
46	C₂×D₃	—	18	4	no	—
47	C₃×C₄	—	2	4	no	—
48	C₃×C₄	D₃²	1	4	yes	—
49	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	Dic₃	—	9	4	no	—
51	Dic₃	—	9	1	no	—
52	Dic₃	—	18	4	no	—
53	C₃²	—	2	4	no	—
54	C₃²	ΓL(1,𝔽₂₅)	1	4	yes	complemented
55	C₃²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
56	C₂×C₄	—	27	1	no	—
57	C₈	—	3	1	no	—
58	C₈	—	9	1	no	—
59	C₂×C₃	—	2	4	no	—
60	C₂×C₃	#624	1	4	yes	—
61	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
62	D₃	—	18	8	no	—
63	D₃	—	18	4	no	—
64	D₃	—	18	1	no	—
65	C₂²	—	27	1	no	—
66	C₄	—	27	1	no	—
67	C₄	D₃×C₃⋊D₃	1	1	yes	characteristic
68	C₃	—	2	4	no	—
69	C₃	#3712	1	4	yes	complemented
70	C₃	#3743	1	1	yes	characteristic, complemented
71	C₂	—	54	1	no	—
72	C₂	#6289	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×C₃⋊(C₃⋊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. C₃²×C₃⋊C₈	C₂	semidirect
13. C₃⋊(C₃⋊C₈)	D₃	semidirect
15. C₃×C₃⋊C₈	D₃	semidirect
29. C₃³	M₁₆	semidirect
36. C₃⋊C₈	C₃⋊D₃	semidirect
54. C₃²	ΓL(1,𝔽₂₅)	semidirect
55. C₃²	ΓL(1,𝔽₂₅)	semidirect
69. C₃	#3712	semidirect
70. C₃	#3743	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃⋊(C₃⋊D₃)	2
C₃×C₃⋊(C₃⋊C₈)	2
C₃²×C₃⋊C₈	2
#3743	3
#3712	3