Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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. C2×C4⋊C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#95 | C2 | semidirect |
| 4. C7×#396 | C3 | semidirect |
| 10. C2×C7×C4⋊C4 | C2×C3 | semidirect |
| 12. C7×#95 | C2×C3 | semidirect |
| 40. #396 | ΓL(1,𝔽8) | direct (N × H) |
| 60. C2×C4⋊C4 | C2×ΓL(1,𝔽8) | semidirect |
| 62. #95 | C2×ΓL(1,𝔽8) | semidirect |
| 81. ΓL(1,𝔽8) | #396 | direct (N × H) |
| 109. C7 | C3×#396 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| C2×C22⋊C4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#95 | 2 |
| C7×#396 | 3 |
| C3×#396 | 7 |