Subgroups — C₃²×C₂²⋊H(3,𝔽₃) #11758109

Order 972.

1112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⁴	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃³×A₄	C₃	1	3	yes	maximal, complemented
3	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	36	yes	maximal, direct product factor
4	C₃²×H(3,𝔽₃)	—	4	1	no	maximal, Sylow
5	C₂×C₃⁴	—	3	1	no	—
6	C₂²×C₃³	—	3	9	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₂²×C₃³	C₃²	1	12	yes	complemented
9	C₃²×A₄	—	3	27	no	—
10	C₃²×A₄	C₃²	1	36	yes	complemented
11	C₂²⋊H(3,𝔽₃)	C₃²	1	81	yes	direct product factor
12	C₃⁴	—	4	3	no	—
13	C₃⁴	A₄	1	1	yes	characteristic, complemented
14	C₃×H(3,𝔽₃)	—	4	36	no	—
15	C₂×C₃³	—	3	27	no	—
16	C₂×C₃³	—	3	1	no	—
17	C₂×C₃³	—	3	12	no	—
18	C₂²×C₃²	—	3	36	no	—
19	C₂²×C₃²	H(3,𝔽₃)	1	9	yes	complemented
20	C₂²×C₃²	C₃³	1	4	yes	—
21	C₂²×C₃²	C₃³	1	9	yes	complemented
22	C₃×A₄	—	3	108	no	—
23	C₃×A₄	C₃³	1	27	yes	complemented
24	C₃³	—	3	9	no	—
25	C₃³	—	4	36	no	—
26	C₃³	—	12	27	no	—
27	C₃³	C₃×A₄	1	1	yes	characteristic
28	C₃³	C₃×A₄	1	12	yes	complemented
29	H(3,𝔽₃)	—	4	81	no	—
30	C₂×C₃²	—	3	108	no	—
31	C₂×C₃²	—	3	9	no	—
32	C₂×C₃²	—	3	4	no	—
33	C₂×C₃²	—	3	9	no	—
34	C₂²×C₃	—	3	9	no	—
35	C₂²×C₃	C₃×H(3,𝔽₃)	1	12	yes	complemented
36	C₂²×C₃	C₃⁴	1	1	yes	characteristic
37	A₄	—	3	27	no	—
38	C₃²	—	3	36	no	—
39	C₃²	—	4	27	no	—
40	C₃²	—	12	108	no	—
41	C₃²	C₂²⋊H(3,𝔽₃)	1	9	yes	direct product factor
42	C₃²	C₃²×A₄	1	4	yes	—
43	C₃²	C₃²×A₄	1	9	yes	complemented
44	C₂×C₃	—	3	27	no	—
45	C₂×C₃	—	3	12	no	—
46	C₂×C₃	—	3	1	no	—
47	C₂²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
48	C₃	—	3	9	no	—
49	C₃	—	12	27	no	—
50	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	12	yes	direct product factor
51	C₃	C₃³×A₄	1	1	yes	characteristic
52	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃⁴	C₃	semidirect
2. C₃³×A₄	C₃	semidirect
3. C₃×C₂²⋊H(3,𝔽₃)	C₃	direct (N × H)
8. C₂²×C₃³	C₃²	semidirect
10. C₃²×A₄	C₃²	semidirect
11. C₂²⋊H(3,𝔽₃)	C₃²	direct (N × H)
13. C₃⁴	A₄	semidirect
19. C₂²×C₃²	H(3,𝔽₃)	semidirect
21. C₂²×C₃²	C₃³	semidirect
23. C₃×A₄	C₃³	semidirect
28. C₃³	C₃×A₄	semidirect
35. C₂²×C₃	C₃×H(3,𝔽₃)	semidirect
41. C₃²	C₂²⋊H(3,𝔽₃)	direct (N × H)
43. C₃²	C₃²×A₄	semidirect
47. C₂²	C₃²×H(3,𝔽₃)	semidirect
50. C₃	C₃×C₂²⋊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
C₂²×C₃⁴	3
C₃³×A₄	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃²×H(3,𝔽₃)	4

Subgroups — C32×C22⋊H(3,𝔽3) #11758109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₂²⋊H(3,𝔽₃) #11758109