Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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. C2×M16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 6. C7×#469 | C3 | semidirect |
| 16. C2×C7×D8 | C2×C3 | semidirect |
| 17. C2×C7×M16 | C2×C3 | semidirect |
| 18. C7×#121 | C2×C3 | semidirect |
| 19. C7×#106 | C2×C3 | semidirect |
| 20. C7×#102 | C2×C3 | semidirect |
| 33. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 58. #469 | ΓL(1,𝔽8) | direct (N × H) |
| 70. C7×C8 | C3×D4 | semidirect |
| 84. C2×D8 | C2×ΓL(1,𝔽8) | semidirect |
| 85. C2×M16 | C2×ΓL(1,𝔽8) | semidirect |
| 86. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 87. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 88. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 111. ΓL(1,𝔽8) | #469 | direct (N × H) |
| 146. C8 | D4×ΓL(1,𝔽8) | semidirect |
| 147. C7 | C3×#469 | 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 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#469 | 3 |
| C3×#469 | 7 |