Subgroups — C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃)) #92397

Order 486.

456 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃⁴	C₂×C₃
Frattini	C₃	C₃×C₃⋊(C₃⋊D₃)
Fitting	C₃²×H(3,𝔽₃)	C₂
socle	C₃³	C₃×D₃
LCS	G ▷ C₃⁴
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃²×H(3,𝔽₃)	C₂	semidirect
2. C₃⋊(C₃⋊(C₃⋊D₃))	C₃	semidirect
6. C₃⁴	D₃	semidirect
7. C₃⁴	C₂×C₃	semidirect
9. C₃×H(3,𝔽₃)	D₃	semidirect
22. C₃³	C₃×D₃	semidirect
23. C₃³	C₃⋊D₃	semidirect
25. H(3,𝔽₃)	C₃⋊D₃	semidirect
38. C₃²	(C₃⋊D₃)⋊C₃	semidirect
40. C₃²	C₃×C₃⋊D₃	semidirect
41. C₃²	C₃⋊(C₃⋊D₃)	semidirect
50. C₃	C₃⋊((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
C₃²×H(3,𝔽₃)	2
C₃⋊(C₃⋊(C₃⋊D₃))	3
C₃×C₃⋊(C₃⋊D₃)	3
C₃⋊((C₃⋊D₃)⋊C₃)	3

Subgroups — C3⋊(C3⋊((C3⋊D3)⋊C3)) #92397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃)) #92397