Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
284 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 183 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 |
|---|---|---|
| 3. ΓL(1,𝔽8)×#97 | C2 | semidirect |
| 4. C4×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 5. C4×C7×M16 | C3 | semidirect |
| 8. M16×ΓL(1,𝔽8) | C4 | direct (N × H) |
| 15. C7×#97 | C2×C3 | semidirect |
| 16. C4×C7×C8 | C2×C3 | semidirect |
| 25. C8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 28. C7×M16 | C3×C4 | semidirect |
| 42. C4×ΓL(1,𝔽8) | M16 | direct (N × H) |
| 44. C4×M16 | ΓL(1,𝔽8) | direct (N × H) |
| 52. C7×C8 | C2×C3×C4 | semidirect |
| 65. #97 | C2×ΓL(1,𝔽8) | semidirect |
| 66. C4×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 72. C4×C7 | C3×M16 | semidirect |
| 82. ΓL(1,𝔽8) | C4×M16 | direct (N × H) |
| 85. M16 | C4×ΓL(1,𝔽8) | direct (N × H) |
| 107. C8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 108. C7 | C3×C4×M16 | semidirect |
| 117. C4 | M16×ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M16×ΓL(1,𝔽8) | 2 |
| C2×C42×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#97 | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C4×C7×M16 | 3 |
| C3×C4×M16 | 7 |