Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
316 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 123 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. C2×M16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 5. C7×#408 | C3 | semidirect |
| 18. C2×C7×M16 | C2×C3 | semidirect |
| 19. C22×C7×C8 | C2×C3 | semidirect |
| 20. C7×C22⋊C8 | C2×C3 | semidirect |
| 60. #408 | ΓL(1,𝔽8) | direct (N × H) |
| 91. C2×M16 | C2×ΓL(1,𝔽8) | semidirect |
| 92. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 93. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 117. ΓL(1,𝔽8) | #408 | direct (N × H) |
| 156. C7 | C3×#408 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×ΓL(1,𝔽8) | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#408 | 3 |
| C3×#408 | 7 |