Subgroups — #11757856

Order 972.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×A₄	C₃	1	1	yes	maximal, characteristic
2	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#1094	C₃	1	2	yes	maximal, complemented
4	(C₃×A₄)⋊C₉	C₃	1	9	yes	maximal, complemented
5	#6891	—	4	1	no	maximal, Sylow
6	C₂×C₃×3₋¹⁺²	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₃²×A₄	—	3	3	no	—
9	C₃²×A₄	—	3	9	no	—
10	C₃²×A₄	C₃²	1	3	yes	—
11	C₂²×3₋¹⁺²	—	3	3	no	—
12	C₂²×C₃×C₉	C₃²	1	3	yes	complemented
13	#1094	—	3	6	no	—
14	C₃×C₂²⋊C₉	C₃²	1	6	yes	complemented
15	C₃⁴	—	4	1	no	—
16	C₃×3₋¹⁺²	—	4	2	no	—
17	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic, complemented
18	C₃²⋊C₉	—	4	9	no	—
19	C₂×C₃³	—	3	1	no	—
20	C₂×3₋¹⁺²	—	3	9	no	—
21	C₂×C₃×C₉	—	3	3	no	—
22	C₂²×C₃²	—	3	3	no	—
23	C₂²×C₃²	H(3,𝔽₃)	1	3	yes	—
24	C₂²×C₃²	C₃³	1	1	yes	characteristic
25	C₃×A₄	—	3	27	no	—
26	C₃×A₄	—	3	3	no	—
27	C₃×A₄	—	3	6	no	—
28	C₃×A₄	3₋¹⁺²	1	9	yes	complemented
29	C₂²×C₉	—	3	3	no	—
30	C₂²⋊C₉	—	3	6	no	—
31	C₃³	—	4	3	no	—
32	C₃³	—	12	9	no	—
33	C₃³	—	12	3	no	—
34	C₃³	C₃×A₄	1	1	yes	characteristic
35	3₋¹⁺²	—	3	3	no	—
36	3₋¹⁺²	—	12	6	no	—
37	C₃×C₉	—	4	6	no	—
38	C₃×C₉	C₃×A₄	1	3	yes	complemented
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	3	3	no	—
41	C₂×C₃²	—	9	3	no	—
42	C₂×C₉	—	3	9	no	—
43	C₂²×C₃	—	3	3	no	—
44	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	C₂²×C₃	C₃×3₋¹⁺²	1	3	yes	—
46	A₄	—	3	9	no	—
47	C₃²	—	3	3	no	—
48	C₃²	—	4	9	no	—
49	C₃²	—	12	27	no	—
50	C₃²	—	12	6	no	—
51	C₃²	—	12	3	no	—
52	C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
53	C₃²	C₃²×A₄	1	1	yes	characteristic
54	C₉	—	3	3	no	—
55	C₉	—	12	6	no	—
56	C₂×C₃	—	3	3	no	—
57	C₂×C₃	—	3	1	no	—
58	C₂×C₃	—	9	3	no	—
59	C₂²	#6891	1	1	yes	characteristic, complemented, Sylow
60	C₃	—	3	3	no	—
61	C₃	—	12	9	no	—
62	C₃	A₄×3₋¹⁺²	1	3	yes	—
63	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
64	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₃×3₋¹⁺²	C₃
prime radical	C₃³×A₄	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6891
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. (C₃×A₄)⋊C₉	C₃	semidirect
12. C₂²×C₃×C₉	C₃²	semidirect
14. C₃×C₂²⋊C₉	C₃²	semidirect
17. C₃×3₋¹⁺²	A₄	semidirect
28. C₃×A₄	3₋¹⁺²	semidirect
38. C₃×C₉	C₃×A₄	semidirect
59. C₂²	#6891	semidirect

Maximal subgroups

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

subgroup	index
C₃³×A₄	3
C₂²×C₃×3₋¹⁺²	3
C₃×#1094	3
(C₃×A₄)⋊C₉	3
#6891	4