Subgroups — C₂×D₃×SL(2,𝔽₃) #64274

Order 288.

129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×SL(2,𝔽₃)	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₃×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×SL(2,𝔽₃)	—	3	1	no	maximal
5	C₂²×C₃×D₃	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
8	D₃×Q₈	—	3	2	no	—
9	D₃×Q₈	C₂×C₃	1	2	yes	complemented
10	C₂×C₄×D₃	—	3	1	no	—
11	C₂×Dic₆	—	3	1	no	—
12	C₂×SL(2,𝔽₃)	—	2	1	no	—
13	C₂×SL(2,𝔽₃)	—	3	2	no	—
14	C₂×SL(2,𝔽₃)	D₃	1	1	yes	characteristic, direct product factor
15	C₂²×C₃²	—	4	1	no	—
16	C₂×C₃×D₃	—	4	4	no	—
17	C₂×C₃×D₃	—	4	2	no	—
18	C₂²×Q₈	—	3	1	no	Sylow
19	C₂³×C₃	—	12	1	no	—
20	C₂²×D₃	A₄	1	1	yes	characteristic
21	C₃×Q₈	—	3	1	no	—
22	C₃×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₄	—	3	1	no	—
24	C₂×Dic₃	—	3	1	no	—
25	C₄×D₃	—	3	2	no	—
26	C₄×D₃	—	3	2	no	—
27	Dic₆	—	3	2	no	—
28	Dic₆	—	3	2	no	—
29	SL(2,𝔽₃)	—	2	1	no	—
30	SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic, direct product factor
31	C₂×C₃²	—	4	2	no	—
32	C₂×C₃²	—	4	1	no	—
33	C₃×D₃	—	4	4	no	—
34	C₂×Q₈	—	3	2	no	—
35	C₂×Q₈	—	9	1	no	—
36	C₂×Q₈	—	9	2	no	—
37	C₂×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
38	C₂²×C₄	—	9	1	no	—
39	C₂²×C₃	—	4	1	no	—
40	C₂²×C₃	—	8	1	no	—
41	C₂²×C₃	—	12	4	no	—
42	C₂²×C₃	—	12	2	no	—
43	C₂²×C₃	C₂×A₄	1	1	yes	characteristic
44	C₂×D₃	SL(2,𝔽₃)	1	4	yes	direct product factor
45	C₂×D₃	C₂×A₄	1	2	yes	—
46	C₃×C₄	—	3	1	no	—
47	C₃×C₄	—	3	1	no	—
48	Dic₃	—	3	2	no	—
49	C₃²	—	4	1	no	Sylow
50	C₂³	—	3	1	no	—
51	Q₈	—	3	1	no	—
52	Q₈	—	9	2	no	—
53	Q₈	—	9	2	no	—
54	Q₈	C₂×C₃×D₃	1	1	yes	characteristic, complemented
55	C₂×C₄	—	3	1	no	—
56	C₂×C₄	—	9	2	no	—
57	C₂×C₄	—	9	2	no	—
58	C₂×C₄	—	9	1	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	—	4	2	no	—
61	C₂×C₃	—	8	1	no	—
62	C₂×C₃	—	8	2	no	—
63	C₂×C₃	—	12	4	no	—
64	C₂×C₃	C₂×SL(2,𝔽₃)	1	2	yes	complemented
65	C₂×C₃	C₂²×A₄	1	1	yes	characteristic
66	D₃	C₂×SL(2,𝔽₃)	1	4	yes	direct product factor
67	C₂²	—	3	2	no	—
68	C₂²	—	3	4	no	—
69	C₂²	D₃×A₄	1	1	yes	characteristic
70	C₄	—	3	1	no	—
71	C₄	—	3	1	no	—
72	C₄	—	9	2	no	—
73	C₃	—	4	1	no	—
74	C₃	—	8	1	no	—
75	C₃	C₂²×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
76	C₂	—	3	4	no	—
77	C₂	D₃×SL(2,𝔽₃)	1	2	yes	direct product factor
78	C₂	C₂×D₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	D₃×A₄
commutator, cosocle, hypocenter	C₃×Q₈	C₂²×C₃
Frattini	C₂	C₂×D₃×A₄
Fitting	C₂×C₃×Q₈	C₂×C₃
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ C₃×Q₈
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×SL(2,𝔽₃)	C₂	semidirect
2. D₃×SL(2,𝔽₃)	C₂	direct (N × H)
3. C₂×D₃×Q₈	C₃	semidirect
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
7. C₂×C₃×Q₈	C₂×C₃	semidirect
9. D₃×Q₈	C₂×C₃	semidirect
14. C₂×SL(2,𝔽₃)	D₃	direct (N × H)
22. C₃×Q₈	C₂²×C₃	semidirect
30. SL(2,𝔽₃)	C₂×D₃	direct (N × H)
37. C₂×Q₈	C₃×D₃	semidirect
44. C₂×D₃	SL(2,𝔽₃)	direct (N × H)
54. Q₈	C₂×C₃×D₃	semidirect
64. C₂×C₃	C₂×SL(2,𝔽₃)	semidirect
66. D₃	C₂×SL(2,𝔽₃)	direct (N × H)
75. C₃	C₂²×SL(2,𝔽₃)	semidirect
77. C₂	D₃×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
D₃×SL(2,𝔽₃)	2
C₂×D₃×Q₈	3
C₂²×SL(2,𝔽₃)	3
C₂²×C₃×D₃	4

Subgroups — C2×D3×SL(2,𝔽3) #64274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×SL(2,𝔽₃) #64274