Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 57 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. C4×C7×SL(2,𝔽3) | C2 | semidirect |
| 3. SL(2,𝔽3)×Dic7 | C2 | semidirect |
| 4. C4×Q8×D7 | C3 | semidirect |
| 7. D7×SL(2,𝔽3) | C4 | direct (N × H) |
| 9. C4×C7×Q8 | C2×C3 | semidirect |
| 10. Q8×Dic7 | C2×C3 | semidirect |
| 20. C7×SL(2,𝔽3) | C2×C4 | semidirect |
| 23. Q8×D7 | C3×C4 | semidirect |
| 37. C4×SL(2,𝔽3) | D7 | direct (N × H) |
| 47. C7×Q8 | C2×C3×C4 | semidirect |
| 57. C4×D7 | SL(2,𝔽3) | direct (N × H) |
| 71. C4×Q8 | C3×D7 | semidirect |
| 81. C4×C7 | C2×SL(2,𝔽3) | semidirect |
| 84. Dic7 | C2×SL(2,𝔽3) | semidirect |
| 89. SL(2,𝔽3) | C4×D7 | direct (N × H) |
| 108. D7 | C4×SL(2,𝔽3) | direct (N × H) |
| 119. Q8 | C3×C4×D7 | semidirect |
| 130. C7 | C2×C4×SL(2,𝔽3) | semidirect |
| 145. C4 | D7×SL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D7×SL(2,𝔽3) | 2 |
| C4×C7×SL(2,𝔽3) | 2 |
| SL(2,𝔽3)×Dic7 | 2 |
| C4×Q8×D7 | 3 |
| C2×C3×C4×D7 | 4 |
| C2×C4×SL(2,𝔽3) | 7 |