Subgroups — D₄×ΓL(1,𝔽₈) #4250

Order 168.

32 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
2	C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂×ΓL(1,𝔽₈)	—	2	2	no	—
5	C₂×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
6	C₂²×C₇	C₂×C₃	1	2	yes	complemented
7	C₄×C₇	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×D₄	—	7	1	no	maximal, Hall
9	ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic, direct product factor, Hall
10	C₂×C₇	—	2	2	no	—
11	C₂×C₇	C₂²×C₃	1	1	yes	characteristic
12	C₂²×C₃	—	7	2	no	—
13	C₃×C₄	—	7	1	no	—
14	D₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
15	C₇	C₃×D₄	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃	—	7	1	no	—
17	C₂×C₃	—	14	2	no	—
18	C₂²	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
19	C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
20	C₃	—	7	1	no	Sylow
21	C₂	—	2	2	no	—
22	C₂	C₂²×ΓL(1,𝔽₈)	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.