Subgroups — Q₈×C₂²⋊ΓL(1,𝔽₈) #10621613

Order 672.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊ΓL(1,𝔽₈)	C₂	1	3	yes	maximal
2	C₂²×C₇×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
4	Q₈×ΓL(1,𝔽₈)	—	4	1	no	maximal
5	C₂×C₇×Q₈	—	3	1	no	—
6	C₂×C₇×Q₈	—	3	3	no	—
7	C₂²×C₄×C₇	C₂×C₃	1	3	yes	—
8	Q₈×A₄	—	7	1	no	maximal, Hall
9	C₂²⋊ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic, direct product factor
10	C₄×ΓL(1,𝔽₈)	—	4	3	no	—
11	C₂³×C₇	C₂²×C₃	1	1	yes	characteristic
12	C₇×Q₈	—	3	2	no	—
13	C₇×Q₈	—	3	3	no	—
14	C₇×Q₈	A₄	1	1	yes	characteristic, complemented
15	C₂×C₄×C₇	—	3	3	no	—
16	C₂×C₄×C₇	—	3	3	no	—
17	C₄×A₄	—	7	3	no	—
18	C₂×ΓL(1,𝔽₈)	—	4	1	no	—
19	C₂²×Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
20	C₂²×C₇	—	3	1	no	—
21	C₂²×C₇	—	3	1	no	—
22	C₂²×C₇	C₃×Q₈	1	1	yes	characteristic, complemented
23	C₄×C₇	—	3	3	no	—
24	C₄×C₇	C₂×A₄	1	3	yes	—
25	C₂×A₄	—	7	1	no	—
26	C₃×Q₈	—	28	1	no	—
27	ΓL(1,𝔽₈)	—	4	1	no	Hall
28	C₂×Q₈	—	3	3	no	—
29	C₂×Q₈	—	3	1	no	—
30	C₂²×C₄	C₂×ΓL(1,𝔽₈)	1	3	yes	—
31	C₂×C₇	—	3	1	no	—
32	C₂×C₇	—	3	1	no	—
33	C₂×C₇	C₂²×A₄	1	1	yes	characteristic
34	A₄	—	7	1	no	—
35	C₃×C₄	—	28	3	no	—
36	C₂³	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	Q₈	—	3	2	no	—
38	Q₈	—	3	3	no	—
39	Q₈	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
40	C₂×C₄	—	3	3	no	—
41	C₂×C₄	—	3	3	no	—
42	C₇	Q₈×A₄	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₃	—	28	1	no	—
44	C₂²	—	3	1	no	—
45	C₂²	—	3	1	no	—
46	C₂²	Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
47	C₄	—	3	3	no	—
48	C₄	C₂×C₂²⋊ΓL(1,𝔽₈)	1	3	yes	—
49	C₃	—	28	1	no	Sylow
50	C₂	—	3	1	no	—
51	C₂	—	3	1	no	—
52	C₂	C₂²×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₂²⋊ΓL(1,𝔽₈)
commutator, socle, cosocle	C₂³×C₇	C₂²×C₃
Fitting	C₂²×C₇×Q₈	C₃
prime radical	C₂×C₂²⋊ΓL(1,𝔽₈)	C₂²
hypercenter	Q₈	C₂²⋊ΓL(1,𝔽₈)
hypocenter	C₂²×C₇	C₃×Q₈
LCS	G ▷ C₂³×C₇ ▷ C₂²×C₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×Q₈	C₃	semidirect
9. C₂²⋊ΓL(1,𝔽₈)	Q₈	direct (N × H)
14. C₇×Q₈	A₄	semidirect
19. C₂²×Q₈	ΓL(1,𝔽₈)	semidirect
22. C₂²×C₇	C₃×Q₈	semidirect
39. Q₈	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
42. C₇	Q₈×A₄	semidirect
46. C₂²	Q₈×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊ΓL(1,𝔽₈)	2
C₂²×C₇×Q₈	3
Q₈×ΓL(1,𝔽₈)	4
Q₈×A₄	7

Subgroups — Q8×C22⋊ΓL(1,𝔽8) #10621613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₂²⋊ΓL(1,𝔽₈) #10621613