Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
404 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 130 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×C8×AGL(1,𝔽7) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 4. (C22⋊(C7⋊C8))⋊C3 | C2 | semidirect |
| 5. (C2×AGL(1,𝔽7))⋊C8 | C2 | semidirect |
| 6. D7×C22⋊C8 | C3 | semidirect |
| 15. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 16. C2×ΓL(1,𝔽8)⋊C8 | C22 | semidirect |
| 19. C2×C8×D7 | C2×C3 | semidirect |
| 20. C7×C22⋊C8 | C2×C3 | semidirect |
| 21. C22⋊(C7⋊C8) | C2×C3 | semidirect |
| 22. (C2×D7)⋊C8 | C2×C3 | semidirect |
| 27. C22×AGL(1,𝔽7) | C8 | semidirect |
| 50. C2×C7×C8 | C22×C3 | semidirect |
| 51. C2×C7⋊C8 | C22×C3 | semidirect |
| 60. C22×ΓL(1,𝔽8) | C2×C8 | semidirect |
| 67. C2×AGL(1,𝔽7) | C2×C8 | semidirect |
| 77. C22×D7 | C3×C8 | semidirect |
| 108. AGL(1,𝔽7) | C22⋊C8 | direct (N × H) |
| 113. C22⋊C8 | AGL(1,𝔽7) | direct (N × H) |
| 116. C22×C7 | C2×C3×C8 | semidirect |
| 123. C2×D7 | C2×C3×C8 | semidirect |
| 147. ΓL(1,𝔽8) | C2×C22⋊C8 | semidirect |
| 158. C2×C8 | C2×AGL(1,𝔽7) | semidirect |
| 164. D7 | C3×C22⋊C8 | semidirect |
| 198. C7 | C2×C3×C22⋊C8 | semidirect |
| 214. 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×C4×AGL(1,𝔽7) | 2 |
| C2×C8×AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| (C22⋊(C7⋊C8))⋊C3 | 2 |
| (C2×AGL(1,𝔽7))⋊C8 | 2 |
| D7×C22⋊C8 | 3 |
| C2×C3×C22⋊C8 | 7 |