Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 86 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 | direct (N × H) |
| 3. C2×C4×C7×Q8 | C3 | semidirect |
| 5. C2×C7×SL(2,𝔽3) | C4 | direct (N × H) |
| 9. C4×C7×Q8 | C2×C3 | semidirect |
| 12. C2×C4×SL(2,𝔽3) | C7 | direct (N × H) |
| 16. C7×SL(2,𝔽3) | C2×C4 | direct (N × H) |
| 19. C2×C7×Q8 | C3×C4 | semidirect |
| 29. C4×SL(2,𝔽3) | C2×C7 | direct (N × H) |
| 35. C2×C4×Q8 | C3×C7 | semidirect |
| 40. C7×Q8 | C2×C3×C4 | semidirect |
| 47. C2×C4×C7 | SL(2,𝔽3) | direct (N × H) |
| 50. C2×SL(2,𝔽3) | C4×C7 | direct (N × H) |
| 58. C4×Q8 | C2×C3×C7 | semidirect |
| 69. C4×C7 | C2×SL(2,𝔽3) | direct (N × H) |
| 73. SL(2,𝔽3) | C2×C4×C7 | direct (N × H) |
| 77. C2×Q8 | C3×C4×C7 | semidirect |
| 86. C2×C7 | C4×SL(2,𝔽3) | direct (N × H) |
| 99. Q8 | C2×C3×C4×C7 | semidirect |
| 106. C2×C4 | C7×SL(2,𝔽3) | direct (N × H) |
| 108. C7 | C2×C4×SL(2,𝔽3) | direct (N × H) |
| 121. C4 | C2×C7×SL(2,𝔽3) | direct (N × H) |
| 123. C2 | C4×C7×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 |
|---|---|
| C22×C7×SL(2,𝔽3) | 2 |
| C4×C7×SL(2,𝔽3) | 2 |
| C2×C4×C7×Q8 | 3 |
| C22×C3×C4×C7 | 4 |
| C2×C4×SL(2,𝔽3) | 7 |