Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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. ΓL(1,𝔽8)×#117 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#107 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 7. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 8. C7×#482 | C3 | semidirect |
| 21. C7×#117 | C2×C3 | semidirect |
| 22. C7×#107 | C2×C3 | semidirect |
| 23. C7×#106 | C2×C3 | semidirect |
| 24. C7×#102 | C2×C3 | semidirect |
| 25. C7×#102 | C2×C3 | semidirect |
| 26. C7×C22⋊C8 | C2×C3 | semidirect |
| 73. #482 | ΓL(1,𝔽8) | direct (N × H) |
| 105. #117 | C2×ΓL(1,𝔽8) | semidirect |
| 106. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 107. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 108. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 109. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 110. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 138. ΓL(1,𝔽8) | #482 | direct (N × H) |
| 182. C7 | C3×#482 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#121 | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#482 | 3 |
| C3×#482 | 7 |