Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
492 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 192 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×C42×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | C2 | semidirect |
| 4. #10620712 | C2 | direct (N × H) |
| 5. C2×#6439 | C3 | semidirect |
| 9. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 11. (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | semidirect |
| 12. C2×C42×C7 | C2×C3 | semidirect |
| 14. C2×Dic7⋊C4 | C2×C3 | semidirect |
| 15. #6439 | C2×C3 | semidirect |
| 27. C42×C7 | C22×C3 | semidirect |
| 29. Dic7⋊C4 | C22×C3 | semidirect |
| 54. C2×ΓL(1,𝔽8) | #126 | semidirect |
| 60. C2×C42 | AGL(1,𝔽7) | semidirect |
| 74. ΓL(1,𝔽8) | C2×#126 | semidirect |
| 80. C42 | C2×AGL(1,𝔽7) | semidirect |
| 81. C2×C7 | C3×#126 | semidirect |
| 97. C7 | C2×C3×#126 | semidirect |
| 109. C2 | #10620712 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×ΓL(1,𝔽8) | 2 |
| C2×(C2×AGL(1,𝔽7))⋊C4 | 2 |
| C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | 2 |
| #10620712 | 2 |
| C2×#6439 | 3 |
| C2×C3×#126 | 7 |