Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
182 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 182 conjugacy classes of subgroups — too many to draw. Only its 34 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×C7×GL(2,𝔽3) | C2 | semidirect |
| 2. C4×C7×SL(2,𝔽3) | C2 | semidirect |
| 9. C4×C7×Q8 | D3 | semidirect |
| 12. C4⋊GL(2,𝔽3) | C7 | direct (N × H) |
| 16. C7×SL(2,𝔽3) | D4 | semidirect |
| 26. C2×GL(2,𝔽3) | C2×C7 | semidirect |
| 27. C4×SL(2,𝔽3) | C2×C7 | semidirect |
| 36. C7×Q8 | D12 | semidirect |
| 54. C4×Q8 | D3×C7 | semidirect |
| 64. C4×C7 | GL(2,𝔽3) | semidirect |
| 68. SL(2,𝔽3) | C7×D4 | semidirect |
| 90. Q8 | C7×D12 | semidirect |
| 99. C7 | C4⋊GL(2,𝔽3) | direct (N × H) |
| 111. C4 | C7×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×GL(2,𝔽3) | 2 |
| C4×C7×SL(2,𝔽3) | 2 |
| C7×#460 | 3 |
| C2×C7×D12 | 4 |
| C4⋊GL(2,𝔽3) | 7 |