Subgroups — C₃×C₄×ΓL(1,𝔽₈) #6980

Order 252.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₄×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
4	C₃×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, direct product factor, Hall
5	C₂×C₃×C₇	C₂×C₃	1	1	yes	characteristic
6	C₂×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	—
7	C₃²×C₄	—	7	1	no	maximal, Hall
8	C₄×C₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	C₃×C₇	C₃×C₄	1	1	yes	characteristic, complemented
10	ΓL(1,𝔽₈)	C₃×C₄	1	3	yes	direct product factor
11	C₂×C₃²	—	7	1	no	—
12	C₂×C₇	C₂×C₃²	1	1	yes	characteristic
13	C₃×C₄	—	7	3	no	—
14	C₃×C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
15	C₃²	—	7	1	no	Sylow
16	C₇	C₃²×C₄	1	1	yes	characteristic, complemented, Sylow
17	C₂×C₃	—	7	3	no	—
18	C₂×C₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
19	C₄	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
20	C₃	—	7	3	no	—
21	C₃	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
22	C₂	C₂×C₃×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.