Subgroups — #6289

Order 216.

136 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
2	C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃⋊D₃	—	3	1	no	maximal
5	#624	—	3	4	no	maximal
6	C₂×C₃³	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊D₃)	C₄	1	2	yes	complemented
8	C₂×C₃⋊D₃	—	3	1	no	—
9	C₂×C₃⋊D₃	—	3	4	no	—
10	C₂×C₃⋊D₃	—	6	4	no	—
11	C₃²×C₄	—	3	1	no	—
12	C₃⋊Dic₃	D₃	1	1	yes	characteristic, complemented
13	C₃×Dic₃	—	3	4	no	—
14	C₃×Dic₃	D₃	1	4	yes	complemented
15	C₃³	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
16	C₄×D₃	—	9	4	no	—
17	C₄×D₃	—	9	1	no	—
18	C₂×C₃²	—	2	4	no	—
19	C₂×C₃²	C₂×D₃	1	4	yes	—
20	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
21	C₃⋊D₃	—	3	2	no	—
22	C₃⋊D₃	—	3	8	no	—
23	C₃⋊D₃	—	6	8	no	—
24	C₂×D₃	—	9	4	no	—
25	C₂×D₃	—	9	1	no	—
26	C₂×D₃	—	18	4	no	—
27	C₃×C₄	—	3	4	no	—
28	C₃×C₄	—	9	1	no	—
29	Dic₃	—	3	4	no	—
30	Dic₃	C₃⋊D₃	1	1	yes	characteristic, complemented
31	C₃²	—	2	4	no	—
32	C₃²	C₄×D₃	1	4	yes	complemented
33	C₃²	C₄×D₃	1	1	yes	characteristic, complemented
34	C₂×C₄	—	27	1	no	Sylow
35	C₂×C₃	—	2	4	no	—
36	C₂×C₃	D₃²	1	4	yes	—
37	C₂×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
38	D₃	—	9	8	no	—
39	D₃	—	9	2	no	—
40	D₃	—	18	8	no	—
41	C₂²	—	27	1	no	—
42	C₄	—	3	1	no	—
43	C₄	—	9	1	no	—
44	C₃	—	2	4	no	—
45	C₃	#624	1	4	yes	complemented
46	C₃	C₄×C₃⋊D₃	1	1	yes	characteristic, complemented
47	C₂	—	27	2	no	—
48	C₂	D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊D₃
commutator, hypocenter	C₃³	C₂×C₄
Fitting, socle, cosocle	C₂×C₃³	C₂²
prime radical	C₂×C₃⋊(C₃⋊D₃)	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊Dic₃	C₂	semidirect
3. C₃²×Dic₃	C₂	semidirect
7. C₃⋊(C₃⋊D₃)	C₄	semidirect
12. C₃⋊Dic₃	D₃	semidirect
14. C₃×Dic₃	D₃	semidirect
15. C₃³	C₂×C₄	semidirect
30. Dic₃	C₃⋊D₃	semidirect
32. C₃²	C₄×D₃	semidirect
33. C₃²	C₄×D₃	semidirect
45. C₃	#624	semidirect
46. C₃	C₄×C₃⋊D₃	semidirect

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
C₃×C₃⋊Dic₃	2
C₃²×Dic₃	2
C₄×C₃⋊D₃	3
#624	3