Subgroups — #92299

Order 486.

91 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊AGL(1,ℤ/9)	—	3	1	no	maximal
3	C₃×C₃⋊D₉	—	3	1	no	maximal
4	C₉⋊D₉	C₃	1	1	yes	maximal, characteristic, complemented
5	#4169	—	3	2	no	maximal
6	C₃×3₋¹⁺²	D₃	1	1	yes	characteristic
7	C₃²×C₉	D₃	1	1	yes	characteristic
8	C₉⋊C₉	—	2	2	no	—
9	C₉⋊C₉	—	2	1	no	—
10	C₃²⋊C₉	D₃	1	2	yes	—
11	C₉²	—	2	1	no	—
12	C₉²	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×C₃⋊D₃	—	9	1	no	—
14	C₃⋊D₉	—	3	1	no	—
15	C₃⋊D₉	—	3	1	no	—
16	C₃⋊D₉	—	3	2	no	—
17	AGL(1,ℤ/9)	—	9	3	no	—
18	C₃×D₉	—	9	3	no	—
19	C₃³	C₃⋊D₃	1	1	yes	characteristic
20	3₋¹⁺²	—	2	3	no	—
21	3₋¹⁺²	D₉	1	3	yes	complemented
22	C₃×C₉	—	2	1	no	—
23	C₃×C₉	—	2	2	no	—
24	C₃×C₉	—	2	1	no	—
25	C₃×C₉	—	3	3	no	—
26	C₃×C₉	C₃×D₃	1	2	yes	—
27	C₃×C₉	C₃×D₃	1	1	yes	characteristic
28	C₃×C₉	C₃×D₃	1	1	yes	characteristic
29	C₃⋊D₃	—	9	1	no	—
30	C₃×D₃	—	27	1	no	—
31	C₃×D₃	—	27	2	no	—
32	C₃×D₃	—	27	1	no	—
33	D₉	—	9	3	no	—
34	D₉	—	9	3	no	—
35	D₉	—	27	2	no	—
36	C₃²	—	2	1	no	—
37	C₃²	—	3	1	no	—
38	C₃²	—	3	2	no	—
39	C₃²	C₃⋊D₉	1	1	yes	characteristic
40	C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
41	C₉	—	2	3	no	—
42	C₉	—	3	2	no	—
43	C₉	—	6	1	no	—
44	C₉	—	6	2	no	—
45	C₉	C₃×D₉	1	3	yes	complemented
46	C₉	AGL(1,ℤ/9)	1	3	yes	complemented
47	C₂×C₃	—	81	1	no	—
48	D₃	—	27	1	no	—
49	D₃	—	27	1	no	—
50	D₃	—	27	2	no	—
51	C₃	—	3	1	no	—
52	C₃	—	6	1	no	—
53	C₃	C₃×C₃⋊D₉	1	1	yes	characteristic
54	C₃	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
55	C₃	#4207	1	2	yes	—
56	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₉²	C₂×C₃
Frattini, socle	C₃²	C₃×C₃⋊D₃
Fitting	C₉×3₋¹⁺²	C₂
LCS	G ▷ C₉²
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₉×3₋¹⁺²	C₂	semidirect
4. C₉⋊D₉	C₃	semidirect
12. C₉²	C₂×C₃	semidirect
21. 3₋¹⁺²	D₉	semidirect
45. C₉	C₃×D₉	semidirect
46. C₉	AGL(1,ℤ/9)	semidirect

Maximal subgroups

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

subgroup	index
C₉×3₋¹⁺²	2
C₃⋊AGL(1,ℤ/9)	3
C₃×C₃⋊D₉	3
C₉⋊D₉	3
#4169	3