Subgroups — #92259

Order 486.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6891	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃²×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic
3	D₃×3₋¹⁺²	—	3	1	no	maximal
4	(C₃⋊D₃)⋊C₉	C₃	1	3	yes	maximal, complemented
5	C₃⁴	C₂×C₃	1	1	yes	characteristic
6	C₃×3₋¹⁺²	—	2	1	no	—
7	C₃×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
8	C₃²⋊C₉	—	2	3	no	—
9	C₃²⋊C₉	C₂×C₃	1	3	yes	complemented
10	C₃×C₃⋊D₃	—	3	1	no	—
11	C₃×C₃⋊D₃	C₃²	1	1	yes	characteristic
12	C₃²×D₃	—	3	1	no	—
13	C₃²×D₃	—	9	1	no	—
14	C₂×3₋¹⁺²	—	9	1	no	—
15	D₃×C₉	—	3	3	no	—
16	C₃³	—	2	1	no	—
17	C₃³	—	3	1	no	—
18	C₃³	—	3	1	no	—
19	C₃³	—	6	3	no	—
20	C₃³	—	6	1	no	—
21	C₃³	—	6	1	no	—
22	C₃³	C₃×D₃	1	1	yes	characteristic
23	C₃³	C₂×C₃²	1	1	yes	characteristic
24	3₋¹⁺²	—	3	1	no	—
25	3₋¹⁺²	—	6	1	no	—
26	3₋¹⁺²	—	6	2	no	—
27	3₋¹⁺²	—	6	1	no	—
28	C₃×C₉	—	2	3	no	—
29	C₃×C₉	C₃×D₃	1	3	yes	complemented
30	C₂×C₃²	—	9	1	no	—
31	C₃⋊D₃	3₋¹⁺²	1	1	yes	characteristic, complemented
32	C₃×D₃	—	3	1	no	—
33	C₃×D₃	—	9	1	no	—
34	C₃×D₃	—	9	1	no	—
35	C₃×D₃	—	9	3	no	—
36	C₂×C₉	—	9	3	no	—
37	C₃²	—	2	3	no	—
38	C₃²	—	2	1	no	—
39	C₃²	—	2	1	no	—
40	C₃²	—	3	3	no	—
41	C₃²	—	3	1	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	6	1	no	—
44	C₃²	—	6	1	no	—
45	C₃²	—	6	1	no	—
46	C₃²	—	6	3	no	—
47	C₃²	—	6	3	no	—
48	C₃²	—	6	3	no	—
49	C₃²	—	6	3	no	—
50	C₃²	—	6	1	no	—
51	C₃²	—	6	1	no	—
52	C₃²	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
53	C₃²	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
54	C₃²	C₃²×D₃	1	1	yes	characteristic
55	C₉	—	3	3	no	—
56	C₉	—	6	3	no	—
57	C₂×C₃	—	9	1	no	—
58	C₂×C₃	—	27	1	no	—
59	D₃	—	3	1	no	—
60	D₃	—	9	1	no	—
61	C₃	—	2	1	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	1	no	—
64	C₃	—	6	3	no	—
65	C₃	—	6	1	no	—
66	C₃	—	6	1	no	—
67	C₃	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
68	C₃	D₃×3₋¹⁺²	1	1	yes	characteristic
69	C₂	—	9	1	no	Sylow

	subgroup	G/H
center	C₃	C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃³	C₂×C₃²
Frattini, socle	C₃²	C₃²×D₃
Fitting	#6891	C₂
prime radical	C₃²×C₃⋊D₃	C₃
hypercenter	C₃²	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×3₋¹⁺²
LCS	G ▷ C₃³ ▷ C₃²
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. #6891	C₂	semidirect
4. (C₃⋊D₃)⋊C₉	C₃	semidirect
7. C₃×3₋¹⁺²	D₃	semidirect
9. C₃²⋊C₉	C₂×C₃	semidirect
29. C₃×C₉	C₃×D₃	semidirect
31. C₃⋊D₃	3₋¹⁺²	semidirect
53. C₃²	C₂×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
#6891	2
C₃²×C₃⋊D₃	3
D₃×3₋¹⁺²	3
(C₃⋊D₃)⋊C₉	3