Subgroups — #88420

Order 432.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3711	—	3	1	no	maximal
5	#3667	—	3	1	no	maximal
6	C₂×D₃×C₉	C₄	1	1	yes	characteristic
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₉×Dic₃	C₄	1	1	yes	characteristic
9	C₃×C₄×D₃	D₃	1	1	yes	characteristic
10	C₃⋊(C₃⋊C₈)	—	3	1	no	—
11	C₃×C₃⋊C₈	—	3	1	no	—
12	C₂×C₄×C₉	—	3	1	no	—
13	C₉⋊C₈	—	3	1	no	—
14	C₉⋊C₈	—	6	1	no	—
15	C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
17	D₃×C₉	—	2	1	no	—
18	#208	—	9	1	no	—
19	ΓL(1,𝔽₂₅)	—	9	1	no	—
20	C₂×C₃×D₃	Dic₃	1	1	yes	characteristic
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃×Dic₃	Dic₃	1	1	yes	characteristic
23	C₂²×C₉	—	3	1	no	—
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	—	3	1	no	—
26	C₄×C₉	C₂×D₃	1	1	yes	characteristic
27	C₃×C₉	M₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃×C₄	—	3	1	no	—
29	C₄×D₃	D₉	1	1	yes	characteristic
30	C₃×C₈	—	9	1	no	—
31	C₃⋊C₈	—	3	1	no	—
32	C₃⋊C₈	—	9	1	no	—
33	C₃⋊C₈	—	9	1	no	—
34	C₃⋊C₈	—	18	1	no	—
35	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
36	C₃×D₃	—	2	1	no	—
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	—	6	1	no	—
39	C₂×C₉	C₄×D₃	1	1	yes	characteristic
40	M₁₆	—	27	1	no	Sylow
41	C₂²×C₃	—	3	1	no	—
42	C₂×D₃	Dic₉	1	1	yes	characteristic
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	C₂×D₉	1	1	yes	characteristic
46	C₃×C₄	D₃²	1	1	yes	characteristic
47	Dic₃	Dic₉	1	1	yes	characteristic
48	C₃²	#208	1	1	yes	characteristic
49	C₉	—	2	1	no	—
50	C₉	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
51	C₂×C₄	—	3	1	no	—
52	C₈	—	9	1	no	—
53	C₈	—	27	1	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	6	1	no	—
56	C₂×C₃	C₂×Dic₉	1	1	yes	characteristic
57	C₂×C₃	D₃×Dic₃	1	1	yes	characteristic
58	D₃	—	2	1	no	—
59	C₂²	—	3	1	no	—
60	C₄	—	3	1	no	—
61	C₄	D₃×D₉	1	1	yes	characteristic
62	C₃	—	2	1	no	—
63	C₃	#3667	1	1	yes	characteristic, complemented
64	C₃	#3711	1	1	yes	characteristic
65	C₂	—	6	1	no	—
66	C₂	D₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄
Frattini	C₃×C₄	D₃²
Fitting, cosocle	C₃×C₄×C₉	C₂²
socle	C₂×C₃²	C₂×Dic₃
prime radical	C₂×C₃×D₃	Dic₃
hypocenter	C₃×C₉	M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₉⋊C₈)	C₂	semidirect
3. C₃×C₉⋊C₈	C₂	semidirect
15. C₉⋊C₈	D₃	semidirect
27. C₃×C₉	M₁₆	semidirect
50. C₉	ΓL(1,𝔽₂₅)	semidirect
63. C₃	#3667	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₃×C₉	2
C₃⋊(C₉⋊C₈)	2
C₃×C₉⋊C₈	2
#3711	3
#3667	3