Subgroups — C₃×#66365 #11757848

Order 972.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#1094	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	#66365	C₃	1	9	yes	maximal, direct product factor
6	C₃×#731	—	4	1	no	maximal, Sylow
7	C₂×C₃×3₋¹⁺²	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₃²×A₄	—	3	1	no	—
10	C₂²×3₋¹⁺²	—	3	2	no	—
11	C₂²×3₋¹⁺²	C₃²	1	3	yes	complemented
12	C₂²×C₃×C₉	—	3	1	no	—
13	C₂²⋊H(3,𝔽₃)	—	3	2	no	—
14	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
15	#1094	—	3	2	no	—
16	#1094	C₃²	1	3	yes	complemented
17	C₃×C₂²⋊C₉	—	3	2	no	—
18	C₃×C₂²⋊C₉	—	3	1	no	—
19	C₃×C₂²⋊C₉	—	3	1	no	—
20	C₃×C₂²⋊C₉	C₃²	1	3	yes	complemented
21	C₃×3₋¹⁺²	—	4	1	no	—
22	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
23	C₃×H(3,𝔽₃)	—	4	1	no	—
24	C₃²×C₉	—	4	1	no	—
25	#731	—	4	9	no	—
26	C₂×C₃³	—	3	1	no	—
27	C₂×3₋¹⁺²	—	3	6	no	—
28	C₂×3₋¹⁺²	—	3	3	no	—
29	C₂×C₃×C₉	—	3	3	no	—
30	C₂²×C₃²	—	3	3	no	—
31	C₂²×C₃²	H(3,𝔽₃)	1	2	yes	—
32	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
33	C₂²×C₃²	C₃³	1	1	yes	characteristic
34	C₃×A₄	—	3	3	no	—
35	C₃×A₄	—	9	3	no	—
36	C₂²×C₉	—	3	3	no	—
37	C₂²⋊C₉	—	3	3	no	—
38	C₂²⋊C₉	—	3	3	no	—
39	C₃³	—	12	1	no	—
40	C₃³	C₃×A₄	1	1	yes	characteristic
41	3₋¹⁺²	—	3	2	no	—
42	3₋¹⁺²	—	4	3	no	—
43	3₋¹⁺²	—	12	2	no	—
44	3₋¹⁺²	C₃×A₄	1	3	yes	complemented
45	H(3,𝔽₃)	—	4	3	no	—
46	H(3,𝔽₃)	—	12	2	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	4	3	no	—
49	C₃×C₉	—	12	2	no	—
50	C₃×C₉	—	12	1	no	—
51	C₃×C₉	—	12	1	no	—
52	C₂×C₃²	—	3	1	no	—
53	C₂×C₃²	—	3	2	no	—
54	C₂×C₃²	—	3	1	no	—
55	C₂×C₃²	—	9	3	no	—
56	C₂×C₉	—	3	9	no	—
57	C₂²×C₃	—	3	2	no	—
58	C₂²×C₃	—	3	1	no	—
59	C₂²×C₃	#731	1	3	yes	complemented
60	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	A₄	—	9	3	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	12	3	no	—
64	C₃²	—	36	3	no	—
65	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
66	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
67	C₃²	C₃²×A₄	1	1	yes	characteristic
68	C₉	—	3	3	no	—
69	C₉	—	12	3	no	—
70	C₉	—	12	3	no	—
71	C₂×C₃	—	3	3	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	9	2	no	—
74	C₂×C₃	—	9	1	no	—
75	C₂²	C₃×#731	1	1	yes	characteristic, complemented, Sylow
76	C₃	—	3	2	no	—
77	C₃	—	3	1	no	—
78	C₃	—	36	3	no	—
79	C₃	#66365	1	3	yes	direct product factor
80	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
81	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₃×#731
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. C₃²×C₂²⋊C₉	C₃	semidirect
5. #66365	C₃	direct (N × H)
11. C₂²×3₋¹⁺²	C₃²	semidirect
16. #1094	C₃²	semidirect
20. C₃×C₂²⋊C₉	C₃²	semidirect
22. C₃×3₋¹⁺²	A₄	semidirect
44. 3₋¹⁺²	C₃×A₄	semidirect
59. C₂²×C₃	#731	semidirect
75. C₂²	C₃×#731	semidirect
79. C₃	#66365	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×C₃×3₋¹⁺²	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
C₃²×C₂²⋊C₉	3
#66365	3
C₃×#731	4

Subgroups — C3×#66365 #11757848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66365 #11757848