Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
588 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 272 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. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. Q8×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 4. #6456⋊C3 | C2 | semidirect |
| 5. C4⋊C4×AGL(1,𝔽7) | C2 | semidirect |
| 6. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 7. #6435⋊C3 | C2 | semidirect |
| 8. #90265 | C3 | semidirect |
| 10. #66659 | C4 | semidirect |
| 23. C4×C7×Q8 | C2×C3 | semidirect |
| 24. Q8×Dic7 | C2×C3 | semidirect |
| 25. #6456 | C2×C3 | semidirect |
| 26. D7×C4⋊C4 | C2×C3 | semidirect |
| 27. C4×D28 | C2×C3 | semidirect |
| 28. #6435 | C2×C3 | semidirect |
| 31. Q8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 38. C4⋊AGL(1,𝔽7) | C2×C4 | semidirect |
| 40. C4×AGL(1,𝔽7) | C2×C4 | semidirect |
| 42. #1161 | C3×C4 | semidirect |
| 73. C7×Q8 | C2×C3×C4 | semidirect |
| 80. D28 | C2×C3×C4 | semidirect |
| 82. C4×D7 | C2×C3×C4 | semidirect |
| 102. C4×Q8 | AGL(1,𝔽7) | semidirect |
| 128. ΓL(1,𝔽8) | #520 | semidirect |
| 157. Q8 | C4×AGL(1,𝔽7) | semidirect |
| 167. C7 | C3×#520 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66659 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| Q8×ΓL(1,𝔽8)⋊C4 | 2 |
| #6456⋊C3 | 2 |
| C4⋊C4×AGL(1,𝔽7) | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #6435⋊C3 | 2 |
| #90265 | 3 |
| C3×#520 | 7 |