Subgroups — H(3,𝔽₃)×#226 #13048362

Order 1296.

194 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
2	C₃²×#226	C₃	1	4	yes	maximal, complemented
3	Q₁₆×H(3,𝔽₃)	—	3	1	no	maximal
4	Dic₃×H(3,𝔽₃)	—	4	1	no	maximal
5	C₃²×SL(2,𝔽₃)	C₂×C₃	1	4	yes	—
6	Q₈×H(3,𝔽₃)	—	3	1	no	—
7	Q₈×H(3,𝔽₃)	D₃	1	1	yes	characteristic
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	C₃×#226	—	3	4	no	—
12	C₃×#226	C₃²	1	1	yes	characteristic
13	C₃²×Q₁₆	—	3	4	no	—
14	C₃²×Dic₃	—	4	4	no	—
15	C₄×H(3,𝔽₃)	—	3	1	no	—
16	C₄×H(3,𝔽₃)	—	6	1	no	—
17	C₃×H(3,𝔽₃)	—	4	1	no	Sylow
18	C₃²×Q₈	—	3	4	no	—
19	C₃²×Q₈	C₃×D₃	1	4	yes	—
20	C₃×SL(2,𝔽₃)	—	2	4	no	—
21	C₃×SL(2,𝔽₃)	—	3	4	no	—
22	C₃×SL(2,𝔽₃)	—	6	4	no	—
23	C₃×SL(2,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
24	C₃²×C₈	—	3	4	no	—
25	C₂×C₃³	—	4	4	no	—
26	C₂×H(3,𝔽₃)	—	8	4	no	—
27	C₂×H(3,𝔽₃)	S₄	1	1	yes	characteristic
28	#226	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor
29	C₃×Q₁₆	—	3	1	no	—
30	C₃×Q₁₆	—	9	4	no	—
31	C₃²×C₄	—	3	4	no	—
32	C₃²×C₄	—	6	4	no	—
33	C₃×Dic₃	—	4	1	no	—
34	C₃×Dic₃	—	12	4	no	—
35	C₃³	—	4	4	no	—
36	H(3,𝔽₃)	—	8	4	no	—
37	H(3,𝔽₃)	#226	1	1	yes	characteristic, direct product factor
38	C₃×Q₈	—	3	1	no	—
39	C₃×Q₈	—	3	4	no	—
40	C₃×Q₈	—	9	4	no	—
41	C₃×Q₈	C₃²×D₃	1	1	yes	characteristic
42	SL(2,𝔽₃)	—	2	1	no	—
43	SL(2,𝔽₃)	—	6	4	no	—
44	SL(2,𝔽₃)	C₂×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₈	—	3	1	no	—
46	C₃×C₈	—	9	4	no	—
47	C₂×C₃²	—	4	1	no	—
48	C₂×C₃²	—	8	4	no	—
49	C₂×C₃²	—	12	4	no	—
50	C₂×C₃²	—	24	4	no	—
51	C₂×C₃²	C₃×S₄	1	4	yes	—
52	Q₁₆	—	3	1	no	Sylow
53	C₃×C₄	—	3	1	no	—
54	C₃×C₄	—	6	1	no	—
55	C₃×C₄	—	9	4	no	—
56	C₃×C₄	—	18	4	no	—
57	Dic₃	—	4	1	no	—
58	C₃²	—	4	1	no	—
59	C₃²	—	8	4	no	—
60	C₃²	—	12	4	no	—
61	C₃²	—	24	4	no	—
62	C₃²	C₃×#226	1	4	yes	complemented
63	Q₈	—	3	1	no	—
64	Q₈	D₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₈	—	3	1	no	—
66	C₂×C₃	—	3	4	no	—
67	C₂×C₃	—	4	1	no	—
68	C₂×C₃	—	8	1	no	—
69	C₂×C₃	—	24	4	no	—
70	C₂×C₃	C₃²×S₄	1	1	yes	characteristic
71	C₄	—	3	1	no	—
72	C₄	—	6	1	no	—
73	C₃	—	3	4	no	—
74	C₃	—	4	1	no	—
75	C₃	—	8	1	no	—
76	C₃	—	24	4	no	—
77	C₃	C₃²×#226	1	1	yes	characteristic
78	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
prime radical	SL(2,𝔽₃)×H(3,𝔽₃)	C₂
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
2. C₃²×#226	C₃	semidirect
28. #226	H(3,𝔽₃)	direct (N × H)
37. H(3,𝔽₃)	#226	direct (N × H)
62. C₃²	C₃×#226	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×#226	3
Q₁₆×H(3,𝔽₃)	3
Dic₃×H(3,𝔽₃)	4

Subgroups — H(3,𝔽3)×#226 #13048362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#226 #13048362