Subgroups — C₂²×C₃×SL(2,𝔽₃) #64333

Order 288.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×SL(2,𝔽₃)	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₃×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×SL(2,𝔽₃)	C₃	1	3	yes	maximal, direct product factor
4	C₂³×C₃²	—	4	1	no	maximal
5	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, direct product factor
6	C₂×C₃×Q₈	—	3	3	no	—
7	C₂×C₃×Q₈	C₂×C₃	1	3	yes	complemented
8	C₂²×C₃×C₄	—	3	1	no	—
9	C₂×SL(2,𝔽₃)	C₂×C₃	1	9	yes	direct product factor
10	C₂²×C₃²	—	4	4	no	—
11	C₂²×C₃²	—	4	3	no	—
12	C₂²×Q₈	C₃²	1	1	yes	characteristic, complemented, Sylow
13	C₂³×C₃	—	4	3	no	—
14	C₂³×C₃	A₄	1	1	yes	characteristic
15	C₃×Q₈	—	3	3	no	—
16	C₃×Q₈	—	3	2	no	—
17	C₃×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃×C₄	—	3	3	no	—
19	C₂×C₃×C₄	—	3	3	no	—
20	SL(2,𝔽₃)	C₂²×C₃	1	3	yes	direct product factor
21	C₂×C₃²	—	4	6	no	—
22	C₂×C₃²	—	4	1	no	—
23	C₂×Q₈	—	3	3	no	—
24	C₂×Q₈	C₂×C₃²	1	3	yes	complemented
25	C₂²×C₄	—	3	1	no	—
26	C₂²×C₃	—	4	12	no	—
27	C₂²×C₃	—	4	9	no	—
28	C₂²×C₃	SL(2,𝔽₃)	1	4	yes	direct product factor
29	C₂²×C₃	C₂×A₄	1	3	yes	—
30	C₃×C₄	—	3	3	no	—
31	C₃×C₄	—	3	1	no	—
32	C₃²	—	4	1	no	Sylow
33	C₂³	C₃×A₄	1	1	yes	characteristic
34	Q₈	—	3	3	no	—
35	Q₈	—	3	2	no	—
36	Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
37	C₂×C₄	—	3	3	no	—
38	C₂×C₄	—	3	3	no	—
39	C₂×C₃	—	4	3	no	—
40	C₂×C₃	—	4	18	no	—
41	C₂×C₃	C₂×SL(2,𝔽₃)	1	6	yes	direct product factor
42	C₂×C₃	C₂²×A₄	1	1	yes	characteristic
43	C₂²	C₃×SL(2,𝔽₃)	1	4	yes	direct product factor
44	C₂²	C₂×C₃×A₄	1	3	yes	—
45	C₄	—	3	1	no	—
46	C₄	—	3	3	no	—
47	C₃	—	4	3	no	—
48	C₃	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
49	C₂	C₂×C₃×SL(2,𝔽₃)	1	6	yes	direct product factor
50	C₂	C₂²×C₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂³×C₃	A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃²
Frattini	C₂	C₂²×C₃×A₄
Fitting	C₂²×C₃×Q₈	C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×SL(2,𝔽₃)	C₂	direct (N × H)
2. C₂²×C₃×Q₈	C₃	semidirect
3. C₂²×SL(2,𝔽₃)	C₃	direct (N × H)
5. C₃×SL(2,𝔽₃)	C₂²	direct (N × H)
7. C₂×C₃×Q₈	C₂×C₃	semidirect
9. C₂×SL(2,𝔽₃)	C₂×C₃	direct (N × H)
12. C₂²×Q₈	C₃²	semidirect
17. C₃×Q₈	C₂²×C₃	semidirect
20. SL(2,𝔽₃)	C₂²×C₃	direct (N × H)
24. C₂×Q₈	C₂×C₃²	semidirect
28. C₂²×C₃	SL(2,𝔽₃)	direct (N × H)
36. Q₈	C₂²×C₃²	semidirect
41. C₂×C₃	C₂×SL(2,𝔽₃)	direct (N × H)
43. C₂²	C₃×SL(2,𝔽₃)	direct (N × H)
48. C₃	C₂²×SL(2,𝔽₃)	direct (N × H)
49. C₂	C₂×C₃×SL(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
C₂×C₃×SL(2,𝔽₃)	2
C₂²×C₃×Q₈	3
C₂²×SL(2,𝔽₃)	3
C₂³×C₃²	4

Subgroups — C22×C3×SL(2,𝔽3) #64333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×SL(2,𝔽₃) #64333