Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
256 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 72 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. C2×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 2. Q8×ΓL(1,𝔽8)⋊C4 | C2 | semidirect |
| 3. C2×#1143⋊C3 | C2 | semidirect |
| 5. #6406⋊C3 | C2 | semidirect |
| 6. #6395⋊C3 | C2 | semidirect |
| 7. (ΓL(1,𝔽8)⋊C4)⋊C8 | C2 | semidirect |
| 8. #89935 | C3 | semidirect |
| 22. C2×C7×SD16 | C2×C3 | semidirect |
| 23. Q8×Dic7 | C2×C3 | semidirect |
| 24. C2×#1143 | C2×C3 | semidirect |
| 26. #6406 | C2×C3 | semidirect |
| 27. #6395 | C2×C3 | semidirect |
| 28. Dic7⋊C8 | C2×C3 | semidirect |
| 33. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 79. ΓL(1,𝔽8)⋊C4 | SD16 | semidirect |
| 84. C7×Q8 | C3×D4 | semidirect |
| 117. C2×SD16 | AGL(1,𝔽7) | semidirect |
| 135. Dic7 | C3×SD16 | semidirect |
| 153. ΓL(1,𝔽8) | #460 | semidirect |
| 188. Q8 | (C22⋊D7)⋊C3 | semidirect |
| 202. C7 | C3×#460 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD16×ΓL(1,𝔽8) | 2 |
| Q8×ΓL(1,𝔽8)⋊C4 | 2 |
| C2×#1143⋊C3 | 2 |
| #6503⋊C3 | 2 |
| #6406⋊C3 | 2 |
| #6395⋊C3 | 2 |
| (ΓL(1,𝔽8)⋊C4)⋊C8 | 2 |
| #89935 | 3 |
| C3×#460 | 7 |