Subgroups — #88534

Order 432.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	#3750	—	3	1	no	maximal
5	Dic₉⋊C₄	—	3	3	no	maximal
6	C₂²×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₉	—	2	1	no	—
8	C₃⋊Dic₉	—	2	1	no	—
9	C₃⋊Dic₉	C₄	1	2	yes	complemented
10	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	—	3	1	no	—
12	C₂×C₃⋊Dic₃	—	3	1	no	—
13	C₂×C₄×C₉	D₃	1	3	yes	—
14	C₂×Dic₉	—	3	3	no	—
15	C₂×Dic₉	—	3	3	no	—
16	C₂×C₃×C₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₉	D₄	1	1	yes	characteristic
18	C₂×C₃×C₉	Q₈	1	1	yes	characteristic
19	Dic₃⋊C₄	—	9	3	no	—
20	Dic₃⋊C₄	—	9	1	no	—
21	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
22	C₃²×C₄	—	2	1	no	—
23	C₃⋊Dic₃	—	3	2	no	—
24	C₃⋊Dic₃	—	6	1	no	—
25	C₂²×C₉	C₂×D₃	1	3	yes	—
26	C₄×C₉	—	2	3	no	—
27	Dic₉	—	3	6	no	—
28	Dic₉	—	6	3	no	—
29	C₃×C₉	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃×C₄	D₉	1	3	yes	—
31	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
32	C₂×Dic₃	—	9	3	no	—
33	C₂×Dic₃	—	9	1	no	—
34	C₂×Dic₃	—	9	3	no	—
35	C₂×Dic₃	—	9	1	no	—
36	C₂×C₃²	Dic₆	1	1	yes	characteristic
37	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
38	C₂×C₃²	#67	1	1	yes	characteristic
39	C₂×C₉	Dic₆	1	3	yes	—
40	C₂×C₉	C₄×D₃	1	3	yes	—
41	C₂×C₉	#67	1	3	yes	—
42	C₄⋊C₄	—	27	1	no	Sylow
43	C₂²×C₃	C₂×D₉	1	3	yes	—
44	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₃×C₄	—	2	3	no	—
46	C₃×C₄	—	2	1	no	—
47	Dic₃	—	9	6	no	—
48	Dic₃	—	9	2	no	—
49	Dic₃	—	18	3	no	—
50	Dic₃	—	18	1	no	—
51	C₃²	Dic₃⋊C₄	1	1	yes	characteristic
52	C₉	Dic₃⋊C₄	1	3	yes	complemented
53	C₂×C₄	—	27	1	no	—
54	C₂×C₄	—	27	1	no	—
55	C₂×C₄	C₃⋊D₉	1	1	yes	characteristic
56	C₂×C₃	Dic₁₈	1	3	yes	—
57	C₂×C₃	C₄×D₉	1	3	yes	—
58	C₂×C₃	#611	1	3	yes	—
59	C₂×C₃	#634	1	1	yes	characteristic
60	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
61	C₂×C₃	#638	1	1	yes	characteristic
62	C₂²	C₂×C₃⋊D₉	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	27	2	no	—
65	C₄	—	54	1	no	—
66	C₃	Dic₉⋊C₄	1	3	yes	complemented
67	C₃	#3750	1	1	yes	characteristic
68	C₂	#6226	1	1	yes	characteristic
69	C₂	C₄×C₃⋊D₉	1	1	yes	characteristic
70	C₂	#6233	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
9. C₃⋊Dic₉	C₄	semidirect
29. C₃×C₉	C₄⋊C₄	semidirect
52. C₉	Dic₃⋊C₄	semidirect
66. C₃	Dic₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₉	2
C₂×C₃⋊Dic₉	2
#3750	3
Dic₉⋊C₄	3