Subgroups — Q₈⋊H(5,𝔽₃) #14518863

Order 1944.

935 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
3	SL(2,𝔽₃)×H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
4	C₂×H(5,𝔽₃)	—	4	1	no	maximal
5	C₃×C₄×H(3,𝔽₃)	—	3	1	no	—
6	H(5,𝔽₃)	—	4	1	no	Sylow
7	C₃³×Q₈	C₃²	1	4	yes	complemented
8	C₃²×SL(2,𝔽₃)	C₃²	1	36	yes	complemented
9	Q₈×H(3,𝔽₃)	C₃²	1	9	yes	complemented
10	Q₈⋊H(3,𝔽₃)	C₃²	1	72	yes	complemented
11	Q₈⋊H(3,𝔽₃)	C₃²	1	9	yes	complemented
12	C₂×C₃×H(3,𝔽₃)	—	4	12	no	—
13	C₂×C₃×H(3,𝔽₃)	—	4	27	no	—
14	C₂×C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
15	C₃³×C₄	—	3	4	no	—
16	C₄×H(3,𝔽₃)	—	3	9	no	—
17	C₃×H(3,𝔽₃)	—	4	12	no	—
18	C₃×H(3,𝔽₃)	—	4	27	no	—
19	C₃×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
20	C₃²×Q₈	—	9	4	no	—
21	C₃²×Q₈	C₃³	1	1	yes	characteristic
22	C₃²×Q₈	C₃³	1	12	yes	—
23	C₃×SL(2,𝔽₃)	—	9	36	no	—
24	C₃×SL(2,𝔽₃)	C₃³	1	27	yes	—
25	C₂×C₃³	—	4	36	no	—
26	C₂×C₃³	C₃×A₄	1	4	yes	—
27	C₂×H(3,𝔽₃)	—	4	9	no	—
28	C₂×H(3,𝔽₃)	—	4	72	no	—
29	C₂×H(3,𝔽₃)	C₃×A₄	1	9	yes	—
30	C₃²×C₄	—	3	1	no	—
31	C₃²×C₄	—	3	12	no	—
32	C₃²×C₄	—	9	12	no	—
33	C₃³	—	4	36	no	—
34	C₃³	C₃×SL(2,𝔽₃)	1	4	yes	complemented
35	H(3,𝔽₃)	—	4	9	no	—
36	H(3,𝔽₃)	—	4	72	no	—
37	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	9	yes	complemented
38	C₃×Q₈	—	3	1	no	—
39	C₃×Q₈	—	3	12	no	—
40	C₃×Q₈	C₃⁴	1	1	yes	characteristic
41	SL(2,𝔽₃)	—	3	27	no	—
42	C₂×C₃²	—	4	27	no	—
43	C₂×C₃²	—	9	4	no	—
44	C₂×C₃²	—	36	36	no	—
45	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
46	C₂×C₃²	C₃²×A₄	1	12	yes	—
47	C₃×C₄	—	3	3	no	—
48	C₃×C₄	—	3	1	no	—
49	C₃×C₄	—	9	12	no	—
50	C₃²	—	4	27	no	—
51	C₃²	—	9	4	no	—
52	C₃²	—	36	36	no	—
53	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
54	C₃²	C₃²×SL(2,𝔽₃)	1	12	yes	—
55	Q₈	H(5,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₃	—	3	1	no	—
57	C₂×C₃	—	3	12	no	—
58	C₂×C₃	—	12	27	no	—
59	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
60	C₄	—	3	1	no	—
61	C₃	—	3	1	no	—
62	C₃	—	3	12	no	—
63	C₃	—	12	27	no	—
64	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
65	C₂	C₂²⋊H(5,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃³×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Fitting	C₃×Q₈×H(3,𝔽₃)	C₃
hypercenter	C₂×C₃×H(3,𝔽₃)	A₄
hypocenter	Q₈	H(5,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×H(3,𝔽₃)	C₃	semidirect
2. C₃×Q₈⋊H(3,𝔽₃)	C₃	semidirect
3. SL(2,𝔽₃)×H(3,𝔽₃)	C₃	semidirect
7. C₃³×Q₈	C₃²	semidirect
8. C₃²×SL(2,𝔽₃)	C₃²	semidirect
9. Q₈×H(3,𝔽₃)	C₃²	semidirect
10. Q₈⋊H(3,𝔽₃)	C₃²	semidirect
11. Q₈⋊H(3,𝔽₃)	C₃²	semidirect
19. C₃×H(3,𝔽₃)	SL(2,𝔽₃)	semidirect
34. C₃³	C₃×SL(2,𝔽₃)	semidirect
37. H(3,𝔽₃)	C₃×SL(2,𝔽₃)	semidirect
55. Q₈	H(5,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×H(3,𝔽₃)	3
C₃×Q₈⋊H(3,𝔽₃)	3
SL(2,𝔽₃)×H(3,𝔽₃)	3
C₂×H(5,𝔽₃)	4

Subgroups — Q8⋊H(5,𝔽3) #14518863

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊H(5,𝔽₃) #14518863