Subgroups — #88544

Order 432.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊(C₃⋊D₉)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#3753	—	3	1	no	maximal
5	#3673	—	3	3	no	maximal
6	C₂²×C₃×C₉	—	2	1	no	—
7	C₂×C₃⋊D₉	—	2	1	no	—
8	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃²×D₄	D₃	1	1	yes	characteristic
10	C₄⋊(C₃⋊D₃)	—	3	1	no	—
11	C₃⋊(C₃⋊C₈)	—	3	1	no	—
12	D₄×C₉	D₃	1	3	yes	complemented
13	D₃₆	—	3	3	no	—
14	C₉⋊C₈	—	3	3	no	—
15	C₂×C₃×C₉	—	4	1	no	—
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	C₃⋊D₉	—	4	1	no	—
18	#213	—	9	3	no	—
19	#213	—	9	1	no	—
20	C₂²×C₃²	—	2	1	no	—
21	C₂×C₃⋊D₃	—	6	1	no	—
22	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
23	C₂²×C₉	—	2	3	no	—
24	C₂×D₉	—	6	3	no	—
25	C₄×C₉	C₂×D₃	1	3	yes	—
26	C₃×C₉	D₈	1	1	yes	characteristic, complemented, Sylow
27	C₃×D₄	D₉	1	3	yes	complemented
28	C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
29	D₁₂	—	9	3	no	—
30	D₁₂	—	9	1	no	—
31	C₃⋊C₈	—	9	3	no	—
32	C₃⋊C₈	—	9	1	no	—
33	C₂×C₃²	—	4	1	no	—
34	C₂×C₃²	#67	1	1	yes	characteristic
35	C₃⋊D₃	—	12	1	no	—
36	C₂×C₉	—	4	3	no	—
37	C₂×C₉	#67	1	3	yes	—
38	D₉	—	12	3	no	—
39	D₈	—	27	1	no	Sylow
40	C₂²×C₃	—	2	3	no	—
41	C₂²×C₃	—	2	1	no	—
42	C₂×D₃	—	18	3	no	—
43	C₂×D₃	—	18	1	no	—
44	C₃×C₄	C₂×D₉	1	3	yes	—
45	C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₃²	#213	1	1	yes	characteristic
47	C₉	#213	1	3	yes	complemented
48	D₄	—	27	1	no	—
49	D₄	C₃⋊D₉	1	1	yes	characteristic, complemented
50	C₈	—	27	1	no	—
51	C₂×C₃	—	4	3	no	—
52	C₂×C₃	—	4	1	no	—
53	C₂×C₃	#611	1	3	yes	—
54	C₂×C₃	#638	1	1	yes	characteristic
55	D₃	—	36	3	no	—
56	D₃	—	36	1	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	54	1	no	—
59	C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
60	C₃	#3673	1	3	yes	complemented
61	C₃	#3753	1	1	yes	characteristic
62	C₂	—	4	1	no	—
63	C₂	—	108	1	no	—
64	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×D₄×C₉	C₂
socle	C₂×C₃²	#67
hypercenter	D₄	C₃⋊D₉
hypocenter	C₃×C₉	D₈
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₃×D₄×C₉	C₂	semidirect
2. C₄⋊(C₃⋊D₉)	C₂	semidirect
3. C₃⋊(C₉⋊C₈)	C₂	semidirect
12. D₄×C₉	D₃	semidirect
26. C₃×C₉	D₈	semidirect
27. C₃×D₄	D₉	semidirect
47. C₉	#213	semidirect
49. D₄	C₃⋊D₉	semidirect
60. C₃	#3673	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₄×C₉	2
C₄⋊(C₃⋊D₉)	2
C₃⋊(C₉⋊C₈)	2
#3753	3
#3673	3