Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
192 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 80 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)×C4⋊C8 | C2 | semidirect |
| 5. #6389⋊C3 | C2 | semidirect |
| 6. C8×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 7. C4⋊(ΓL(1,𝔽8)⋊C8) | C2 | semidirect |
| 8. #89612 | C3 | semidirect |
| 21. C7×C4⋊C8 | C2×C3 | semidirect |
| 24. #6389 | C2×C3 | semidirect |
| 25. C8×Dic7 | C2×C3 | semidirect |
| 26. C4⋊(C7⋊C8) | C2×C3 | semidirect |
| 101. C4⋊C8 | AGL(1,𝔽7) | semidirect |
| 130. ΓL(1,𝔽8) | #433 | semidirect |
| 170. C7 | C3×#433 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6435⋊C3 | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| (C2×AGL(1,𝔽7))⋊C8 | 2 |
| #6389⋊C3 | 2 |
| C8×ΓL(1,𝔽8)⋊C4 | 2 |
| C4⋊(ΓL(1,𝔽8)⋊C8) | 2 |
| #89612 | 3 |
| C3×#433 | 7 |