Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
248 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 92 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×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×#1131⋊C3 | C2 | semidirect |
| 4. (C22⋊(C7⋊C8))⋊C3 | C2 | semidirect |
| 5. (C2×AGL(1,𝔽7))⋊C8 | C2 | semidirect |
| 6. #6389⋊C3 | C2 | semidirect |
| 7. (ΓL(1,𝔽8)⋊C4)⋊C8 | C2 | semidirect |
| 8. #89883 | C3 | semidirect |
| 22. C22×C7×C8 | C2×C3 | semidirect |
| 24. C2×#1131 | C2×C3 | semidirect |
| 25. C22⋊(C7⋊C8) | C2×C3 | semidirect |
| 26. (C2×D7)⋊C8 | C2×C3 | semidirect |
| 27. #6389 | C2×C3 | semidirect |
| 28. Dic7⋊C8 | C2×C3 | semidirect |
| 43. C8×ΓL(1,𝔽8) | D4 | semidirect |
| 68. C22×ΓL(1,𝔽8) | M16 | semidirect |
| 93. C7×C8 | C3×D4 | semidirect |
| 115. C22×C8 | AGL(1,𝔽7) | semidirect |
| 123. C22×C7 | C3×M16 | semidirect |
| 150. ΓL(1,𝔽8) | #435 | semidirect |
| 197. C8 | (C22⋊D7)⋊C3 | semidirect |
| 198. C7 | C3×#435 | semidirect |
| 209. C22 | #1131⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C8×ΓL(1,𝔽8) | 2 |
| C4×(C22⋊D7)⋊C3 | 2 |
| C2×#1131⋊C3 | 2 |
| (C22⋊(C7⋊C8))⋊C3 | 2 |
| (C2×AGL(1,𝔽7))⋊C8 | 2 |
| #6389⋊C3 | 2 |
| (ΓL(1,𝔽8)⋊C4)⋊C8 | 2 |
| #89883 | 3 |
| C3×#435 | 7 |