Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
668 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C2×C7×#41 | C2×C3 |
| hypercenter | C2×#41 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#535 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#41×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C22×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 3. #6510⋊C3 | C2 | semidirect |
| 4. #6503⋊C3 | C2 | semidirect |
| 5. #6500⋊C3 | C2 | semidirect |
| 6. #6493⋊C3 | C2 | semidirect |
| 7. #6489⋊C3 | C2 | semidirect |
| 8. #90521 | C3 | semidirect |
| 11. C2×Q8×ΓL(1,𝔽8) | C22 | semidirect |
| 13. C2×D4×ΓL(1,𝔽8) | C22 | semidirect |
| 15. C22×C4×ΓL(1,𝔽8) | C22 | semidirect |
| 16. C2×(C22⋊D7)⋊C3 | C22 | semidirect |
| 21. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 22. C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | semidirect |
| 23. C4×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 24. C2×C7×#41 | C2×C3 | semidirect |
| 25. C22×D28 | C2×C3 | semidirect |
| 26. #6510 | C2×C3 | semidirect |
| 27. #6503 | C2×C3 | semidirect |
| 28. #6500 | C2×C3 | semidirect |
| 29. #6493 | C2×C3 | semidirect |
| 30. #6489 | C2×C3 | semidirect |
| 41. C2×C4×ΓL(1,𝔽8) | D4 | semidirect |
| 51. C2×C7×Q8 | C22×C3 | semidirect |
| 53. C2×C7×D4 | C22×C3 | semidirect |
| 55. C22×C4×C7 | C22×C3 | semidirect |
| 56. C2×C22⋊D7 | C22×C3 | semidirect |
| 61. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 62. C4⋊Dic7 | C22×C3 | semidirect |
| 63. C4×Dic7 | C22×C3 | semidirect |
| 93. C2×C4×C7 | C3×D4 | semidirect |
| 121. C2×#41 | AGL(1,𝔽7) | semidirect |
| 157. ΓL(1,𝔽8) | #535 | semidirect |
| 160. C2×Q8 | C2×AGL(1,𝔽7) | semidirect |
| 166. C2×D4 | C2×AGL(1,𝔽7) | semidirect |
| 168. C22×C4 | C2×AGL(1,𝔽7) | semidirect |
| 203. C2×C4 | (C22⋊D7)⋊C3 | semidirect |
| 207. C7 | C3×#535 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×ΓL(1,𝔽8) | 2 |
| C22×C4⋊AGL(1,𝔽7) | 2 |
| #6510⋊C3 | 2 |
| #6503⋊C3 | 2 |
| #6500⋊C3 | 2 |
| #6493⋊C3 | 2 |
| #6489⋊C3 | 2 |
| #90521 | 3 |
| C3×#535 | 7 |