Subgroups — Q₈×AGL(1,𝔽₇) #66658

Order 336.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×AGL(1,𝔽₇)	C₂	1	3	yes	maximal
3	ΓL(1,𝔽₈)⋊Q₈	C₂	1	3	yes	maximal, complemented
4	Q₈×D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
5	C₂×AGL(1,𝔽₇)	C₂²	1	1	yes	characteristic
6	C₄×ΓL(1,𝔽₈)	C₂²	1	3	yes	—
7	ΓL(1,𝔽₈)⋊C₄	C₂²	1	3	yes	—
8	C₇×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
9	C₄×D₇	C₂×C₃	1	3	yes	—
10	Dic₁₄	C₂×C₃	1	3	yes	complemented
11	C₂×C₃×Q₈	—	7	1	no	maximal, Hall
12	C₂×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
13	AGL(1,𝔽₇)	Q₈	1	2	yes	direct product factor
14	C₂×D₇	C₂²×C₃	1	1	yes	characteristic
15	C₄×C₇	C₂²×C₃	1	3	yes	—
16	Dic₇	C₂²×C₃	1	3	yes	—
17	C₃×Q₈	—	7	3	no	—
18	C₃×Q₈	—	7	1	no	—
19	C₂×C₃×C₄	—	7	3	no	—
20	ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
21	C₂×Q₈	—	7	1	no	Sylow
22	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
23	D₇	C₃×Q₈	1	2	yes	complemented
24	C₂²×C₃	—	7	1	no	—
25	C₃×C₄	—	7	3	no	—
26	C₃×C₄	—	7	3	no	—
27	Q₈	—	7	3	no	—
28	Q₈	AGL(1,𝔽₇)	1	1	yes	characteristic, direct product factor
29	C₂×C₄	—	7	3	no	—
30	C₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
31	C₂×C₃	—	7	2	no	—
32	C₂×C₃	—	7	1	no	—
33	C₂²	—	7	1	no	—
34	C₄	—	7	3	no	—
35	C₄	C₂×AGL(1,𝔽₇)	1	3	yes	—
36	C₃	—	7	1	no	Sylow
37	C₂	—	7	2	no	—
38	C₂	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic

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

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