Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
400 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 249 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 |
|---|---|---|
| 6. C7×#557 | C3 | semidirect |
| 23. Q8×ΓL(1,𝔽8) | Q8 | semidirect |
| 49. #557 | ΓL(1,𝔽8) | direct (N × H) |
| 51. C7×Q8 | C3×Q8 | semidirect |
| 87. ΓL(1,𝔽8) | #557 | direct (N × H) |
| 107. Q8 | Q8×ΓL(1,𝔽8) | semidirect |
| 113. C7 | C3×#557 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| ΓL(1,𝔽8)×#125 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| C7×#557 | 3 |
| C3×#557 | 7 |