Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C2×D8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#131 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#108 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#100 | C2 | semidirect |
| 6. C7×#472 | C3 | semidirect |
| 18. C7×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 19. C2×C7×D8 | C2×C3 | semidirect |
| 20. C7×#131 | C2×C3 | semidirect |
| 21. C7×#108 | C2×C3 | semidirect |
| 22. C7×#100 | C2×C3 | semidirect |
| 59. #472 | ΓL(1,𝔽8) | direct (N × H) |
| 87. AGL(1,ℤ/8) | C2×ΓL(1,𝔽8) | semidirect |
| 88. C2×D8 | C2×ΓL(1,𝔽8) | semidirect |
| 89. #131 | C2×ΓL(1,𝔽8) | semidirect |
| 90. #108 | C2×ΓL(1,𝔽8) | semidirect |
| 91. #100 | C2×ΓL(1,𝔽8) | semidirect |
| 111. ΓL(1,𝔽8) | #472 | direct (N × H) |
| 146. C7 | C3×#472 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×AGL(1,ℤ/8) | 2 |
| C2×D8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#131 | 2 |
| ΓL(1,𝔽8)×#108 | 2 |
| ΓL(1,𝔽8)×#100 | 2 |
| C7×#472 | 3 |
| C3×#472 | 7 |