Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
300 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 117 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. C2×M16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#101 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#100 | C2 | semidirect |
| 5. C7×#413 | C3 | semidirect |
| 17. C2×C7×M16 | C2×C3 | semidirect |
| 18. C7×#101 | C2×C3 | semidirect |
| 19. C7×#100 | C2×C3 | semidirect |
| 57. #413 | ΓL(1,𝔽8) | direct (N × H) |
| 86. C2×M16 | C2×ΓL(1,𝔽8) | semidirect |
| 87. #101 | C2×ΓL(1,𝔽8) | semidirect |
| 88. #100 | C2×ΓL(1,𝔽8) | semidirect |
| 111. ΓL(1,𝔽8) | #413 | direct (N × H) |
| 148. C7 | C3×#413 | 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 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#101 | 2 |
| ΓL(1,𝔽8)×#100 | 2 |
| C7×#413 | 3 |
| C3×#413 | 7 |