Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
146 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 35 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 |
|---|---|---|
| 3. C22×C7×Q16 | C3 | semidirect |
| 24. C7×Q16 | A4 | semidirect |
| 29. C22⋊ΓL(1,𝔽8) | Q16 | direct (N × H) |
| 32. C22×Q16 | ΓL(1,𝔽8) | semidirect |
| 55. C22×C7 | C3×Q16 | semidirect |
| 72. Q16 | C22⋊ΓL(1,𝔽8) | direct (N × H) |
| 92. C7 | A4×Q16 | semidirect |
| 96. C22 | Q16×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×C22⋊ΓL(1,𝔽8) | 2 |
| C8×C22⋊ΓL(1,𝔽8) | 2 |
| C22×C7×Q16 | 3 |
| Q16×ΓL(1,𝔽8) | 4 |
| A4×Q16 | 7 |