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 | C7×#117 | C2×C3 |
| hypercenter | #117 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#535 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1157⋊C3 | C2 | semidirect |
| 2. C22×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#117 | C2 | semidirect |
| 4. #6458⋊C3 | C2 | semidirect |
| 5. #6445⋊C3 | C2 | semidirect |
| 6. #6438⋊C3 | C2 | semidirect |
| 7. #10620710 | C2 | semidirect |
| 8. #90216 | C3 | semidirect |
| 16. C2×C4⋊AGL(1,𝔽7) | C22 | semidirect |
| 17. C2×C4⋊AGL(1,𝔽7) | C22 | semidirect |
| 18. C2×C4×AGL(1,𝔽7) | C22 | semidirect |
| 19. C2×ΓL(1,𝔽8)⋊Q8 | C22 | semidirect |
| 20. C4⋊C4×ΓL(1,𝔽8) | C22 | semidirect |
| 22. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 23. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 24. C2×#1157 | C2×C3 | semidirect |
| 25. C22×D28 | C2×C3 | semidirect |
| 26. C7×#117 | C2×C3 | semidirect |
| 27. #6458 | C2×C3 | semidirect |
| 28. #6445 | C2×C3 | semidirect |
| 29. #6438 | C2×C3 | semidirect |
| 30. #6437 | C2×C3 | semidirect |
| 37. C2×C4×ΓL(1,𝔽8) | D4 | semidirect |
| 56. C2×D28 | C22×C3 | semidirect |
| 57. C2×D28 | C22×C3 | semidirect |
| 58. C2×C4×D7 | C22×C3 | semidirect |
| 59. C2×Dic14 | C22×C3 | semidirect |
| 60. C7×C4⋊C4 | C22×C3 | semidirect |
| 62. C42×C7 | C22×C3 | semidirect |
| 63. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 89. C2×C4×C7 | C3×D4 | semidirect |
| 127. #117 | AGL(1,𝔽7) | semidirect |
| 157. ΓL(1,𝔽8) | #535 | semidirect |
| 169. C4⋊C4 | C2×AGL(1,𝔽7) | semidirect |
| 172. C42 | C2×AGL(1,𝔽7) | semidirect |
| 203. C2×C4 | C4⋊AGL(1,𝔽7) | 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×#1157⋊C3 | 2 |
| C22×C4⋊AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| #6458⋊C3 | 2 |
| #6445⋊C3 | 2 |
| #6438⋊C3 | 2 |
| #10620710 | 2 |
| #90216 | 3 |
| C3×#535 | 7 |