Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
212 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 104 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 |
|---|---|---|
| 4. #6396⋊C3 | C2 | direct (N × H) |
| 5. C2×#6396 | C3 | semidirect |
| 19. #6396 | C2×C3 | semidirect |
| 76. C2×ΓL(1,𝔽8) | #108 | semidirect |
| 100. ΓL(1,𝔽8) | C2×#108 | semidirect |
| 113. C2×C7 | C3×#108 | semidirect |
| 133. C7 | C2×C3×#108 | semidirect |
| 149. C2 | #6396⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×M16×ΓL(1,𝔽8) | 2 |
| C2×#1136⋊C3 | 2 |
| C22×ΓL(1,𝔽8)⋊C8 | 2 |
| #6396⋊C3 | 2 |
| C2×#6396 | 3 |
| C2×C3×#108 | 7 |