Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
228 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 228 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 |
|---|---|---|
| 2. C2×C4⋊C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 6. C7×#481 | C3 | semidirect |
| 17. C2×C7×C4⋊C4 | C2×C3 | semidirect |
| 18. C7×#106 | C2×C3 | semidirect |
| 19. C7×#102 | C2×C3 | semidirect |
| 20. C7×C22⋊C8 | C2×C3 | semidirect |
| 58. #481 | ΓL(1,𝔽8) | direct (N × H) |
| 85. C2×C4⋊C4 | C2×ΓL(1,𝔽8) | semidirect |
| 86. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 87. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 88. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 111. ΓL(1,𝔽8) | #481 | direct (N × H) |
| 148. C7 | C3×#481 | 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 |
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#481 | 3 |
| C3×#481 | 7 |