Subgroups — #88523

Order 432.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3743	—	3	1	no	maximal
5	#3663	—	3	3	no	maximal
6	C₂×C₃⋊D₉	C₄	1	1	yes	characteristic
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃⋊Dic₉	C₄	1	1	yes	characteristic
9	C₄×C₃⋊D₃	—	3	1	no	—
10	C₃²×C₈	D₃	1	1	yes	characteristic
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	C₄×D₉	—	3	3	no	—
13	C₈×C₉	D₃	1	3	yes	complemented
14	C₉⋊C₈	—	3	3	no	—
15	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
16	C₃⋊D₉	—	2	1	no	—
17	ΓL(1,𝔽₂₅)	—	9	3	no	—
18	ΓL(1,𝔽₂₅)	—	9	1	no	—
19	C₂×C₃⋊D₃	—	3	1	no	—
20	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
21	C₃⋊Dic₃	—	3	1	no	—
22	C₂×D₉	—	3	3	no	—
23	C₄×C₉	C₂×D₃	1	3	yes	—
24	Dic₉	—	3	3	no	—
25	C₃×C₉	M₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₄×D₃	—	9	3	no	—
27	C₄×D₃	—	9	1	no	—
28	C₃×C₈	D₉	1	3	yes	complemented
29	C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
30	C₃⋊C₈	—	9	3	no	—
31	C₃⋊C₈	—	9	1	no	—
32	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
33	C₃⋊D₃	—	6	1	no	—
34	C₂×C₉	C₄×D₃	1	3	yes	—
35	D₉	—	6	3	no	—
36	M₁₆	—	27	1	no	Sylow
37	C₂×D₃	—	9	3	no	—
38	C₂×D₃	—	9	1	no	—
39	C₃×C₄	C₂×D₉	1	3	yes	—
40	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
41	Dic₃	—	9	3	no	—
42	Dic₃	—	9	1	no	—
43	C₃²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
44	C₉	ΓL(1,𝔽₂₅)	1	3	yes	complemented
45	C₂×C₄	—	27	1	no	—
46	C₈	—	27	1	no	—
47	C₈	C₃⋊D₉	1	1	yes	characteristic, complemented
48	C₂×C₃	C₄×D₉	1	3	yes	—
49	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
50	D₃	—	18	3	no	—
51	D₃	—	18	1	no	—
52	C₂²	—	27	1	no	—
53	C₄	—	27	1	no	—
54	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
55	C₃	#3663	1	3	yes	complemented
56	C₃	#3743	1	1	yes	characteristic
57	C₂	—	54	1	no	—
58	C₂	C₄×C₃⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₃×C₈×C₉	C₂
socle	C₂×C₃²	C₄×D₃
prime radical	C₂×C₃⋊D₉	C₄
cosocle	C₃×C₄×C₉	C₂²
hypercenter	C₈	C₃⋊D₉
hypocenter	C₃×C₉	M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×C₉	C₂	semidirect
3. C₃⋊(C₉⋊C₈)	C₂	semidirect
13. C₈×C₉	D₃	semidirect
25. C₃×C₉	M₁₆	semidirect
28. C₃×C₈	D₉	semidirect
44. C₉	ΓL(1,𝔽₂₅)	semidirect
47. C₈	C₃⋊D₉	semidirect
55. C₃	#3663	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃⋊D₉	2
C₃×C₈×C₉	2
C₃⋊(C₉⋊C₈)	2
#3743	3
#3663	3