Subgroups — #13091538

Order 1458.

506 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₉)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92301	C₃	1	4	yes	maximal, complemented
3	#92258	—	3	4	no	maximal
4	C₃²⋊H(3,𝔽₃)	—	2	16	no	—
5	C₃²⋊H(3,𝔽₃)	D₃	1	4	yes	complemented
6	C₃²⋊H(3,𝔽₃)	C₂×C₃	1	4	yes	complemented
7	C₃⋊(C₃⋊(C₃⋊D₃))	C₃²	1	1	yes	characteristic, complemented
8	C₃²×C₃⋊D₃	—	9	1	no	—
9	C₃⋊((C₃⋊D₃)⋊C₃)	—	3	16	no	—
10	C₃⁴	—	2	4	no	—
11	C₃⁴	C₃⋊D₃	1	1	yes	characteristic, complemented
12	C₃⁴	C₂×C₃²	1	1	yes	characteristic, complemented
13	C₃×H(3,𝔽₃)	—	2	16	no	—
14	C₃×H(3,𝔽₃)	—	2	4	no	—
15	C₃×H(3,𝔽₃)	—	2	16	no	—
16	C₃×H(3,𝔽₃)	—	2	16	no	—
17	C₃×H(3,𝔽₃)	C₃×D₃	1	16	yes	—
18	C₃⋊(C₃⋊D₃)	—	3	4	no	—
19	C₃⋊(C₃⋊D₃)	—	27	4	no	—
20	C₃×C₃⋊D₃	—	9	4	no	—
21	C₃²×D₃	—	27	4	no	—
22	(C₃⋊D₃)⋊C₃	—	27	16	no	—
23	C₃³	—	2	16	no	—
24	C₃³	—	9	4	no	—
25	C₃³	—	9	4	no	—
26	C₃³	—	18	8	no	—
27	C₃³	—	18	8	no	—
28	C₃³	C₃²×D₃	1	4	yes	—
29	C₃³	C₃×C₃⋊D₃	1	4	yes	—
30	H(3,𝔽₃)	—	9	16	no	—
31	H(3,𝔽₃)	—	18	16	no	—
32	H(3,𝔽₃)	—	18	16	no	—
33	H(3,𝔽₃)	—	18	16	no	—
34	H(3,𝔽₃)	—	18	4	no	—
35	C₂×C₃²	—	81	1	no	—
36	C₃⋊D₃	—	9	1	no	—
37	C₃⋊D₃	—	27	16	no	—
38	C₃⋊D₃	—	81	9	no	—
39	C₃×D₃	—	27	16	no	—
40	C₃²	—	3	16	no	—
41	C₃²	—	3	16	no	—
42	C₃²	—	6	16	no	—
43	C₃²	—	6	32	no	—
44	C₃²	—	6	16	no	—
45	C₃²	—	9	9	no	—
46	C₃²	—	9	1	no	—
47	C₃²	—	18	4	no	—
48	C₃²	—	18	36	no	—
49	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
50	C₂×C₃	—	81	4	no	—
51	D₃	—	27	4	no	—
52	D₃	—	243	4	no	—
53	C₃	—	9	4	no	—
54	C₃	—	9	4	no	—
55	C₃	—	18	16	no	—
56	C₃	#92391	1	4	yes	—
57	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃⁴	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	H(3,𝔽₉)	C₂
LCS	G ▷ C₃⁴
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₉)	C₂	semidirect
2. #92301	C₃	semidirect
5. C₃²⋊H(3,𝔽₃)	D₃	semidirect
6. C₃²⋊H(3,𝔽₃)	C₂×C₃	semidirect
7. C₃⋊(C₃⋊(C₃⋊D₃))	C₃²	semidirect
11. C₃⁴	C₃⋊D₃	semidirect
12. C₃⁴	C₂×C₃²	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₉)	2
#92301	3
#92258	3