Subgroups — M₁₆×ΓL(1,𝔽₈) #66583

Order 336.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
2	C₈×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal, complemented
3	C₇×M₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
5	C₄×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic
6	C₄×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic
8	C₇×C₈	C₂×C₃	1	2	yes	complemented
9	C₃×M₁₆	—	7	1	no	maximal, Hall
10	C₂×ΓL(1,𝔽₈)	—	2	1	no	—
11	C₂×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₇	C₃×C₄	1	1	yes	characteristic
13	C₄×C₇	C₃×C₄	1	1	yes	characteristic
14	C₄×C₇	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×C₄	—	7	1	no	—
16	C₃×C₈	—	7	2	no	—
17	ΓL(1,𝔽₈)	M₁₆	1	1	yes	characteristic, direct product factor, Hall
18	M₁₆	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
19	C₂×C₇	—	2	1	no	—
20	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂²×C₃	—	7	1	no	—
22	C₃×C₄	—	7	1	no	—
23	C₃×C₄	—	7	1	no	—
24	C₂×C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
25	C₈	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
26	C₇	C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃	—	7	1	no	—
28	C₂×C₃	—	14	1	no	—
29	C₂²	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
30	C₄	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
31	C₄	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
32	C₃	—	7	1	no	Sylow
33	C₂	—	2	1	no	—
34	C₂	C₂×C₄×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.