Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
176 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 31 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. SL(2,𝔽3)×Dic7 | C2 | semidirect |
| 3. C7⋊#859 | C2 | semidirect |
| 7. C7×GL(2,𝔽3) | C4 | semidirect |
| 9. Q8×Dic7 | D3 | semidirect |
| 19. C7×SL(2,𝔽3) | C2×C4 | semidirect |
| 44. C7×Q8 | C4×D3 | semidirect |
| 56. GL(2,𝔽3) | Dic7 | direct (N × H) |
| 76. Dic7 | GL(2,𝔽3) | direct (N × H) |
| 81. SL(2,𝔽3) | C2×Dic7 | semidirect |
| 106. Q8 | D3×Dic7 | semidirect |
| 116. C7 | C4×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 |
| SL(2,𝔽3)×Dic7 | 2 |
| C7⋊#859 | 2 |
| SD16×Dic7 | 3 |
| C2×D3×Dic7 | 4 |
| C4×GL(2,𝔽3) | 7 |