Subgroups — #92303

Order 486.

109 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(C₃⋊D₉)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊AGL(1,ℤ/9)	—	3	1	no	maximal
4	C₃⋊((C₃⋊D₃)⋊C₃)	—	3	1	no	maximal
5	#4169	—	3	2	no	maximal
6	C₃×3₋¹⁺²	—	2	1	no	—
7	C₃×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
8	C₃×H(3,𝔽₃)	D₃	1	1	yes	characteristic
9	C₃²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃²⋊C₉	—	2	1	no	—
11	C₃²⋊C₉	—	2	2	no	—
12	C₃²⋊C₉	D₃	1	2	yes	complemented
13	C₃⋊(C₃⋊D₃)	—	3	1	no	—
14	C₃×C₃⋊D₃	—	9	1	no	—
15	C₃⋊D₉	—	3	1	no	—
16	C₃⋊D₉	—	3	2	no	—
17	C₃⋊D₉	—	9	3	no	—
18	AGL(1,ℤ/9)	—	9	3	no	—
19	(C₃⋊D₃)⋊C₃	—	9	3	no	—
20	C₃³	—	2	1	no	—
21	C₃³	C₃×D₃	1	1	yes	characteristic
22	C₃³	C₃⋊D₃	1	1	yes	characteristic
23	3₋¹⁺²	—	3	3	no	—
24	3₋¹⁺²	—	6	3	no	—
25	H(3,𝔽₃)	—	3	3	no	—
26	C₃×C₉	—	2	1	no	—
27	C₃×C₉	—	2	2	no	—
28	C₃×C₉	—	3	3	no	—
29	C₃×C₉	C₃×D₃	1	2	yes	—
30	C₃×C₉	C₃×D₃	1	1	yes	characteristic
31	C₃⋊D₃	—	9	1	no	—
32	C₃⋊D₃	—	9	3	no	—
33	C₃⋊D₃	—	27	1	no	—
34	C₃⋊D₃	—	27	2	no	—
35	C₃×D₃	—	27	1	no	—
36	C₃×D₃	—	27	1	no	—
37	C₃×D₃	—	27	2	no	—
38	D₉	—	9	3	no	—
39	D₉	—	27	2	no	—
40	C₃²	—	3	1	no	—
41	C₃²	—	3	2	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	2	no	—
45	C₃²	—	6	1	no	—
46	C₃²	—	6	1	no	—
47	C₃²	—	6	2	no	—
48	C₃²	AGL(1,ℤ/9)	1	3	yes	complemented
49	C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
50	C₉	—	3	2	no	—
51	C₉	—	6	1	no	—
52	C₉	—	6	2	no	—
53	C₉	(C₃⋊D₃)⋊C₃	1	3	yes	complemented
54	C₂×C₃	—	81	1	no	—
55	D₃	—	27	1	no	—
56	D₃	—	27	1	no	—
57	D₃	—	27	2	no	—
58	D₃	—	81	3	no	—
59	C₃	—	3	3	no	—
60	C₃	—	9	1	no	—
61	C₃	—	18	1	no	—
62	C₃	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
63	C₃	C₃⋊AGL(1,ℤ/9)	1	1	yes	characteristic
64	C₃	#4207	1	2	yes	—
65	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉	C₂×C₃
Frattini, socle	C₃²	C₃×C₃⋊D₃
Fitting	C₉⋊H(3,𝔽₃)	C₂
LCS	G ▷ C₃²×C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₉⋊H(3,𝔽₃)	C₂	semidirect
2. C₃⋊(C₃⋊D₉)	C₃	semidirect
7. C₃×3₋¹⁺²	D₃	semidirect
9. C₃²×C₉	C₂×C₃	semidirect
12. C₃²⋊C₉	D₃	semidirect
48. C₃²	AGL(1,ℤ/9)	semidirect
53. C₉	(C₃⋊D₃)⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₉⋊H(3,𝔽₃)	2
C₃⋊(C₃⋊D₉)	3
C₃⋊AGL(1,ℤ/9)	3
C₃⋊((C₃⋊D₃)⋊C₃)	3
#4169	3