Subgroups — #88424

Order 432.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6226	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3716	—	3	1	no	maximal
5	#3672	—	3	1	no	maximal
6	C₂×D₃×C₉	—	2	1	no	—
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃⋊Dic₉	—	2	1	no	—
9	#634	—	3	1	no	—
10	C₃×D₁₂	D₃	1	1	yes	characteristic
11	C₃×C₃⋊C₈	—	3	1	no	—
12	D₄×C₉	—	3	1	no	—
13	Dic₁₈	—	3	1	no	—
14	Dic₁₈	—	6	1	no	—
15	C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	D₃×C₉	—	4	1	no	—
18	#214	—	9	1	no	—
19	#204	—	9	1	no	—
20	C₂×C₃×D₃	—	2	1	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃⋊Dic₃	—	6	1	no	—
23	C₂²×C₉	—	6	1	no	—
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	C₂×D₃	1	1	yes	characteristic
26	Dic₉	—	6	2	no	—
27	Dic₉	—	6	1	no	—
28	C₃×C₉	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×D₄	—	3	1	no	—
30	D₁₂	D₉	1	1	yes	characteristic
31	Dic₆	—	9	1	no	—
32	Dic₆	—	9	1	no	—
33	Dic₆	—	18	1	no	—
34	C₃×C₈	—	9	1	no	—
35	C₃⋊C₈	—	3	1	no	—
36	C₂×C₃²	#67	1	1	yes	characteristic
37	C₃×D₃	—	4	1	no	—
38	C₂×C₉	—	2	1	no	—
39	C₂×C₉	—	12	1	no	—
40	C₂×C₉	D₁₂	1	1	yes	characteristic
41	SD₁₆	—	27	1	no	Sylow
42	C₂²×C₃	—	6	1	no	—
43	C₂×D₃	—	2	1	no	—
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	C₂×D₉	1	1	yes	characteristic
46	C₃×C₄	D₃²	1	1	yes	characteristic
47	Dic₃	—	18	2	no	—
48	Dic₃	—	18	1	no	—
49	Dic₃	—	18	1	no	—
50	C₃²	#214	1	1	yes	characteristic
51	C₉	—	2	1	no	—
52	C₉	#204	1	1	yes	characteristic, complemented
53	Q₈	—	27	1	no	—
54	D₄	—	3	1	no	—
55	C₈	—	9	1	no	—
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	12	1	no	—
58	C₂×C₃	#611	1	1	yes	characteristic
59	C₂×C₃	#626	1	1	yes	characteristic
60	D₃	—	4	1	no	—
61	C₂²	—	6	1	no	—
62	C₄	—	54	1	no	—
63	C₄	D₃×D₉	1	1	yes	characteristic
64	C₃	—	2	1	no	—
65	C₃	#3672	1	1	yes	characteristic, complemented
66	C₃	#3716	1	1	yes	characteristic
67	C₂	—	12	1	no	—
68	C₂	#6195	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6195
commutator, Fitting, cosocle	C₃×C₄×C₉	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃²	#67
prime radical	C₃×D₁₂	D₃
hypercenter	C₄	D₃×D₉
hypocenter	C₃×C₉	SD₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. #6226	C₂	semidirect
3. C₃×C₉⋊C₈	C₂	semidirect
15. C₉⋊C₈	D₃	semidirect
28. C₃×C₉	SD₁₆	semidirect
52. C₉	#204	semidirect
65. C₃	#3672	semidirect

Maximal subgroups

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

subgroup	index
#6226	2
C₉×D₁₂	2
C₃×C₉⋊C₈	2
#3716	3
#3672	3