Subgroups — C₂×D₃×C₃⋊D₃ #6334

Order 216.

232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₃⋊D₃	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₃⋊D₃	—	3	1	no	maximal
6	C₂×D₃²	—	3	4	no	maximal
7	C₂×C₃³	C₂²	1	1	yes	characteristic, complemented
8	C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
9	C₃×C₃⋊D₃	C₂²	1	2	yes	complemented
10	C₃²×D₃	C₂²	1	2	yes	complemented
11	C₂²×C₃²	—	3	1	no	—
12	C₂×C₃⋊D₃	—	3	1	no	—
13	C₂×C₃⋊D₃	—	3	4	no	—
14	C₂×C₃⋊D₃	—	3	4	no	—
15	C₂×C₃⋊D₃	—	6	4	no	—
16	C₂×C₃⋊D₃	D₃	1	1	yes	characteristic, direct product factor
17	C₂×C₃×D₃	—	3	4	no	—
18	C₂×C₃×D₃	D₃	1	4	yes	complemented
19	D₃²	—	3	16	no	—
20	C₃³	C₂³	1	1	yes	characteristic, complemented, Sylow
21	C₂²×D₃	—	9	4	no	—
22	C₂²×D₃	—	9	1	no	—
23	C₂×C₃²	—	2	4	no	—
24	C₂×C₃²	—	3	2	no	—
25	C₂×C₃²	C₂×D₃	1	4	yes	complemented
26	C₂×C₃²	C₂×D₃	1	1	yes	characteristic, complemented
27	C₃⋊D₃	—	3	2	no	—
28	C₃⋊D₃	—	3	8	no	—
29	C₃⋊D₃	—	6	8	no	—
30	C₃⋊D₃	C₂×D₃	1	2	yes	direct product factor
31	C₃×D₃	—	3	8	no	—
32	C₃×D₃	C₂×D₃	1	8	yes	complemented
33	C₂²×C₃	—	3	4	no	—
34	C₂²×C₃	—	9	1	no	—
35	C₂×D₃	—	3	4	no	—
36	C₂×D₃	—	9	16	no	—
37	C₂×D₃	—	9	4	no	—
38	C₂×D₃	—	9	1	no	—
39	C₂×D₃	—	9	4	no	—
40	C₂×D₃	—	18	4	no	—
41	C₂×D₃	C₃⋊D₃	1	1	yes	characteristic, direct product factor
42	C₃²	—	2	4	no	—
43	C₃²	C₂²×D₃	1	4	yes	complemented
44	C₃²	C₂²×D₃	1	1	yes	characteristic, complemented
45	C₂³	—	27	1	no	Sylow
46	C₂×C₃	—	2	4	no	—
47	C₂×C₃	—	3	8	no	—
48	C₂×C₃	—	9	2	no	—
49	C₂×C₃	D₃²	1	4	yes	complemented
50	C₂×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented
51	D₃	—	3	8	no	—
52	D₃	—	9	8	no	—
53	D₃	—	9	2	no	—
54	D₃	—	18	8	no	—
55	D₃	C₂×C₃⋊D₃	1	2	yes	direct product factor
56	C₂²	—	3	1	no	—
57	C₂²	—	9	1	no	—
58	C₂²	—	27	4	no	—
59	C₂²	—	27	1	no	—
60	C₃	—	2	4	no	—
61	C₃	C₂×D₃²	1	4	yes	complemented
62	C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic, complemented
63	C₂	—	3	2	no	—
64	C₂	—	9	2	no	—
65	C₂	—	27	2	no	—
66	C₂	D₃×C₃⋊D₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×C₃⋊D₃
commutator, cosocle, hypocenter	C₃³	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃³	C₂²
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⋊(C₃⋊D₃)	C₂	semidirect
2. C₂×C₃×C₃⋊D₃	C₂	semidirect
3. C₂×C₃²×D₃	C₂	semidirect
4. D₃×C₃⋊D₃	C₂	direct (N × H)
7. C₂×C₃³	C₂²	semidirect
8. C₃⋊(C₃⋊D₃)	C₂²	semidirect
9. C₃×C₃⋊D₃	C₂²	semidirect
10. C₃²×D₃	C₂²	semidirect
16. C₂×C₃⋊D₃	D₃	direct (N × H)
18. C₂×C₃×D₃	D₃	semidirect
20. C₃³	C₂³	semidirect
25. C₂×C₃²	C₂×D₃	semidirect
26. C₂×C₃²	C₂×D₃	semidirect
30. C₃⋊D₃	C₂×D₃	direct (N × H)
32. C₃×D₃	C₂×D₃	semidirect
41. C₂×D₃	C₃⋊D₃	direct (N × H)
43. C₃²	C₂²×D₃	semidirect
44. C₃²	C₂²×D₃	semidirect
49. C₂×C₃	D₃²	semidirect
50. C₂×C₃	C₂×C₃⋊D₃	semidirect
55. D₃	C₂×C₃⋊D₃	direct (N × H)
61. C₃	C₂×D₃²	semidirect
62. C₃	C₂²×C₃⋊D₃	semidirect
66. C₂	D₃×C₃⋊D₃	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⋊(C₃⋊D₃)	2
C₂×C₃×C₃⋊D₃	2
C₂×C₃²×D₃	2
D₃×C₃⋊D₃	2
C₂²×C₃⋊D₃	3
C₂×D₃²	3

Subgroups — C2×D3×C3⋊D3 #6334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×C₃⋊D₃ #6334