Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
268 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 111 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×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#107 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 5. C4×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C4×C7×D8 | C3 | semidirect |
| 9. D8×ΓL(1,𝔽8) | C4 | direct (N × H) |
| 16. C4×C7×D4 | C2×C3 | semidirect |
| 17. C7×#107 | C2×C3 | semidirect |
| 18. C7×#102 | C2×C3 | semidirect |
| 19. C4×C7×C8 | C2×C3 | semidirect |
| 23. D4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 30. C8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 33. C7×D8 | C3×C4 | semidirect |
| 50. C4×ΓL(1,𝔽8) | D8 | direct (N × H) |
| 53. C4×D8 | ΓL(1,𝔽8) | direct (N × H) |
| 56. C7×D4 | C2×C3×C4 | semidirect |
| 63. C7×C8 | C2×C3×C4 | semidirect |
| 77. C4×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 78. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 79. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 80. C4×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 87. C4×C7 | C3×D8 | semidirect |
| 100. ΓL(1,𝔽8) | C4×D8 | direct (N × H) |
| 103. D8 | C4×ΓL(1,𝔽8) | direct (N × H) |
| 124. D4 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 131. C8 | C2×C4×ΓL(1,𝔽8) | semidirect |
| 132. C7 | C3×C4×D8 | semidirect |
| 143. C4 | D8×ΓL(1,𝔽8) | direct (N × H) |
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 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C4×C7×D8 | 3 |
| C3×C4×D8 | 7 |