Subgroups — #88422

Order 432.

82 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, complemented
2	C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3714	—	3	1	no	maximal
5	#3673	—	3	1	no	maximal
6	C₂×C₃⋊D₉	—	2	1	no	—
7	C₂×D₃×C₉	—	2	1	no	—
8	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊D₃)	—	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	D₃₆	—	3	1	no	—
14	D₃₆	—	6	1	no	—
15	C₉⋊C₈	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	C₃⋊D₉	—	4	1	no	—
18	D₃×C₉	—	4	1	no	—
19	#213	—	9	1	no	—
20	D₂₄	—	9	1	no	—
21	C₂×C₃⋊D₃	—	6	1	no	—
22	C₂×C₃×D₃	—	2	1	no	—
23	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
24	C₂²×C₉	—	6	1	no	—
25	C₂×D₉	—	6	2	no	—
26	C₂×D₉	—	6	1	no	—
27	C₄×C₉	—	2	1	no	—
28	C₄×C₉	C₂×D₃	1	1	yes	characteristic
29	C₃×C₉	D₈	1	1	yes	characteristic, complemented, Sylow
30	C₃×D₄	—	3	1	no	—
31	D₁₂	—	9	1	no	—
32	D₁₂	—	9	1	no	—
33	D₁₂	—	18	1	no	—
34	D₁₂	D₉	1	1	yes	characteristic, complemented
35	C₃×C₈	—	9	1	no	—
36	C₃⋊C₈	—	3	1	no	—
37	C₂×C₃²	#67	1	1	yes	characteristic
38	C₃⋊D₃	—	12	1	no	—
39	C₃×D₃	—	4	1	no	—
40	C₂×C₉	—	2	1	no	—
41	C₂×C₉	—	12	1	no	—
42	C₂×C₉	D₁₂	1	1	yes	characteristic
43	D₉	—	12	2	no	—
44	D₉	—	12	1	no	—
45	D₈	—	27	1	no	Sylow
46	C₂²×C₃	—	6	1	no	—
47	C₂×D₃	—	2	1	no	—
48	C₂×D₃	—	18	2	no	—
49	C₂×D₃	—	18	1	no	—
50	C₂×D₃	—	18	1	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	C₂×D₉	1	1	yes	characteristic
53	C₃×C₄	D₃²	1	1	yes	characteristic
54	C₃²	#213	1	1	yes	characteristic
55	C₉	—	2	1	no	—
56	C₉	D₂₄	1	1	yes	characteristic, complemented
57	D₄	—	3	1	no	—
58	D₄	—	27	1	no	—
59	C₈	—	9	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	12	1	no	—
62	C₂×C₃	#611	1	1	yes	characteristic
63	C₂×C₃	#626	1	1	yes	characteristic
64	D₃	—	4	1	no	—
65	D₃	—	36	2	no	—
66	D₃	—	36	1	no	—
67	D₃	—	36	1	no	—
68	C₂²	—	6	1	no	—
69	C₂²	—	54	1	no	—
70	C₄	D₃×D₉	1	1	yes	characteristic
71	C₃	—	2	1	no	—
72	C₃	#3673	1	1	yes	characteristic, complemented
73	C₃	#3714	1	1	yes	characteristic
74	C₂	—	12	1	no	—
75	C₂	—	108	1	no	—
76	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
hypercenter	C₄	D₃×D₉
hypocenter	C₃×C₉	D₈
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. C₄⋊(C₃⋊D₉)	C₂	semidirect
2. C₉×D₁₂	C₂	semidirect
3. C₃×C₉⋊C₈	C₂	semidirect
15. C₉⋊C₈	D₃	semidirect
29. C₃×C₉	D₈	semidirect
34. D₁₂	D₉	semidirect
56. C₉	D₂₄	semidirect
72. C₃	#3673	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
C₉×D₁₂	2
C₃×C₉⋊C₈	2
#3714	3
#3673	3