Subgroups — C₃×A₄×H(3,𝔽₃) #11758100

Order 972.

956 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×A₄	C₃	1	4	yes	maximal, complemented
2	C₂²×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	8	yes	maximal, complemented
4	A₄×H(3,𝔽₃)	C₃	1	27	yes	maximal, direct product factor
5	C₃²×H(3,𝔽₃)	—	4	1	no	maximal, Sylow
6	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	4	yes	complemented
8	C₃²×A₄	—	3	36	no	—
9	C₃²×A₄	C₃²	1	1	yes	characteristic
10	C₃²×A₄	C₃²	1	36	yes	complemented
11	C₃²×A₄	C₃²	1	8	yes	complemented
12	C₂²×H(3,𝔽₃)	C₃²	1	9	yes	complemented
13	C₂²⋊H(3,𝔽₃)	C₃²	1	72	yes	complemented
14	C₃⁴	—	4	4	no	—
15	C₃×H(3,𝔽₃)	—	4	8	no	—
16	C₃×H(3,𝔽₃)	—	4	27	no	—
17	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic, direct product factor
18	C₂×C₃³	—	3	4	no	—
19	C₂×H(3,𝔽₃)	—	3	9	no	—
20	C₂²×C₃²	—	3	12	no	—
21	C₂²×C₃²	C₃³	1	1	yes	characteristic
22	C₂²×C₃²	C₃³	1	12	yes	complemented
23	C₃×A₄	—	3	24	no	—
24	C₃×A₄	—	3	108	no	—
25	C₃×A₄	H(3,𝔽₃)	1	9	yes	direct product factor
26	C₃×A₄	C₃³	1	3	yes	—
27	C₃×A₄	C₃³	1	24	yes	complemented
28	C₃³	—	4	1	no	—
29	C₃³	—	4	8	no	—
30	C₃³	—	4	36	no	—
31	C₃³	—	12	36	no	—
32	C₃³	C₃×A₄	1	4	yes	complemented
33	H(3,𝔽₃)	—	4	72	no	—
34	H(3,𝔽₃)	C₃×A₄	1	9	yes	direct product factor
35	C₂×C₃²	—	3	1	no	—
36	C₂×C₃²	—	3	12	no	—
37	C₂×C₃²	—	9	12	no	—
38	C₂²×C₃	—	3	12	no	—
39	C₂²×C₃	C₃×H(3,𝔽₃)	1	3	yes	complemented
40	C₂²×C₃	C₃⁴	1	1	yes	characteristic
41	A₄	—	3	24	no	—
42	A₄	C₃×H(3,𝔽₃)	1	9	yes	direct product factor
43	C₃²	—	3	12	no	—
44	C₃²	—	4	9	no	—
45	C₃²	—	4	3	no	—
46	C₃²	—	4	24	no	—
47	C₃²	—	12	108	no	—
48	C₃²	—	12	24	no	—
49	C₃²	C₃²×A₄	1	1	yes	characteristic
50	C₃²	C₃²×A₄	1	12	yes	complemented
51	C₂×C₃	—	3	3	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	9	12	no	—
54	C₂²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
55	C₃	—	3	12	no	—
56	C₃	—	4	9	no	—
57	C₃	—	12	24	no	—
58	C₃	A₄×H(3,𝔽₃)	1	3	yes	direct product factor
59	C₃	C₃³×A₄	1	1	yes	characteristic
60	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₃²×A₄
commutator, cosocle	C₂²×C₃	C₃⁴
Frattini	C₃	C₃³×A₄
Fitting	C₂²×C₃×H(3,𝔽₃)	C₃
socle	C₂²×C₃²	C₃³
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	C₃²×H(3,𝔽₃)
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃³×A₄	C₃	semidirect
2. C₂²×C₃×H(3,𝔽₃)	C₃	semidirect
3. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
4. A₄×H(3,𝔽₃)	C₃	direct (N × H)
7. C₂²×C₃³	C₃²	semidirect
10. C₃²×A₄	C₃²	semidirect
11. C₃²×A₄	C₃²	semidirect
12. C₂²×H(3,𝔽₃)	C₃²	semidirect
13. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
17. C₃×H(3,𝔽₃)	A₄	direct (N × H)
22. C₂²×C₃²	C₃³	semidirect
25. C₃×A₄	H(3,𝔽₃)	direct (N × H)
27. C₃×A₄	C₃³	semidirect
32. C₃³	C₃×A₄	semidirect
34. H(3,𝔽₃)	C₃×A₄	direct (N × H)
39. C₂²×C₃	C₃×H(3,𝔽₃)	semidirect
42. A₄	C₃×H(3,𝔽₃)	direct (N × H)
50. C₃²	C₃²×A₄	semidirect
54. C₂²	C₃²×H(3,𝔽₃)	semidirect
58. C₃	A₄×H(3,𝔽₃)	direct (N × H)

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×H(3,𝔽₃)	3
C₃×C₂²⋊H(3,𝔽₃)	3
A₄×H(3,𝔽₃)	3
C₃²×H(3,𝔽₃)	4

Subgroups — C3×A4×H(3,𝔽3) #11758100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×A₄×H(3,𝔽₃) #11758100