Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
320 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 81 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 |
|---|---|---|
| 2. ΓL(1,𝔽8)×#123 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#99 | C2 | semidirect |
| 4. C7×#458 | C3 | semidirect |
| 8. C22×C4×ΓL(1,𝔽8) | C22 | semidirect |
| 11. C22⋊C4×ΓL(1,𝔽8) | C22 | semidirect |
| 13. C7×#123 | C2×C3 | semidirect |
| 14. C7×#99 | C2×C3 | semidirect |
| 29. C22×C4×C7 | C22×C3 | semidirect |
| 32. C7×C22⋊C4 | C22×C3 | semidirect |
| 44. #458 | ΓL(1,𝔽8) | direct (N × H) |
| 66. #123 | C2×ΓL(1,𝔽8) | semidirect |
| 67. #99 | C2×ΓL(1,𝔽8) | semidirect |
| 86. ΓL(1,𝔽8) | #458 | direct (N × H) |
| 90. C22×C4 | C22×ΓL(1,𝔽8) | semidirect |
| 93. C22⋊C4 | C22×ΓL(1,𝔽8) | semidirect |
| 115. C7 | C3×#458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×H(5,𝔽2) | 2 |
| ΓL(1,𝔽8)×#123 | 2 |
| ΓL(1,𝔽8)×#99 | 2 |
| C7×#458 | 3 |
| C3×#458 | 7 |