Subgroups — C₃×SL(2,𝔽₃)×H(3,𝔽₃) #14518862

Order 1944.

1406 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×SL(2,𝔽₃)	C₃	1	4	yes	maximal, complemented
2	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	8	yes	maximal, complemented
4	SL(2,𝔽₃)×H(3,𝔽₃)	C₃	1	27	yes	maximal, direct product factor
5	C₂×C₃²×H(3,𝔽₃)	—	4	1	no	maximal
6	C₃×C₄×H(3,𝔽₃)	—	3	1	no	—
7	C₃²×H(3,𝔽₃)	—	4	1	no	Sylow
8	C₃³×Q₈	C₃²	1	4	yes	complemented
9	C₃²×SL(2,𝔽₃)	—	3	36	no	—
10	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃²×SL(2,𝔽₃)	C₃²	1	36	yes	complemented
12	C₃²×SL(2,𝔽₃)	C₃²	1	8	yes	complemented
13	Q₈×H(3,𝔽₃)	C₃²	1	9	yes	complemented
14	Q₈⋊H(3,𝔽₃)	C₃²	1	72	yes	complemented
15	C₂×C₃⁴	—	4	4	no	—
16	C₂×C₃×H(3,𝔽₃)	—	4	8	no	—
17	C₂×C₃×H(3,𝔽₃)	—	4	27	no	—
18	C₂×C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
19	C₃³×C₄	—	3	4	no	—
20	C₄×H(3,𝔽₃)	—	3	9	no	—
21	C₃⁴	—	4	4	no	—
22	C₃×H(3,𝔽₃)	—	4	8	no	—
23	C₃×H(3,𝔽₃)	—	4	27	no	—
24	C₃×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
25	C₃²×Q₈	—	3	12	no	—
26	C₃²×Q₈	C₃³	1	1	yes	characteristic
27	C₃²×Q₈	C₃³	1	12	yes	complemented
28	C₃×SL(2,𝔽₃)	—	3	24	no	—
29	C₃×SL(2,𝔽₃)	—	3	108	no	—
30	C₃×SL(2,𝔽₃)	H(3,𝔽₃)	1	9	yes	direct product factor
31	C₃×SL(2,𝔽₃)	C₃³	1	3	yes	—
32	C₃×SL(2,𝔽₃)	C₃³	1	24	yes	complemented
33	C₂×C₃³	—	4	1	no	—
34	C₂×C₃³	—	4	36	no	—
35	C₂×C₃³	—	4	8	no	—
36	C₂×C₃³	—	12	36	no	—
37	C₂×C₃³	C₃×A₄	1	4	yes	—
38	C₂×H(3,𝔽₃)	—	4	72	no	—
39	C₂×H(3,𝔽₃)	C₃×A₄	1	9	yes	—
40	C₃²×C₄	—	3	1	no	—
41	C₃²×C₄	—	3	12	no	—
42	C₃²×C₄	—	9	12	no	—
43	C₃³	—	4	1	no	—
44	C₃³	—	4	8	no	—
45	C₃³	—	4	36	no	—
46	C₃³	—	12	36	no	—
47	C₃³	C₃×SL(2,𝔽₃)	1	4	yes	complemented
48	H(3,𝔽₃)	—	4	72	no	—
49	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	9	yes	direct product factor
50	C₃×Q₈	—	3	12	no	—
51	C₃×Q₈	C₃×H(3,𝔽₃)	1	3	yes	complemented
52	C₃×Q₈	C₃⁴	1	1	yes	characteristic
53	SL(2,𝔽₃)	—	3	24	no	—
54	SL(2,𝔽₃)	C₃×H(3,𝔽₃)	1	9	yes	direct product factor
55	C₂×C₃²	—	3	12	no	—
56	C₂×C₃²	—	4	9	no	—
57	C₂×C₃²	—	4	3	no	—
58	C₂×C₃²	—	4	24	no	—
59	C₂×C₃²	—	12	108	no	—
60	C₂×C₃²	—	12	24	no	—
61	C₂×C₃²	C₃²×A₄	1	12	yes	—
62	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
63	C₃×C₄	—	3	3	no	—
64	C₃×C₄	—	3	1	no	—
65	C₃×C₄	—	9	12	no	—
66	C₃²	—	3	12	no	—
67	C₃²	—	4	9	no	—
68	C₃²	—	4	3	no	—
69	C₃²	—	4	24	no	—
70	C₃²	—	12	108	no	—
71	C₃²	—	12	24	no	—
72	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
73	C₃²	C₃²×SL(2,𝔽₃)	1	12	yes	complemented
74	Q₈	C₃²×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	3	12	no	—
76	C₂×C₃	—	4	9	no	—
77	C₂×C₃	—	12	24	no	—
78	C₂×C₃	A₄×H(3,𝔽₃)	1	3	yes	—
79	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
80	C₄	—	3	1	no	—
81	C₃	—	3	12	no	—
82	C₃	—	4	9	no	—
83	C₃	—	12	24	no	—
84	C₃	SL(2,𝔽₃)×H(3,𝔽₃)	1	3	yes	direct product factor
85	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
86	C₂	C₃×A₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Frattini	C₂×C₃	C₃³×A₄
Fitting	C₃×Q₈×H(3,𝔽₃)	C₃
hypercenter	C₂×C₃×H(3,𝔽₃)	A₄
hypocenter	Q₈	C₃²×H(3,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃³×SL(2,𝔽₃)	C₃	semidirect
2. C₃×Q₈×H(3,𝔽₃)	C₃	semidirect
3. C₃×Q₈⋊H(3,𝔽₃)	C₃	semidirect
4. SL(2,𝔽₃)×H(3,𝔽₃)	C₃	direct (N × H)
8. C₃³×Q₈	C₃²	semidirect
11. C₃²×SL(2,𝔽₃)	C₃²	semidirect
12. C₃²×SL(2,𝔽₃)	C₃²	semidirect
13. Q₈×H(3,𝔽₃)	C₃²	semidirect
14. Q₈⋊H(3,𝔽₃)	C₃²	semidirect
24. C₃×H(3,𝔽₃)	SL(2,𝔽₃)	direct (N × H)
27. C₃²×Q₈	C₃³	semidirect
30. C₃×SL(2,𝔽₃)	H(3,𝔽₃)	direct (N × H)
32. C₃×SL(2,𝔽₃)	C₃³	semidirect
47. C₃³	C₃×SL(2,𝔽₃)	semidirect
49. H(3,𝔽₃)	C₃×SL(2,𝔽₃)	direct (N × H)
51. C₃×Q₈	C₃×H(3,𝔽₃)	semidirect
54. SL(2,𝔽₃)	C₃×H(3,𝔽₃)	direct (N × H)
73. C₃²	C₃²×SL(2,𝔽₃)	semidirect
74. Q₈	C₃²×H(3,𝔽₃)	semidirect
84. C₃	SL(2,𝔽₃)×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×SL(2,𝔽₃)	3
C₃×Q₈×H(3,𝔽₃)	3
C₃×Q₈⋊H(3,𝔽₃)	3
SL(2,𝔽₃)×H(3,𝔽₃)	3
C₂×C₃²×H(3,𝔽₃)	4

Subgroups — C3×SL(2,𝔽3)×H(3,𝔽3) #14518862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×SL(2,𝔽₃)×H(3,𝔽₃) #14518862