Subgroups — C₃×D₃×ΓL(1,𝔽₈) #67395

Order 378.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×ΓL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₃×D₃×C₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×ΓL(1,𝔽₈)	—	3	1	no	maximal
4	D₃×ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
5	C₃²×C₇	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×ΓL(1,𝔽₈)	—	2	1	no	—
7	C₃×ΓL(1,𝔽₈)	—	2	3	no	—
8	C₃×ΓL(1,𝔽₈)	D₃	1	1	yes	characteristic, direct product factor
9	C₃×ΓL(1,𝔽₈)	C₂×C₃	1	3	yes	complemented
10	C₃²×D₃	—	7	1	no	maximal, Hall
11	C₂×C₃×C₇	—	3	1	no	—
12	D₃×C₇	C₃²	1	1	yes	characteristic, complemented
13	C₂×ΓL(1,𝔽₈)	—	3	3	no	—
14	C₃³	—	7	1	no	Sylow
15	C₃×C₇	—	2	1	no	—
16	C₃×C₇	C₃×D₃	1	1	yes	characteristic, complemented
17	C₃×C₇	C₂×C₃²	1	1	yes	characteristic, complemented
18	ΓL(1,𝔽₈)	—	2	3	no	—
19	ΓL(1,𝔽₈)	C₃×D₃	1	3	yes	direct product factor
20	C₂×C₃²	—	21	1	no	—
21	C₃×D₃	—	7	3	no	—
22	C₃×D₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
23	C₂×C₇	—	3	1	no	Hall
24	C₃²	—	7	1	no	—
25	C₃²	—	7	3	no	—
26	C₃²	—	14	3	no	—
27	C₃²	—	14	1	no	—
28	C₃²	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₇	C₃²×D₃	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₃	—	3	1	no	—
31	C₂×C₃	—	21	3	no	—
32	D₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
33	C₃	—	2	1	no	—
34	C₃	—	7	3	no	—
35	C₃	—	14	3	no	—
36	C₃	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
37	C₃	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
38	C₂	—	3	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	D₃×ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₃×C₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₇	C₂×C₃
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×ΓL(1,𝔽₈)	C₂	semidirect
2. C₃×D₃×C₇	C₃	semidirect
4. D₃×ΓL(1,𝔽₈)	C₃	direct (N × H)
5. C₃²×C₇	C₂×C₃	semidirect
8. C₃×ΓL(1,𝔽₈)	D₃	direct (N × H)
9. C₃×ΓL(1,𝔽₈)	C₂×C₃	semidirect
12. D₃×C₇	C₃²	semidirect
16. C₃×C₇	C₃×D₃	semidirect
17. C₃×C₇	C₂×C₃²	semidirect
19. ΓL(1,𝔽₈)	C₃×D₃	direct (N × H)
22. C₃×D₃	ΓL(1,𝔽₈)	direct (N × H)
28. C₃²	C₂×ΓL(1,𝔽₈)	semidirect
29. C₇	C₃²×D₃	semidirect
32. D₃	C₃×ΓL(1,𝔽₈)	direct (N × H)
36. C₃	D₃×ΓL(1,𝔽₈)	direct (N × H)
37. C₃	C₂×C₃×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
C₃²×ΓL(1,𝔽₈)	2
C₃×D₃×C₇	3
C₂×C₃×ΓL(1,𝔽₈)	3
D₃×ΓL(1,𝔽₈)	3
C₃²×D₃	7

Subgroups — C3×D3×ΓL(1,𝔽8) #67395

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×ΓL(1,𝔽₈) #67395