Subgroups — #6294

Order 216.

90 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	2	yes	maximal, complemented
3	#624	—	3	1	no	maximal
4	D₃×Dic₃	—	3	2	no	maximal
5	C₂×C₃³	C₂²	1	1	yes	characteristic
6	C₃×C₃⋊D₃	C₄	1	2	yes	complemented
7	C₂×C₃⋊D₃	D₃	1	1	yes	characteristic
8	C₂×C₃×D₃	—	3	2	no	—
9	C₂×C₃×D₃	—	6	1	no	—
10	C₃⋊Dic₃	D₃	1	2	yes	complemented
11	C₃×Dic₃	—	3	2	no	—
12	C₃×Dic₃	—	3	2	no	—
13	C₃×Dic₃	—	6	2	no	—
14	C₃³	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
15	C₂×Dic₃	—	9	1	no	—
16	C₄×D₃	—	9	2	no	—
17	C₂×C₃²	—	2	2	no	—
18	C₂×C₃²	—	2	1	no	—
19	C₂×C₃²	—	4	1	no	—
20	C₂×C₃²	C₂×D₃	1	2	yes	—
21	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
22	C₃⋊D₃	Dic₃	1	2	yes	complemented
23	C₃×D₃	—	3	4	no	—
24	C₃×D₃	—	6	2	no	—
25	C₂²×C₃	—	9	1	no	—
26	C₂×D₃	—	3	2	no	—
27	C₂×D₃	—	6	1	no	—
28	C₃×C₄	—	9	2	no	—
29	Dic₃	—	3	2	no	—
30	Dic₃	—	3	2	no	—
31	Dic₃	—	6	2	no	—
32	C₃²	—	2	2	no	—
33	C₃²	—	2	1	no	—
34	C₃²	—	4	1	no	—
35	C₃²	C₄×D₃	1	2	yes	complemented
36	C₃²	C₂×Dic₃	1	1	yes	characteristic, complemented
37	C₂×C₄	—	27	1	no	Sylow
38	C₂×C₃	—	2	2	no	—
39	C₂×C₃	—	2	1	no	—
40	C₂×C₃	—	4	1	no	—
41	C₂×C₃	—	9	2	no	—
42	C₂×C₃	D₃²	1	2	yes	—
43	C₂×C₃	D₃²	1	1	yes	characteristic
44	D₃	—	3	4	no	—
45	D₃	—	6	2	no	—
46	C₂²	—	9	1	no	—
47	C₄	—	9	2	no	—
48	C₃	—	2	1	no	—
49	C₃	—	2	2	no	—
50	C₃	—	4	1	no	—
51	C₃	D₃×Dic₃	1	2	yes	complemented
52	C₃	#624	1	1	yes	characteristic, complemented
53	C₂	—	9	2	no	—
54	C₂	#1113	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	#1113
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
6. C₃×C₃⋊D₃	C₄	semidirect
10. C₃⋊Dic₃	D₃	semidirect
14. C₃³	C₂×C₄	semidirect
22. C₃⋊D₃	Dic₃	semidirect
35. C₃²	C₄×D₃	semidirect
36. C₃²	C₂×Dic₃	semidirect
51. C₃	D₃×Dic₃	semidirect
52. C₃	#624	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
#624	3
D₃×Dic₃	3