Subgroups — #92304

Order 486.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6893	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×AGL(1,ℤ/9)	—	3	2	no	maximal
3	C₃×(C₃⋊D₃)⋊C₃	—	3	1	no	maximal
4	C₃²×D₉	—	3	1	no	maximal
5	#4181	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×3₋¹⁺²	D₃	1	2	yes	complemented
7	C₃×H(3,𝔽₃)	D₃	1	1	yes	characteristic
8	C₃²×C₉	D₃	1	1	yes	characteristic, complemented
9	C₃²⋊C₉	—	2	1	no	—
10	C₃²⋊C₉	—	2	2	no	—
11	C₃²⋊C₉	—	2	1	no	—
12	C₃²⋊C₉	C₂×C₃	1	1	yes	characteristic, complemented
13	C₃×C₃⋊D₃	—	3	1	no	—
14	C₃²×D₃	—	9	1	no	—
15	AGL(1,ℤ/9)	—	3	6	no	—
16	(C₃⋊D₃)⋊C₃	—	3	3	no	—
17	C₃×D₉	—	3	3	no	—
18	C₃×D₉	—	3	1	no	—
19	C₃×D₉	—	3	2	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	6	no	—
24	H(3,𝔽₃)	—	3	3	no	—
25	C₃×C₉	—	2	1	no	—
26	C₃×C₉	—	2	2	no	—
27	C₃×C₉	—	3	3	no	—
28	C₃×C₉	C₃×D₃	1	1	yes	characteristic
29	C₃×C₉	C₃×D₃	1	2	yes	—
30	C₂×C₃²	—	27	1	no	—
31	C₃⋊D₃	—	3	1	no	—
32	C₃×D₃	—	9	3	no	—
33	C₃×D₃	—	9	1	no	—
34	C₃×D₃	—	27	1	no	—
35	D₉	—	3	1	no	—
36	D₉	—	3	2	no	—
37	C₃²	—	3	1	no	—
38	C₃²	—	3	1	no	—
39	C₃²	—	3	1	no	—
40	C₃²	—	6	1	no	—
41	C₃²	—	6	1	no	—
42	C₃²	—	6	2	no	—
43	C₃²	—	6	1	no	—
44	C₃²	—	6	1	no	—
45	C₃²	#272	1	3	yes	—
46	C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
47	C₉	—	3	1	no	—
48	C₉	—	3	2	no	—
49	C₉	—	6	1	no	—
50	C₉	—	6	2	no	—
51	C₂×C₃	—	27	3	no	—
52	C₂×C₃	—	27	1	no	—
53	D₃	—	9	1	no	—
54	D₃	—	27	1	no	—
55	C₃	—	2	1	no	—
56	C₃	—	3	3	no	—
57	C₃	—	9	1	no	—
58	C₃	—	18	1	no	—
59	C₃	C₃×#272	1	1	yes	characteristic
60	C₃	#4207	1	1	yes	characteristic
61	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#4207
commutator, cosocle, hypocenter	C₃²⋊C₉	C₂×C₃
Frattini, socle	C₃²	C₃×C₃⋊D₃
Fitting	#6893	C₂
LCS	G ▷ C₃²⋊C₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. #6893	C₂	semidirect
5. #4181	C₃	semidirect
6. C₃×3₋¹⁺²	D₃	semidirect
8. C₃²×C₉	D₃	semidirect
12. C₃²⋊C₉	C₂×C₃	semidirect

Maximal subgroups

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

subgroup	index
#6893	2
C₃×AGL(1,ℤ/9)	3
C₃×(C₃⋊D₃)⋊C₃	3
C₃²×D₉	3
#4181	3