Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
452 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 216 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 |
|---|---|---|
| 6. #6453⋊C3 | C2 | direct (N × H) |
| 7. C2×#6453 | C3 | semidirect |
| 23. #6453 | C2×C3 | semidirect |
| 83. C2×ΓL(1,𝔽8) | #125 | semidirect |
| 113. ΓL(1,𝔽8) | C2×#125 | semidirect |
| 124. C2×C7 | C3×#125 | semidirect |
| 148. C7 | C2×C3×#125 | semidirect |
| 163. C2 | #6453⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | 2 |
| C2×C4×ΓL(1,𝔽8)⋊C4 | 2 |
| #6453⋊C3 | 2 |
| C2×#6453 | 3 |
| C2×C3×#125 | 7 |