Subgroups — #92180

Order 486.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)⋊C₉	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊(C₃⋊D₉)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃⋊((C₃⋊D₃)⋊C₃)	—	3	1	no	maximal
4	C₃×H(3,𝔽₃)	D₃	1	1	yes	characteristic
5	C₃²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃²⋊C₉	—	2	1	no	—
7	C₃⋊(C₃⋊D₃)	—	3	1	no	—
8	C₃×C₃⋊D₃	—	9	1	no	—
9	C₃⋊D₉	—	9	1	no	—
10	C₃⋊D₉	—	9	3	no	—
11	(C₃⋊D₃)⋊C₃	—	9	2	no	—
12	(C₃⋊D₃)⋊C₃	—	9	1	no	—
13	C₃³	—	3	1	no	—
14	C₃³	C₃×D₃	1	1	yes	characteristic
15	H(3,𝔽₃)	—	2	1	no	—
16	H(3,𝔽₃)	—	3	2	no	—
17	H(3,𝔽₃)	D₉	1	1	yes	characteristic, complemented
18	C₃×C₉	—	3	1	no	—
19	C₃×C₉	—	3	3	no	—
20	C₃×C₉	—	6	1	no	—
21	C₃⋊D₃	—	9	1	no	—
22	C₃⋊D₃	—	9	1	no	—
23	C₃⋊D₃	—	9	2	no	—
24	C₃⋊D₃	—	27	1	no	—
25	C₃⋊D₃	—	27	1	no	—
26	C₃⋊D₃	—	27	1	no	—
27	C₃×D₃	—	27	1	no	—
28	C₃×D₃	—	27	1	no	—
29	C₃×D₃	—	27	1	no	—
30	C₃×D₃	—	27	1	no	—
31	D₉	—	27	3	no	—
32	C₃²	—	3	1	no	—
33	C₃²	—	3	1	no	—
34	C₃²	—	3	1	no	—
35	C₃²	—	3	1	no	—
36	C₃²	—	6	1	no	—
37	C₃²	—	9	1	no	—
38	C₃²	—	9	1	no	—
39	C₃²	—	9	1	no	—
40	C₃²	C₃×D₉	1	1	yes	characteristic
41	C₃²	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
42	C₃²	AGL(1,ℤ/9)	1	2	yes	—
43	C₉	—	3	3	no	—
44	C₉	—	18	1	no	—
45	C₂×C₃	—	81	1	no	—
46	D₃	—	27	1	no	—
47	D₃	—	27	1	no	—
48	D₃	—	27	1	no	—
49	D₃	—	27	1	no	—
50	D₃	—	81	1	no	—
51	D₃	—	81	2	no	—
52	C₃	—	3	1	no	—
53	C₃	—	3	2	no	—
54	C₃	—	9	1	no	—
55	C₃	—	18	1	no	—
56	C₃	#4169	1	1	yes	characteristic
57	C₃	#4175	1	1	yes	characteristic
58	C₃	#4177	1	1	yes	characteristic
59	C₃	#4179	1	1	yes	characteristic
60	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃²×C₉	C₂×C₃
Frattini	C₃³	C₃×D₃
Fitting	H(3,𝔽₃)⋊C₉	C₂
socle	C₃²	(C₃⋊D₃)⋊C₃
LCS	G ▷ C₃²×C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)⋊C₉	C₂	semidirect
2. C₃⋊(C₃⋊D₉)	C₃	semidirect
5. C₃²×C₉	C₂×C₃	semidirect
17. H(3,𝔽₃)	D₉	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)⋊C₉	2
C₃⋊(C₃⋊D₉)	3
C₃⋊((C₃⋊D₃)⋊C₃)	3