Subgroups — C₂²×ΓL(1,𝔽₈)⋊C₄ #66660

Order 336.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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