Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
268 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 87 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. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#107 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 6. C7×#466 | C3 | semidirect |
| 16. C2×C7×D8 | C2×C3 | semidirect |
| 17. C22×C7×C8 | C2×C3 | semidirect |
| 18. C7×#121 | C2×C3 | semidirect |
| 19. C7×#107 | C2×C3 | semidirect |
| 20. C7×#102 | C2×C3 | semidirect |
| 33. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 53. C22×ΓL(1,𝔽8) | D8 | semidirect |
| 59. #466 | ΓL(1,𝔽8) | direct (N × H) |
| 71. C7×C8 | C3×D4 | semidirect |
| 86. C2×D8 | C2×ΓL(1,𝔽8) | semidirect |
| 87. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 88. #121 | C2×ΓL(1,𝔽8) | semidirect |
| 89. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 90. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 96. C22×C7 | C3×D8 | semidirect |
| 114. ΓL(1,𝔽8) | #466 | direct (N × H) |
| 151. C8 | D4×ΓL(1,𝔽8) | semidirect |
| 152. C7 | C3×#466 | semidirect |
| 163. C22 | D8×ΓL(1,𝔽8) | 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 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#121 | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#466 | 3 |
| C3×#466 | 7 |