Subgroups — C₃×#66440 #11758103

Order 972.

518 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	#66440	C₃	1	27	yes	maximal, direct product factor
6	C₃×C₉⋊A₄	C₃	1	6	yes	maximal, complemented
7	C₃×C₉×A₄	C₃	1	3	yes	maximal, complemented
8	C₃×#737	—	4	1	no	maximal, Sylow
9	C₂×C₃×3₋¹⁺²	—	3	1	no	—
10	C₂²×C₃³	C₃²	1	1	yes	characteristic
11	C₃²×A₄	C₃²	1	3	yes	—
12	C₂²×3₋¹⁺²	C₃²	1	9	yes	complemented
13	C₂²×C₃×C₉	C₃²	1	3	yes	—
14	C₂²⋊H(3,𝔽₃)	C₃²	1	9	yes	—
15	#1094	C₃²	1	9	yes	complemented
16	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
18	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
19	C₃×C₂²⋊C₉	C₃²	1	9	yes	complemented
20	C₉⋊A₄	C₃²	1	54	yes	complemented
21	C₉×A₄	C₃²	1	27	yes	complemented
22	#737	—	4	27	no	—
23	C₃×3₋¹⁺²	—	4	1	no	—
24	C₃×3₋¹⁺²	—	4	6	no	—
25	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
26	C₃×H(3,𝔽₃)	—	4	1	no	—
27	C₃²×C₉	—	4	1	no	—
28	C₃²×C₉	—	4	3	no	—
29	C₂×C₃³	—	3	1	no	—
30	C₂×3₋¹⁺²	—	3	9	no	—
31	C₂×C₃×C₉	—	3	3	no	—
32	C₂²×C₃²	—	3	3	no	—
33	C₂²×C₃²	C₃³	1	3	yes	—
34	C₂²×C₃²	C₃³	1	1	yes	characteristic
35	C₃×A₄	—	3	9	no	—
36	C₃×A₄	C₃³	1	9	yes	—
37	C₂²×C₉	C₃³	1	9	yes	—
38	C₂²⋊C₉	C₃³	1	3	yes	—
39	C₂²⋊C₉	C₃³	1	9	yes	—
40	C₂²⋊C₉	C₃³	1	6	yes	—
41	C₃³	—	4	3	no	—
42	C₃³	C₃×A₄	1	1	yes	characteristic
43	3₋¹⁺²	—	4	9	no	—
44	3₋¹⁺²	—	4	54	no	—
45	3₋¹⁺²	C₃×A₄	1	9	yes	complemented
46	H(3,𝔽₃)	—	4	9	no	—
47	C₃×C₉	—	4	9	no	—
48	C₃×C₉	—	4	1	no	—
49	C₃×C₉	—	4	3	no	—
50	C₃×C₉	—	4	2	no	—
51	C₃×C₉	—	4	27	no	—
52	C₃×C₉	C₃×A₄	1	3	yes	—
53	C₂×C₃²	—	3	1	no	—
54	C₂×C₃²	—	3	3	no	—
55	C₂×C₃²	—	9	3	no	—
56	C₂×C₉	—	3	9	no	—
57	C₂²×C₃	—	3	3	no	—
58	C₂²×C₃	#737	1	3	yes	complemented
59	C₂²×C₃	C₃⁴	1	1	yes	characteristic
60	A₄	—	3	9	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	4	9	no	—
63	C₃²	—	12	9	no	—
64	C₃²	C₃²×A₄	1	3	yes	—
65	C₃²	C₃²×A₄	1	1	yes	characteristic
66	C₉	—	4	3	no	—
67	C₉	—	4	6	no	—
68	C₉	—	4	9	no	—
69	C₉	C₃²×A₄	1	9	yes	—
70	C₂×C₃	—	3	3	no	—
71	C₂×C₃	—	3	1	no	—
72	C₂×C₃	—	9	3	no	—
73	C₂²	C₃×#737	1	1	yes	characteristic, complemented, Sylow
74	C₃	—	3	3	no	—
75	C₃	—	12	9	no	—
76	C₃	#66440	1	3	yes	direct product factor
77	C₃	C₃³×A₄	1	1	yes	characteristic
78	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₃²×A₄
commutator, cosocle	C₂²×C₃	C₃⁴
Frattini	C₃	C₃³×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	C₃³
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₃×#737
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ 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. #66440	C₃	direct (N × H)
6. C₃×C₉⋊A₄	C₃	semidirect
7. C₃×C₉×A₄	C₃	semidirect
12. C₂²×3₋¹⁺²	C₃²	semidirect
15. #1094	C₃²	semidirect
19. C₃×C₂²⋊C₉	C₃²	semidirect
20. C₉⋊A₄	C₃²	semidirect
21. C₉×A₄	C₃²	semidirect
25. C₃×3₋¹⁺²	A₄	semidirect
45. 3₋¹⁺²	C₃×A₄	semidirect
58. C₂²×C₃	#737	semidirect
73. C₂²	C₃×#737	semidirect
76. C₃	#66440	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
#66440	3
C₃×C₉⋊A₄	3
C₃×C₉×A₄	3
C₃×#737	4

Subgroups — C3×#66440 #11758103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#66440 #11758103