Subgroups — #88416

Order 432.

76 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×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#3717	—	3	1	no	maximal
5	#3664	—	3	1	no	maximal
6	C₂×C₃⋊D₉	—	2	1	no	—
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×Dic₉	—	2	1	no	—
9	C₄⋊(C₃⋊D₃)	—	3	1	no	—
10	C₃×Dic₆	—	3	1	no	—
11	C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
12	D₃₆	—	3	1	no	—
13	D₃₆	—	6	1	no	—
14	Dic₁₈	D₃	1	1	yes	characteristic, complemented
15	C₈×C₉	—	3	1	no	—
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	C₃⋊D₉	—	4	1	no	—
18	#215	—	9	1	no	—
19	#204	—	9	1	no	—
20	C₂×C₃⋊D₃	—	6	1	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃×Dic₃	—	6	1	no	—
23	C₂×D₉	—	6	2	no	—
24	C₂×D₉	—	6	1	no	—
25	C₄×C₉	—	2	1	no	—
26	C₄×C₉	C₂×D₃	1	1	yes	characteristic
27	Dic₉	—	2	1	no	—
28	C₃×C₉	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	9	1	no	—
30	D₁₂	—	9	1	no	—
31	D₁₂	—	9	1	no	—
32	D₁₂	—	18	1	no	—
33	Dic₆	—	3	1	no	—
34	C₃×C₈	—	3	1	no	—
35	C₃⋊C₈	D₉	1	1	yes	characteristic, complemented
36	C₂×C₃²	D₁₂	1	1	yes	characteristic
37	C₃⋊D₃	—	12	1	no	—
38	C₂×C₉	—	2	1	no	—
39	C₂×C₉	#67	1	1	yes	characteristic
40	D₉	—	12	2	no	—
41	D₉	—	12	1	no	—
42	SD₁₆	—	27	1	no	Sylow
43	C₂×D₃	—	18	2	no	—
44	C₂×D₃	—	18	1	no	—
45	C₂×D₃	—	18	1	no	—
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	—	18	1	no	—
48	C₃×C₄	C₂×D₉	1	1	yes	characteristic
49	C₃×C₄	D₃²	1	1	yes	characteristic
50	Dic₃	—	6	1	no	—
51	C₃²	#204	1	1	yes	characteristic
52	C₉	—	2	1	no	—
53	C₉	#215	1	1	yes	characteristic, complemented
54	Q₈	—	9	1	no	—
55	D₄	—	27	1	no	—
56	C₈	—	3	1	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	D₃₆	1	1	yes	characteristic
59	C₂×C₃	#626	1	1	yes	characteristic
60	D₃	—	36	2	no	—
61	D₃	—	36	1	no	—
62	D₃	—	36	1	no	—
63	C₂²	—	54	1	no	—
64	C₄	—	18	1	no	—
65	C₄	D₃×D₉	1	1	yes	characteristic
66	C₃	—	2	1	no	—
67	C₃	#3664	1	1	yes	characteristic, complemented
68	C₃	#3717	1	1	yes	characteristic
69	C₂	—	108	1	no	—
70	C₂	#6192	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6192
commutator, Fitting, cosocle	C₃×C₄×C₉	C₂²
Frattini	C₃×C₄	D₃²
socle	C₂×C₃²	D₁₂
prime radical	C₄⋊(C₃⋊D₉)	C₂
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
2. C₃×Dic₁₈	C₂	semidirect
3. C₉×C₃⋊C₈	C₂	semidirect
14. Dic₁₈	D₃	semidirect
28. C₃×C₉	SD₁₆	semidirect
35. C₃⋊C₈	D₉	semidirect
53. C₉	#215	semidirect
67. C₃	#3664	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×Dic₁₈	2
C₉×C₃⋊C₈	2
#3717	3
#3664	3