Subgroups — C₃×#638 #6307

Order 216.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#638	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×#67	—	3	4	no	maximal
6	C₂×C₃³	—	2	1	no	—
7	C₂×C₃³	C₂²	1	1	yes	characteristic
8	C₃×C₃⋊D₃	—	2	1	no	—
9	C₂²×C₃²	—	2	4	no	—
10	C₂²×C₃²	D₃	1	4	yes	complemented
11	C₂²×C₃²	C₂×C₃	1	1	yes	characteristic, complemented
12	C₂×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×D₃	—	3	4	no	—
14	C₃⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×Dic₃	—	3	4	no	—
16	C₃³	D₄	1	1	yes	characteristic, complemented, Sylow
17	C₃×D₄	—	9	1	no	—
18	#67	—	3	4	no	—
19	C₂×C₃²	—	2	8	no	—
20	C₂×C₃²	—	2	4	no	—
21	C₂×C₃²	—	2	1	no	—
22	C₂×C₃²	—	2	4	no	—
23	C₂×C₃²	C₂×D₃	1	4	yes	—
24	C₂×C₃²	C₂²×C₃	1	1	yes	characteristic
25	C₃⋊D₃	—	2	1	no	—
26	C₃×D₃	—	6	4	no	—
27	C₂²×C₃	—	2	4	no	—
28	C₂²×C₃	—	9	1	no	—
29	C₂²×C₃	C₃×D₃	1	4	yes	complemented
30	C₂²×C₃	C₃⋊D₃	1	1	yes	characteristic, complemented
31	C₂×D₃	—	3	4	no	—
32	C₃×C₄	—	9	1	no	—
33	Dic₃	—	3	4	no	—
34	C₃²	—	2	4	no	—
35	C₃²	#67	1	4	yes	complemented
36	C₃²	C₃×D₄	1	1	yes	characteristic, complemented
37	D₄	—	9	1	no	Sylow
38	C₂×C₃	—	2	8	no	—
39	C₂×C₃	—	2	4	no	—
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	—	2	4	no	—
42	C₂×C₃	—	18	1	no	—
43	C₂×C₃	C₂×C₃×D₃	1	4	yes	—
44	C₂×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
45	D₃	—	6	4	no	—
46	C₂²	—	9	1	no	—
47	C₂²	C₃×C₃⋊D₃	1	1	yes	characteristic, complemented
48	C₄	—	9	1	no	—
49	C₃	—	2	4	no	—
50	C₃	C₃×#67	1	4	yes	complemented
51	C₃	#638	1	1	yes	characteristic, direct product factor
52	C₂	—	2	1	no	—
53	C₂	—	18	1	no	—
54	C₂	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×C₃⋊D₃
commutator, cosocle	C₂×C₃²	C₂²×C₃
Frattini	C₂	C₂×C₃×C₃⋊D₃
Fitting	C₂²×C₃³	C₂
socle	C₂×C₃³	C₂²
hypercenter	C₂²×C₃	C₃⋊D₃
hypocenter	C₃²	C₃×D₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³	C₂	semidirect
2. C₂×C₃×C₃⋊D₃	C₂	semidirect
3. C₃×C₃⋊Dic₃	C₂	semidirect
4. #638	C₃	direct (N × H)
10. C₂²×C₃²	D₃	semidirect
11. C₂²×C₃²	C₂×C₃	semidirect
12. C₂×C₃⋊D₃	C₂×C₃	semidirect
14. C₃⋊Dic₃	C₂×C₃	semidirect
16. C₃³	D₄	semidirect
29. C₂²×C₃	C₃×D₃	semidirect
30. C₂²×C₃	C₃⋊D₃	semidirect
35. C₃²	#67	semidirect
36. C₃²	C₃×D₄	semidirect
47. C₂²	C₃×C₃⋊D₃	semidirect
50. C₃	C₃×#67	semidirect
51. C₃	#638	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³	2
C₂×C₃×C₃⋊D₃	2
C₃×C₃⋊Dic₃	2
#638	3
C₃×#67	3

Subgroups — C3×#638 #6307

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#638 #6307