Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
288 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 72 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×D8×ΓL(1,𝔽8) | C2 | semidirect |
| 2. #6503⋊C3 | C2 | semidirect |
| 3. #6499⋊C3 | C2 | semidirect |
| 4. C2×#1142⋊C3 | C2 | semidirect |
| 5. C2×#1141⋊C3 | C2 | semidirect |
| 6. C2×#1132⋊C3 | C2 | semidirect |
| 7. #6389⋊C3 | C2 | semidirect |
| 8. #89926 | C3 | semidirect |
| 22. C2×C7×D8 | C2×C3 | semidirect |
| 23. #6503 | C2×C3 | semidirect |
| 24. #6499 | C2×C3 | semidirect |
| 25. C2×#1142 | C2×C3 | semidirect |
| 26. C2×#1141 | C2×C3 | semidirect |
| 27. C2×#1132 | C2×C3 | semidirect |
| 28. #6389 | C2×C3 | semidirect |
| 45. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 46. ΓL(1,𝔽8)⋊C8 | D4 | semidirect |
| 94. C7×C8 | C3×D4 | semidirect |
| 95. C7⋊C8 | C3×D4 | semidirect |
| 117. C2×D8 | AGL(1,𝔽7) | semidirect |
| 149. ΓL(1,𝔽8) | #496 | semidirect |
| 195. C8 | (C22⋊D7)⋊C3 | semidirect |
| 196. C7 | C3×#496 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D8×ΓL(1,𝔽8) | 2 |
| #6503⋊C3 | 2 |
| #6499⋊C3 | 2 |
| C2×#1142⋊C3 | 2 |
| C2×#1141⋊C3 | 2 |
| C2×#1132⋊C3 | 2 |
| #6389⋊C3 | 2 |
| #89926 | 3 |
| C3×#496 | 7 |