Subgroups — C₃²×GL(2,𝔽₃) #88967

Order 432.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×GL(2,𝔽₃)	C₃	1	4	yes	maximal, direct product factor
3	C₃²×SD₁₆	—	3	1	no	maximal
4	C₂×C₃²×D₃	—	4	1	no	maximal
5	C₃²×Q₈	D₃	1	1	yes	characteristic, complemented
6	C₃²×D₄	—	3	1	no	—
7	C₃×SL(2,𝔽₃)	—	2	4	no	—
8	C₃×SL(2,𝔽₃)	C₂×C₃	1	4	yes	complemented
9	C₃²×C₈	—	3	1	no	—
10	C₂×C₃³	—	4	1	no	—
11	C₃²×D₃	—	4	2	no	—
12	GL(2,𝔽₃)	C₃²	1	1	yes	characteristic, direct product factor
13	C₃×SD₁₆	—	3	4	no	—
14	C₂²×C₃²	—	6	1	no	—
15	C₂×C₃×D₃	—	4	4	no	—
16	C₃²×C₄	—	3	1	no	—
17	C₃³	—	4	1	no	Sylow
18	C₃×Q₈	C₃×D₃	1	4	yes	complemented
19	C₃×D₄	—	3	4	no	—
20	SL(2,𝔽₃)	—	2	4	no	—
21	SL(2,𝔽₃)	C₂×C₃²	1	1	yes	characteristic, complemented
22	C₃×C₈	—	3	4	no	—
23	C₂×C₃²	—	4	4	no	—
24	C₂×C₃²	—	8	4	no	—
25	C₂×C₃²	—	12	1	no	—
26	C₂×C₃²	S₄	1	1	yes	characteristic
27	C₃×D₃	—	4	8	no	—
28	SD₁₆	—	3	1	no	Sylow
29	C₂²×C₃	—	6	4	no	—
30	C₂×D₃	—	4	1	no	—
31	C₃×C₄	—	3	4	no	—
32	C₃²	—	4	4	no	—
33	C₃²	—	8	4	no	—
34	C₃²	GL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
35	Q₈	C₃²×D₃	1	1	yes	characteristic, complemented
36	D₄	—	3	1	no	—
37	C₈	—	3	1	no	—
38	C₂×C₃	—	4	1	no	—
39	C₂×C₃	—	8	4	no	—
40	C₂×C₃	—	12	4	no	—
41	C₂×C₃	C₃×S₄	1	4	yes	—
42	D₃	—	4	2	no	—
43	C₂²	—	6	1	no	—
44	C₄	—	3	1	no	—
45	C₃	—	4	1	no	—
46	C₃	—	8	4	no	—
47	C₃	C₃×GL(2,𝔽₃)	1	4	yes	direct product factor
48	C₂	—	12	1	no	—
49	C₂	C₃²×S₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₃²	S₄
commutator, cosocle, hypocenter	SL(2,𝔽₃)	C₂×C₃²
Frattini	C₂	C₃²×S₄
Fitting	C₃²×Q₈	D₃
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×SL(2,𝔽₃)	C₂	semidirect
2. C₃×GL(2,𝔽₃)	C₃	direct (N × H)
5. C₃²×Q₈	D₃	semidirect
8. C₃×SL(2,𝔽₃)	C₂×C₃	semidirect
12. GL(2,𝔽₃)	C₃²	direct (N × H)
18. C₃×Q₈	C₃×D₃	semidirect
21. SL(2,𝔽₃)	C₂×C₃²	semidirect
34. C₃²	GL(2,𝔽₃)	direct (N × H)
35. Q₈	C₃²×D₃	semidirect
47. C₃	C₃×GL(2,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
C₃²×SL(2,𝔽₃)	2
C₃×GL(2,𝔽₃)	3
C₃²×SD₁₆	3
C₂×C₃²×D₃	4

Subgroups — C32×GL(2,𝔽3) #88967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×GL(2,𝔽₃) #88967