Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
396 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 78 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×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#121 | C2 | semidirect |
| 3. C2×#1141⋊C3 | C2 | semidirect |
| 4. (C22⋊(C7⋊C8))⋊C3 | C2 | semidirect |
| 5. #6383⋊C3 | C2 | semidirect |
| 6. #89808 | C3 | semidirect |
| 9. C2×D4×ΓL(1,𝔽8) | C22 | semidirect |
| 14. C4⋊C4×ΓL(1,𝔽8) | C22 | semidirect |
| 17. C2×ΓL(1,𝔽8)⋊C8 | C22 | semidirect |
| 18. C22×D28 | C2×C3 | semidirect |
| 19. C7×#121 | C2×C3 | semidirect |
| 20. C2×#1141 | C2×C3 | semidirect |
| 21. C22⋊(C7⋊C8) | C2×C3 | semidirect |
| 22. #6383 | C2×C3 | semidirect |
| 38. C4⋊AGL(1,𝔽7) | D4 | semidirect |
| 42. C2×C7×D4 | C22×C3 | semidirect |
| 47. C7×C4⋊C4 | C22×C3 | semidirect |
| 50. C2×C7⋊C8 | C22×C3 | semidirect |
| 61. C22×ΓL(1,𝔽8) | D8 | semidirect |
| 85. D28 | C3×D4 | semidirect |
| 106. #121 | AGL(1,𝔽7) | semidirect |
| 114. C22×C7 | C3×D8 | semidirect |
| 139. ΓL(1,𝔽8) | #447 | semidirect |
| 145. C2×D4 | C2×AGL(1,𝔽7) | semidirect |
| 149. C4⋊C4 | C2×AGL(1,𝔽7) | semidirect |
| 187. C7 | C3×#447 | semidirect |
| 203. C22 | #1141⋊C3 | 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 |
| ΓL(1,𝔽8)×#121 | 2 |
| C2×#1141⋊C3 | 2 |
| (C22⋊(C7⋊C8))⋊C3 | 2 |
| #6383⋊C3 | 2 |
| #89808 | 3 |
| C3×#447 | 7 |