Subgroups — C₂×Q₈×ΓL(1,𝔽₈) #66697

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	4	yes	maximal, direct product factor
2	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	3	yes	maximal
3	C₂×C₇×Q₈	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
5	C₄×ΓL(1,𝔽₈)	C₂²	1	6	yes	—
6	C₇×Q₈	C₂×C₃	1	4	yes	complemented
7	C₂×C₄×C₇	C₂×C₃	1	3	yes	—
8	C₂×C₃×Q₈	—	7	1	no	maximal, Hall
9	C₂×ΓL(1,𝔽₈)	Q₈	1	2	yes	direct product factor
10	C₂×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
11	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
12	C₄×C₇	C₂²×C₃	1	6	yes	—
13	C₃×Q₈	—	7	4	no	—
14	C₂×C₃×C₄	—	7	3	no	—
15	ΓL(1,𝔽₈)	C₂×Q₈	1	1	yes	characteristic, direct product factor, Hall
16	C₂×Q₈	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
17	C₂×C₇	C₃×Q₈	1	2	yes	complemented
18	C₂×C₇	C₂³×C₃	1	1	yes	characteristic
19	C₂²×C₃	—	7	1	no	—
20	C₃×C₄	—	7	6	no	—
21	Q₈	C₂×ΓL(1,𝔽₈)	1	4	yes	direct product factor
22	C₂×C₄	C₂×ΓL(1,𝔽₈)	1	3	yes	—
23	C₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃	—	7	2	no	—
25	C₂×C₃	—	7	1	no	—
26	C₂²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
27	C₄	C₂²×ΓL(1,𝔽₈)	1	6	yes	—
28	C₃	—	7	1	no	Sylow
29	C₂	Q₈×ΓL(1,𝔽₈)	1	2	yes	direct product factor
30	C₂	C₂³×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.