Subgroups — C₂×C₃⋊((C₃⋊D₃)⋊C₃) #66452

Order 324.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊((C₃⋊D₃)⋊C₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(C₃⋊D₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₃⋊D₃	—	3	1	no	maximal
5	C₂×(C₃⋊D₃)⋊C₃	—	3	3	no	maximal
6	C₃×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented, Sylow
7	C₂×C₃³	—	2	1	no	—
8	C₂×C₃³	D₃	1	1	yes	characteristic, complemented
9	C₂×C₃³	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃⋊(C₃⋊D₃)	C₂×C₃	1	2	yes	complemented
11	C₃×C₃⋊D₃	—	3	2	no	—
12	C₂×H(3,𝔽₃)	—	2	3	no	—
13	C₂×H(3,𝔽₃)	D₃	1	3	yes	complemented
14	(C₃⋊D₃)⋊C₃	—	3	6	no	—
15	C₂×C₃⋊D₃	—	3	1	no	—
16	C₂×C₃⋊D₃	—	3	3	no	—
17	C₂×C₃⋊D₃	—	9	3	no	—
18	C₂×C₃×D₃	—	9	3	no	—
19	C₂×C₃×D₃	—	9	1	no	—
20	C₃³	—	2	1	no	—
21	C₃³	C₂×D₃	1	1	yes	characteristic, complemented
22	C₃³	C₂²×C₃	1	1	yes	characteristic, complemented
23	H(3,𝔽₃)	—	2	3	no	—
24	H(3,𝔽₃)	C₂×D₃	1	3	yes	complemented
25	C₂×C₃²	—	2	1	no	—
26	C₂×C₃²	—	2	3	no	—
27	C₂×C₃²	—	3	3	no	—
28	C₂×C₃²	—	3	3	no	—
29	C₂×C₃²	—	6	3	no	—
30	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
31	C₂×C₃²	C₃×D₃	1	3	yes	complemented
32	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic, complemented
33	C₃⋊D₃	—	3	2	no	—
34	C₃⋊D₃	—	3	6	no	—
35	C₃⋊D₃	—	9	6	no	—
36	C₃×D₃	—	9	6	no	—
37	C₃×D₃	—	9	2	no	—
38	C₂²×C₃	—	27	1	no	—
39	C₂×D₃	—	9	3	no	—
40	C₂×D₃	—	9	1	no	—
41	C₂×D₃	—	27	3	no	—
42	C₃²	—	2	1	no	—
43	C₃²	—	2	3	no	—
44	C₃²	—	3	3	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	6	3	no	—
47	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
48	C₃²	C₂×C₃×D₃	1	3	yes	complemented
49	C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
50	C₂×C₃	—	3	3	no	—
51	C₂×C₃	—	3	1	no	—
52	C₂×C₃	—	6	1	no	—
53	C₂×C₃	—	6	3	no	—
54	C₂×C₃	—	27	2	no	—
55	C₂×C₃	(C₃⋊D₃)⋊C₃	1	3	yes	complemented
56	C₂×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
57	D₃	—	9	6	no	—
58	D₃	—	9	2	no	—
59	D₃	—	27	6	no	—
60	C₂²	—	27	1	no	Sylow
61	C₃	—	3	3	no	—
62	C₃	—	3	1	no	—
63	C₃	—	6	1	no	—
64	C₃	—	6	3	no	—
65	C₃	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	complemented
66	C₃	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
67	C₂	—	27	2	no	—
68	C₂	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃³	C₂²×C₃
Frattini	C₃	C₂×C₃×C₃⋊D₃
Fitting	C₂×C₃×H(3,𝔽₃)	C₂
socle	C₂×C₃²	C₃×D₃
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×H(3,𝔽₃)	C₂	semidirect
2. C₃⋊((C₃⋊D₃)⋊C₃)	C₂	direct (N × H)
3. C₂×C₃⋊(C₃⋊D₃)	C₃	semidirect
6. C₃×H(3,𝔽₃)	C₂²	semidirect
8. C₂×C₃³	D₃	semidirect
9. C₂×C₃³	C₂×C₃	semidirect
10. C₃⋊(C₃⋊D₃)	C₂×C₃	semidirect
13. C₂×H(3,𝔽₃)	D₃	semidirect
21. C₃³	C₂×D₃	semidirect
22. C₃³	C₂²×C₃	semidirect
24. H(3,𝔽₃)	C₂×D₃	semidirect
31. C₂×C₃²	C₃×D₃	semidirect
32. C₂×C₃²	C₃⋊D₃	semidirect
48. C₃²	C₂×C₃×D₃	semidirect
49. C₃²	C₂×C₃⋊D₃	semidirect
55. C₂×C₃	(C₃⋊D₃)⋊C₃	semidirect
65. C₃	C₂×(C₃⋊D₃)⋊C₃	semidirect
68. C₂	C₃⋊((C₃⋊D₃)⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×H(3,𝔽₃)	2
C₃⋊((C₃⋊D₃)⋊C₃)	2
C₂×C₃⋊(C₃⋊D₃)	3
C₂×C₃×C₃⋊D₃	3
C₂×(C₃⋊D₃)⋊C₃	3

Subgroups — C2×C3⋊((C3⋊D3)⋊C3) #66452

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊((C₃⋊D₃)⋊C₃) #66452