Subgroups — #88548

Order 432.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3755	—	3	1	no	maximal
5	#3675	—	3	3	no	maximal
6	C₂×C₃⋊D₉	—	2	1	no	—
7	C₃×C₄×C₉	—	2	1	no	—
8	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃²×Q₈	D₃	1	1	yes	characteristic
10	C₄⋊(C₃⋊D₃)	—	3	1	no	—
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	Q₈×C₉	D₃	1	3	yes	complemented
13	D₃₆	—	3	3	no	—
14	C₉⋊C₈	—	3	3	no	—
15	C₂×C₃×C₉	D₄	1	1	yes	characteristic
16	C₃⋊D₉	—	4	1	no	—
17	#215	—	9	3	no	—
18	#215	—	9	1	no	—
19	C₂×C₃⋊D₃	—	6	1	no	—
20	C₃²×C₄	—	2	1	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₂×D₉	—	6	3	no	—
23	C₄×C₉	—	2	3	no	—
24	C₄×C₉	C₂×D₃	1	3	yes	—
25	C₃×C₉	SD₁₆	1	1	yes	characteristic, complemented, Sylow
26	C₃×Q₈	D₉	1	3	yes	complemented
27	C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
28	D₁₂	—	9	3	no	—
29	D₁₂	—	9	1	no	—
30	C₃⋊C₈	—	9	3	no	—
31	C₃⋊C₈	—	9	1	no	—
32	C₂×C₃²	#67	1	1	yes	characteristic
33	C₃⋊D₃	—	12	1	no	—
34	C₂×C₉	#67	1	3	yes	—
35	D₉	—	12	3	no	—
36	SD₁₆	—	27	1	no	Sylow
37	C₂×D₃	—	18	3	no	—
38	C₂×D₃	—	18	1	no	—
39	C₃×C₄	—	2	3	no	—
40	C₃×C₄	—	2	1	no	—
41	C₃×C₄	C₂×D₉	1	3	yes	—
42	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
43	C₃²	#215	1	1	yes	characteristic
44	C₉	#215	1	3	yes	complemented
45	Q₈	C₃⋊D₉	1	1	yes	characteristic, complemented
46	D₄	—	27	1	no	—
47	C₈	—	27	1	no	—
48	C₂×C₃	#611	1	3	yes	—
49	C₂×C₃	#638	1	1	yes	characteristic
50	D₃	—	36	3	no	—
51	D₃	—	36	1	no	—
52	C₂²	—	54	1	no	—
53	C₄	—	2	1	no	—
54	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
55	C₃	#3675	1	3	yes	complemented
56	C₃	#3755	1	1	yes	characteristic
57	C₂	—	108	1	no	—
58	C₂	#6233	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6233
commutator, cosocle	C₃×C₄×C₉	C₂²
Frattini	C₃×C₄	C₂×C₃⋊D₃
Fitting	C₃×Q₈×C₉	C₂
socle	C₂×C₃²	#67
prime radical	C₄⋊(C₃⋊D₉)	C₂
hypercenter	Q₈	C₃⋊D₉
hypocenter	C₃×C₉	SD₁₆
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×C₉	C₂	semidirect
3. C₃⋊(C₉⋊C₈)	C₂	semidirect
12. Q₈×C₉	D₃	semidirect
25. C₃×C₉	SD₁₆	semidirect
26. C₃×Q₈	D₉	semidirect
44. C₉	#215	semidirect
45. Q₈	C₃⋊D₉	semidirect
55. C₃	#3675	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×C₉	2
C₄⋊(C₃⋊D₉)	2
C₃⋊(C₉⋊C₈)	2
#3755	3
#3675	3