Subgroups — C₃×D₃×AGL(1,𝔽₈) #11759727

Order 1008.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×AGL(1,𝔽₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×AGL(1,𝔽₈)	—	3	1	no	maximal
3	D₃×AGL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×AGL(1,𝔽₈)	—	2	1	no	—
5	C₃×AGL(1,𝔽₈)	D₃	1	1	yes	characteristic, direct product factor
6	C₃×AGL(1,𝔽₈)	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂³×C₃×D₃	C₇	1	1	yes	maximal, characteristic, complemented, Hall
8	C₃×D₃×C₇	—	8	1	no	maximal
9	C₂×AGL(1,𝔽₈)	—	3	1	no	Hall
10	C₂³×C₃²	C₂×C₇	1	1	yes	characteristic, complemented
11	C₂²×C₃×D₃	—	7	1	no	—
12	C₂²×C₃×D₃	—	7	1	no	—
13	C₃²×C₇	—	8	1	no	Hall
14	AGL(1,𝔽₈)	C₃×D₃	1	1	yes	characteristic, direct product factor
15	C₂⁴×C₃	—	3	1	no	—
16	C₂³×D₃	C₃×C₇	1	1	yes	characteristic, complemented
17	C₂×C₃×C₇	—	24	1	no	—
18	D₃×C₇	—	8	1	no	—
19	C₂²×C₃²	—	7	1	no	—
20	C₂×C₃×D₃	—	7	1	no	—
21	C₂×C₃×D₃	—	7	3	no	—
22	C₂³×C₃	—	2	1	no	—
23	C₂³×C₃	—	21	1	no	—
24	C₂³×C₃	—	21	1	no	—
25	C₂³×C₃	D₃×C₇	1	1	yes	characteristic, complemented
26	C₂³×C₃	C₂×C₃×C₇	1	1	yes	characteristic, complemented
27	C₂²×D₃	—	7	1	no	—
28	C₂²×D₃	—	7	1	no	—
29	C₃×C₇	—	8	1	no	—
30	C₃×C₇	—	8	1	no	—
31	C₃×C₇	—	16	1	no	—
32	C₂×C₃²	—	7	1	no	—
33	C₃×D₃	—	7	1	no	—
34	C₃×D₃	AGL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
35	C₂⁴	—	3	1	no	Sylow
36	C₂×C₇	—	24	1	no	—
37	C₂²×C₃	—	7	1	no	—
38	C₂²×C₃	—	7	1	no	—
39	C₂²×C₃	—	14	1	no	—
40	C₂²×C₃	—	21	1	no	—
41	C₂²×C₃	—	21	3	no	—
42	C₂×D₃	—	7	1	no	—
43	C₂×D₃	—	7	3	no	—
44	C₃²	C₂×AGL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
45	C₂³	—	21	1	no	—
46	C₂³	—	21	1	no	—
47	C₂³	C₃×D₃×C₇	1	1	yes	characteristic, complemented
48	C₇	—	8	1	no	Sylow
49	C₂×C₃	—	3	1	no	—
50	C₂×C₃	—	7	1	no	—
51	C₂×C₃	—	7	1	no	—
52	C₂×C₃	—	14	1	no	—
53	C₂×C₃	—	21	1	no	—
54	D₃	—	7	1	no	—
55	D₃	C₃×AGL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
56	C₂²	—	7	1	no	—
57	C₂²	—	21	1	no	—
58	C₂²	—	21	3	no	—
59	C₃	—	2	1	no	—
60	C₃	D₃×AGL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
61	C₃	C₂×C₃×AGL(1,𝔽₈)	1	1	yes	characteristic, complemented
62	C₂	—	3	1	no	—
63	C₂	—	7	1	no	—
64	C₂	—	21	1	no	—

	subgroup	G/H
center, hypercenter	C₃	D₃×AGL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂³×C₃	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂³×C₃²	C₂×C₇
LCS	G ▷ C₂³×C₃
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. C₃²×AGL(1,𝔽₈)	C₂	semidirect
3. D₃×AGL(1,𝔽₈)	C₃	direct (N × H)
5. C₃×AGL(1,𝔽₈)	D₃	direct (N × H)
6. C₃×AGL(1,𝔽₈)	C₂×C₃	semidirect
7. C₂³×C₃×D₃	C₇	semidirect
10. C₂³×C₃²	C₂×C₇	semidirect
14. AGL(1,𝔽₈)	C₃×D₃	direct (N × H)
16. C₂³×D₃	C₃×C₇	semidirect
25. C₂³×C₃	D₃×C₇	semidirect
26. C₂³×C₃	C₂×C₃×C₇	semidirect
34. C₃×D₃	AGL(1,𝔽₈)	direct (N × H)
44. C₃²	C₂×AGL(1,𝔽₈)	semidirect
47. C₂³	C₃×D₃×C₇	semidirect
55. D₃	C₃×AGL(1,𝔽₈)	direct (N × H)
60. C₃	D₃×AGL(1,𝔽₈)	direct (N × H)
61. C₃	C₂×C₃×AGL(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×AGL(1,𝔽₈)	2
C₂×C₃×AGL(1,𝔽₈)	3
D₃×AGL(1,𝔽₈)	3
C₂³×C₃×D₃	7
C₃×D₃×C₇	8

Subgroups — C3×D3×AGL(1,𝔽8) #11759727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×AGL(1,𝔽₈) #11759727