Subgroups — C₃×ΓL(1,𝔽₈)⋊Q₈ #92649

Order 504.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₃×ΓL(1,𝔽₈)⋊C₄	C₂	1	2	yes	maximal
3	C₃×Dic₁₄	C₃	1	1	yes	maximal, characteristic, complemented
4	ΓL(1,𝔽₈)⋊Q₈	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₇	C₂×C₃	1	1	yes	characteristic
7	C₃×Dic₇	C₂×C₃	1	2	yes	—
8	C₄×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	—
9	ΓL(1,𝔽₈)⋊C₄	C₂×C₃	1	6	yes	—
10	C₃²×Q₈	—	7	1	no	maximal, Hall
11	C₃×ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic, complemented, Hall
12	Dic₁₄	C₃²	1	1	yes	characteristic, complemented, Hall
13	C₂×C₃×C₇	C₂²×C₃	1	1	yes	characteristic
14	C₂×ΓL(1,𝔽₈)	C₂²×C₃	1	3	yes	—
15	C₃²×C₄	—	7	2	no	—
16	C₃²×C₄	—	7	1	no	—
17	C₄×C₇	C₂×C₃²	1	1	yes	characteristic
18	Dic₇	C₂×C₃²	1	2	yes	—
19	C₃×Q₈	—	7	3	no	—
20	C₃×Q₈	—	21	1	no	—
21	C₃×C₇	C₃×Q₈	1	1	yes	characteristic, complemented
22	ΓL(1,𝔽₈)	C₃×Q₈	1	3	yes	complemented
23	C₂×C₃²	—	7	1	no	—
24	C₂×C₇	C₂²×C₃²	1	1	yes	characteristic
25	C₃×C₄	—	7	6	no	—
26	C₃×C₄	—	7	3	no	—
27	C₃×C₄	—	21	2	no	—
28	C₃×C₄	AGL(1,𝔽₇)	1	1	yes	characteristic
29	C₃²	—	7	1	no	Sylow
30	Q₈	—	21	1	no	Sylow
31	C₇	C₃²×Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂×C₃	—	7	3	no	—
33	C₂×C₃	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
34	C₄	—	21	2	no	—
35	C₄	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic
36	C₃	—	7	3	no	—
37	C₃	ΓL(1,𝔽₈)⋊Q₈	1	1	yes	characteristic, direct product factor
38	C₂	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₇	C₂²×C₃²
Frattini	C₂	C₂×C₃×AGL(1,𝔽₇)
Fitting	C₃×C₄×C₇	C₂×C₃
socle	C₂×C₃×C₇	C₂²×C₃
prime radical	C₂×C₃×ΓL(1,𝔽₈)	C₂²
hypercenter	C₃×C₄	AGL(1,𝔽₇)
hypocenter	C₇	C₃²×Q₈
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×Dic₁₄	C₃	semidirect
4. ΓL(1,𝔽₈)⋊Q₈	C₃	direct (N × H)
11. C₃×ΓL(1,𝔽₈)	Q₈	semidirect
12. Dic₁₄	C₃²	semidirect
21. C₃×C₇	C₃×Q₈	semidirect
22. ΓL(1,𝔽₈)	C₃×Q₈	semidirect
31. C₇	C₃²×Q₈	semidirect
37. C₃	ΓL(1,𝔽₈)⋊Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×ΓL(1,𝔽₈)	2
C₃×ΓL(1,𝔽₈)⋊C₄	2
C₃×Dic₁₄	3
ΓL(1,𝔽₈)⋊Q₈	3
C₃²×Q₈	7

Subgroups — C3×ΓL(1,𝔽8)⋊Q8 #92649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)⋊Q₈ #92649