Subgroups — C₂×C₃×D₃² #6333

Order 216.

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

	subgroup	G/H
center, hypercenter	C₂×C₃	D₃²
commutator, cosocle, hypocenter	C₃²	C₂³×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃³	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₃

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×C₃²×D₃	C₂	semidirect
3. C₃×D₃²	C₂	direct (N × H)
5. C₂×D₃²	C₃	direct (N × H)
6. C₂×C₃³	C₂²	semidirect
7. C₃×C₃⋊D₃	C₂²	semidirect
8. C₃²×D₃	C₂²	semidirect
10. C₂×C₃⋊D₃	C₂×C₃	semidirect
15. C₂×C₃×D₃	D₃	direct (N × H)
16. C₂×C₃×D₃	C₂×C₃	semidirect
17. D₃²	C₂×C₃	direct (N × H)
18. C₃³	C₂³	semidirect
25. C₂×C₃²	C₂×D₃	semidirect
26. C₂×C₃²	C₂²×C₃	semidirect
27. C₃⋊D₃	C₂²×C₃	semidirect
31. C₃×D₃	C₂×D₃	direct (N × H)
32. C₃×D₃	C₂²×C₃	semidirect
41. C₂×D₃	C₃×D₃	direct (N × H)
45. C₃²	C₂²×D₃	semidirect
46. C₃²	C₂³×C₃	semidirect
55. C₂×C₃	D₃²	direct (N × H)
56. C₂×C₃	C₂×C₃×D₃	semidirect
59. D₃	C₂×C₃×D₃	direct (N × H)
66. C₃	C₂×D₃²	direct (N × H)
67. C₃	C₂²×C₃×D₃	semidirect
70. C₂	C₃×D₃²	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×C₃²×D₃	2
C₃×D₃²	2
C₂²×C₃×D₃	3
C₂×D₃²	3

Subgroups — C2×C3×D32 #6333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃² #6333