Subgroups — C₃⋊#1113 #66478

Order 324.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
2	C₃²×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#1113	—	3	4	no	maximal
4	D₃×C₃⋊D₃	—	3	2	no	maximal
5	C₃⁴	C₂²	1	1	yes	characteristic, complemented, Sylow
6	C₃⋊(C₃⋊D₃)	D₃	1	2	yes	complemented
7	C₃×C₃⋊D₃	—	3	2	no	—
8	C₃×C₃⋊D₃	—	3	8	no	—
9	C₃×C₃⋊D₃	—	6	8	no	—
10	C₃×C₃⋊D₃	D₃	1	4	yes	complemented
11	C₃²×D₃	—	3	2	no	—
12	C₃²×D₃	—	6	1	no	—
13	C₂×C₃⋊D₃	—	9	1	no	—
14	D₃²	—	9	8	no	—
15	D₃²	—	9	1	no	—
16	C₃³	—	2	1	no	—
17	C₃³	—	2	8	no	—
18	C₃³	—	4	4	no	—
19	C₃³	C₂×D₃	1	2	yes	complemented
20	C₃³	C₂×D₃	1	4	yes	complemented
21	C₂×C₃²	—	9	1	no	—
22	C₃⋊D₃	—	3	2	no	—
23	C₃⋊D₃	—	3	8	no	—
24	C₃⋊D₃	—	6	8	no	—
25	C₃⋊D₃	C₃⋊D₃	1	1	yes	characteristic, complemented
26	C₃×D₃	—	3	8	no	—
27	C₃×D₃	—	6	4	no	—
28	C₃×D₃	—	9	8	no	—
29	C₃×D₃	—	9	2	no	—
30	C₃×D₃	—	18	8	no	—
31	C₂×D₃	—	27	2	no	—
32	C₂×D₃	—	27	4	no	—
33	C₃²	—	2	8	no	—
34	C₃²	—	2	8	no	—
35	C₃²	—	2	4	no	—
36	C₃²	—	4	4	no	—
37	C₃²	—	4	12	no	—
38	C₃²	—	4	4	no	—
39	C₃²	D₃²	1	8	yes	complemented
40	C₃²	D₃²	1	1	yes	characteristic, complemented
41	C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
42	C₂×C₃	—	9	4	no	—
43	C₂×C₃	—	27	2	no	—
44	D₃	—	3	2	no	—
45	D₃	—	6	1	no	—
46	D₃	—	9	2	no	—
47	D₃	—	9	8	no	—
48	D₃	—	18	8	no	—
49	C₂²	—	81	1	no	Sylow
50	C₃	—	2	1	no	—
51	C₃	—	2	8	no	—
52	C₃	—	4	4	no	—
53	C₃	D₃×C₃⋊D₃	1	2	yes	complemented
54	C₃	#1113	1	4	yes	complemented
55	C₂	—	9	1	no	—
56	C₂	—	27	2	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃⁴	C₂²
LCS	G ▷ C₃⁴
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃×C₃⋊(C₃⋊D₃)	C₂	semidirect
2. C₃²×C₃⋊D₃	C₂	semidirect
5. C₃⁴	C₂²	semidirect
6. C₃⋊(C₃⋊D₃)	D₃	semidirect
10. C₃×C₃⋊D₃	D₃	semidirect
19. C₃³	C₂×D₃	semidirect
20. C₃³	C₂×D₃	semidirect
25. C₃⋊D₃	C₃⋊D₃	semidirect
39. C₃²	D₃²	semidirect
40. C₃²	D₃²	semidirect
41. C₃²	C₂×C₃⋊D₃	semidirect
53. C₃	D₃×C₃⋊D₃	semidirect
54. C₃	#1113	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃⋊(C₃⋊D₃)	2
C₃²×C₃⋊D₃	2
#1113	3
D₃×C₃⋊D₃	3

Subgroups — C3⋊#1113 #66478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊#1113 #66478