Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
288 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 76 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×D56⋊C3 | C2 | semidirect |
| 2. C2×#1132⋊C3 | C2 | semidirect |
| 3. #6438⋊C3 | C2 | semidirect |
| 4. #10620710 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#97 | C2 | semidirect |
| 6. #89484 | C3 | semidirect |
| 16. C2×D56 | C2×C3 | semidirect |
| 17. C2×#1132 | C2×C3 | semidirect |
| 18. #6438 | C2×C3 | semidirect |
| 19. #6437 | C2×C3 | semidirect |
| 20. C7×#97 | C2×C3 | semidirect |
| 32. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 66. C7×C8 | C3×D4 | semidirect |
| 84. #97 | AGL(1,𝔽7) | semidirect |
| 104. ΓL(1,𝔽8) | #496 | semidirect |
| 136. C8 | C4⋊AGL(1,𝔽7) | semidirect |
| 137. C7 | C3×#496 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D56⋊C3 | 2 |
| C2×#1132⋊C3 | 2 |
| #6438⋊C3 | 2 |
| #10620710 | 2 |
| ΓL(1,𝔽8)×#97 | 2 |
| #89484 | 3 |
| C3×#496 | 7 |