Subgroups — H(3,𝔽₃)×GL(2,𝔽₃) #13048363

Order 1296.

227 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₃)×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×GL(2,𝔽₃)	C₃	1	4	yes	maximal, complemented
3	SD₁₆×H(3,𝔽₃)	—	3	1	no	maximal
4	C₂×D₃×H(3,𝔽₃)	—	4	1	no	maximal
5	C₃²×SL(2,𝔽₃)	C₂×C₃	1	4	yes	complemented
6	Q₈×H(3,𝔽₃)	D₃	1	1	yes	characteristic, complemented
7	D₄×H(3,𝔽₃)	—	3	1	no	—
8	Q₈⋊H(3,𝔽₃)	—	2	4	no	—
9	C₈×H(3,𝔽₃)	—	3	1	no	—
10	C₂×C₃×H(3,𝔽₃)	—	4	1	no	—
11	D₃×H(3,𝔽₃)	—	4	2	no	—
12	C₃×GL(2,𝔽₃)	—	3	4	no	—
13	C₃×GL(2,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃²×SD₁₆	—	3	4	no	—
15	C₂×C₃²×D₃	—	4	4	no	—
16	C₂²×H(3,𝔽₃)	—	6	1	no	—
17	C₄×H(3,𝔽₃)	—	3	1	no	—
18	C₃×H(3,𝔽₃)	—	4	1	no	Sylow
19	C₃²×Q₈	C₃×D₃	1	4	yes	complemented
20	C₃²×D₄	—	3	4	no	—
21	C₃×SL(2,𝔽₃)	—	2	4	no	—
22	C₃×SL(2,𝔽₃)	—	3	4	no	—
23	C₃×SL(2,𝔽₃)	—	6	4	no	—
24	C₃×SL(2,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
25	C₃²×C₈	—	3	4	no	—
26	C₂×C₃³	—	4	4	no	—
27	C₃²×D₃	—	4	8	no	—
28	C₂×H(3,𝔽₃)	—	8	4	no	—
29	C₂×H(3,𝔽₃)	—	12	1	no	—
30	C₂×H(3,𝔽₃)	S₄	1	1	yes	characteristic
31	GL(2,𝔽₃)	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor
32	C₃×SD₁₆	—	3	1	no	—
33	C₃×SD₁₆	—	9	4	no	—
34	C₂²×C₃²	—	6	4	no	—
35	C₂×C₃×D₃	—	4	1	no	—
36	C₂×C₃×D₃	—	12	4	no	—
37	C₃²×C₄	—	3	4	no	—
38	C₃³	—	4	4	no	—
39	H(3,𝔽₃)	—	8	4	no	—
40	H(3,𝔽₃)	GL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
41	C₃×Q₈	—	3	4	no	—
42	C₃×Q₈	C₃²×D₃	1	1	yes	characteristic
43	C₃×D₄	—	3	1	no	—
44	C₃×D₄	—	9	4	no	—
45	SL(2,𝔽₃)	—	2	1	no	—
46	SL(2,𝔽₃)	—	6	4	no	—
47	SL(2,𝔽₃)	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
48	C₃×C₈	—	3	1	no	—
49	C₃×C₈	—	9	4	no	—
50	C₂×C₃²	—	4	1	no	—
51	C₂×C₃²	—	8	4	no	—
52	C₂×C₃²	—	12	4	no	—
53	C₂×C₃²	—	12	4	no	—
54	C₂×C₃²	—	24	4	no	—
55	C₂×C₃²	C₃×S₄	1	4	yes	—
56	C₃×D₃	—	4	2	no	—
57	C₃×D₃	—	12	8	no	—
58	SD₁₆	—	3	1	no	Sylow
59	C₂²×C₃	—	6	1	no	—
60	C₂²×C₃	—	18	4	no	—
61	C₂×D₃	—	4	1	no	—
62	C₃×C₄	—	3	1	no	—
63	C₃×C₄	—	9	4	no	—
64	C₃²	—	4	1	no	—
65	C₃²	—	8	4	no	—
66	C₃²	—	12	4	no	—
67	C₃²	—	24	4	no	—
68	C₃²	C₃×GL(2,𝔽₃)	1	4	yes	complemented
69	Q₈	D₃×H(3,𝔽₃)	1	1	yes	characteristic, complemented
70	D₄	—	3	1	no	—
71	C₈	—	3	1	no	—
72	C₂×C₃	—	3	4	no	—
73	C₂×C₃	—	4	1	no	—
74	C₂×C₃	—	8	1	no	—
75	C₂×C₃	—	12	1	no	—
76	C₂×C₃	—	24	4	no	—
77	C₂×C₃	—	36	4	no	—
78	C₂×C₃	C₃²×S₄	1	1	yes	characteristic
79	D₃	—	4	2	no	—
80	C₂²	—	6	1	no	—
81	C₄	—	3	1	no	—
82	C₃	—	3	4	no	—
83	C₃	—	4	1	no	—
84	C₃	—	8	1	no	—
85	C₃	—	24	4	no	—
86	C₃	C₃²×GL(2,𝔽₃)	1	1	yes	characteristic
87	C₂	—	12	1	no	—
88	C₂	S₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃²×S₄
commutator, cosocle	C₃×SL(2,𝔽₃)	C₂×C₃²
Fitting	Q₈×H(3,𝔽₃)	D₃
hypercenter	C₂×H(3,𝔽₃)	S₄
hypocenter	SL(2,𝔽₃)	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. SL(2,𝔽₃)×H(3,𝔽₃)	C₂	semidirect
2. C₃²×GL(2,𝔽₃)	C₃	semidirect
5. C₃²×SL(2,𝔽₃)	C₂×C₃	semidirect
6. Q₈×H(3,𝔽₃)	D₃	semidirect
19. C₃²×Q₈	C₃×D₃	semidirect
31. GL(2,𝔽₃)	H(3,𝔽₃)	direct (N × H)
40. H(3,𝔽₃)	GL(2,𝔽₃)	direct (N × H)
47. SL(2,𝔽₃)	C₂×H(3,𝔽₃)	semidirect
68. C₃²	C₃×GL(2,𝔽₃)	semidirect
69. Q₈	D₃×H(3,𝔽₃)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
SL(2,𝔽₃)×H(3,𝔽₃)	2
C₃²×GL(2,𝔽₃)	3
SD₁₆×H(3,𝔽₃)	3
C₂×D₃×H(3,𝔽₃)	4

Subgroups — H(3,𝔽3)×GL(2,𝔽3) #13048363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×GL(2,𝔽₃) #13048363