Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
166 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 45 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. C8×C22⋊ΓL(1,𝔽8) | C2 | semidirect |
| 3. C22×C7×M16 | C3 | semidirect |
| 12. C22×C7×C8 | C2×C3 | semidirect |
| 28. C7×M16 | A4 | semidirect |
| 33. C22⋊ΓL(1,𝔽8) | M16 | direct (N × H) |
| 37. C22×M16 | ΓL(1,𝔽8) | semidirect |
| 53. C7×C8 | C2×A4 | semidirect |
| 64. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 72. C22×C7 | C3×M16 | semidirect |
| 93. M16 | C22⋊ΓL(1,𝔽8) | direct (N × H) |
| 120. C8 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 121. C7 | A4×M16 | semidirect |
| 131. C22 | M16×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×C22⋊ΓL(1,𝔽8) | 2 |
| C8×C22⋊ΓL(1,𝔽8) | 2 |
| C22×C7×M16 | 3 |
| M16×ΓL(1,𝔽8) | 4 |
| A4×M16 | 7 |