Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
372 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 104 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×M16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×#1136⋊C3 | C2 | semidirect |
| 4. #6397⋊C3 | C2 | direct (N × H) |
| 5. C2×#6397 | C3 | semidirect |
| 15. M16×ΓL(1,𝔽8) | C22 | semidirect |
| 18. #1136⋊C3 | C22 | semidirect |
| 21. C2×C7×M16 | C2×C3 | semidirect |
| 22. C2×#1136 | C2×C3 | semidirect |
| 23. #6397 | C2×C3 | semidirect |
| 51. C7×M16 | C22×C3 | semidirect |
| 54. #1136 | C22×C3 | semidirect |
| 106. C2×ΓL(1,𝔽8) | #100 | semidirect |
| 111. C2×M16 | AGL(1,𝔽7) | semidirect |
| 142. ΓL(1,𝔽8) | C2×#100 | semidirect |
| 154. M16 | C2×AGL(1,𝔽7) | semidirect |
| 159. C2×C7 | C3×#100 | semidirect |
| 191. C7 | C2×C3×#100 | semidirect |
| 213. C2 | #6397⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊AGL(1,𝔽7) | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| C2×#1136⋊C3 | 2 |
| #6397⋊C3 | 2 |
| C2×#6397 | 3 |
| C2×C3×#100 | 7 |