Subgroups — C₄⋊C₄×ΓL(1,𝔽₈) #66581

Order 336.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	2	yes	maximal
2	C₂×C₄×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊C₄	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
5	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
6	C₄×ΓL(1,𝔽₈)	C₄	1	2	yes	complemented
7	C₂×C₄×C₇	C₂×C₃	1	2	yes	—
8	C₂×C₄×C₇	C₂×C₃	1	1	yes	characteristic
9	C₃×C₄⋊C₄	—	7	1	no	maximal, Hall
10	C₂×ΓL(1,𝔽₈)	C₂×C₄	1	1	yes	characteristic
11	C₂×ΓL(1,𝔽₈)	D₄	1	1	yes	characteristic
12	C₂×ΓL(1,𝔽₈)	Q₈	1	1	yes	characteristic
13	C₂²×C₇	C₂²×C₃	1	1	yes	characteristic
14	C₄×C₇	—	2	2	no	—
15	C₄×C₇	C₃×C₄	1	2	yes	complemented
16	C₂×C₃×C₄	—	7	1	no	—
17	C₂×C₃×C₄	—	7	2	no	—
18	ΓL(1,𝔽₈)	C₄⋊C₄	1	1	yes	characteristic, direct product factor, Hall
19	C₄⋊C₄	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₇	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂×C₇	C₃×D₄	1	1	yes	characteristic
22	C₂×C₇	C₃×Q₈	1	1	yes	characteristic
23	C₂²×C₃	—	7	1	no	—
24	C₃×C₄	—	7	2	no	—
25	C₃×C₄	—	14	2	no	—
26	C₂×C₄	C₂×ΓL(1,𝔽₈)	1	2	yes	—
27	C₂×C₄	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₇	C₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃	—	7	1	no	—
30	C₂×C₃	—	7	1	no	—
31	C₂×C₃	—	7	1	no	—
32	C₂²	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic
33	C₄	—	2	2	no	—
34	C₄	C₄×ΓL(1,𝔽₈)	1	2	yes	complemented
35	C₃	—	7	1	no	Sylow
36	C₂	C₂×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
37	C₂	D₄×ΓL(1,𝔽₈)	1	1	yes	characteristic
38	C₂	Q₈×ΓL(1,𝔽₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×ΓL(1,𝔽₈)
commutator	C₂×C₇	C₂×C₃×C₄
Fitting	C₇×C₄⋊C₄	C₃
socle, cosocle	C₂²×C₇	C₂²×C₃
prime radical	C₂²×ΓL(1,𝔽₈)	C₂²
hypercenter	C₄⋊C₄	ΓL(1,𝔽₈)
hypocenter	C₇	C₃×C₄⋊C₄
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₇×C₄⋊C₄	C₃	semidirect
6. C₄×ΓL(1,𝔽₈)	C₄	semidirect
15. C₄×C₇	C₃×C₄	semidirect
18. ΓL(1,𝔽₈)	C₄⋊C₄	direct (N × H)
19. C₄⋊C₄	ΓL(1,𝔽₈)	direct (N × H)
28. C₇	C₃×C₄⋊C₄	semidirect
34. C₄	C₄×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×ΓL(1,𝔽₈)	2
C₇×C₄⋊C₄	3
C₃×C₄⋊C₄	7

Subgroups — C4⋊C4×ΓL(1,𝔽8) #66581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×ΓL(1,𝔽₈) #66581