Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
268 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 108 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. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×C8×AGL(1,𝔽7) | C2 | semidirect |
| 4. (C22⋊(C7⋊C8))⋊C3 | C2 | semidirect |
| 5. (C2×AGL(1,𝔽7))⋊C8 | C2 | semidirect |
| 6. (ΓL(1,𝔽8)⋊C4)⋊C8 | C2 | semidirect |
| 7. C8×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 8. C8×C22⋊D7 | C3 | semidirect |
| 22. C22×C7×C8 | C2×C3 | semidirect |
| 24. C2×C8×D7 | C2×C3 | semidirect |
| 25. C22⋊(C7⋊C8) | C2×C3 | semidirect |
| 26. (C2×D7)⋊C8 | C2×C3 | semidirect |
| 27. Dic7⋊C8 | C2×C3 | semidirect |
| 28. C8×Dic7 | C2×C3 | semidirect |
| 36. (C22⋊D7)⋊C3 | C8 | direct (N × H) |
| 43. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 68. C22×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 72. C2×AGL(1,𝔽7) | C2×C8 | semidirect |
| 78. ΓL(1,𝔽8)⋊C4 | C2×C8 | semidirect |
| 86. C22⋊D7 | C3×C8 | semidirect |
| 93. C7×C8 | C3×D4 | semidirect |
| 115. C22×C8 | AGL(1,𝔽7) | semidirect |
| 123. C22×C7 | C2×C3×C8 | semidirect |
| 127. C2×D7 | C2×C3×C8 | semidirect |
| 133. Dic7 | C2×C3×C8 | semidirect |
| 150. ΓL(1,𝔽8) | C8×D4 | semidirect |
| 197. C8 | (C22⋊D7)⋊C3 | direct (N × H) |
| 198. C7 | C3×C8×D4 | semidirect |
| 209. C22 | C8×AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C8×ΓL(1,𝔽8) | 2 |
| C4×(C22⋊D7)⋊C3 | 2 |
| C2×C8×AGL(1,𝔽7) | 2 |
| (C22⋊(C7⋊C8))⋊C3 | 2 |
| (C2×AGL(1,𝔽7))⋊C8 | 2 |
| (ΓL(1,𝔽8)⋊C4)⋊C8 | 2 |
| C8×ΓL(1,𝔽8)⋊C4 | 2 |
| C8×C22⋊D7 | 3 |
| C3×C8×D4 | 7 |