Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
516 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 267 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×Q16×ΓL(1,𝔽8) | C2 | direct (N × H) |
| 4. C22×C7×Q16 | C3 | semidirect |
| 9. Q16×ΓL(1,𝔽8) | C22 | direct (N × H) |
| 12. C2×C7×Q16 | C2×C3 | semidirect |
| 26. C7×Q16 | C22×C3 | semidirect |
| 31. C22×ΓL(1,𝔽8) | Q16 | direct (N × H) |
| 36. C22×Q16 | ΓL(1,𝔽8) | direct (N × H) |
| 50. C2×ΓL(1,𝔽8) | C2×Q16 | direct (N × H) |
| 53. C2×Q16 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 55. C22×C7 | C3×Q16 | semidirect |
| 67. ΓL(1,𝔽8) | C22×Q16 | direct (N × H) |
| 72. Q16 | C22×ΓL(1,𝔽8) | direct (N × H) |
| 74. C2×C7 | C2×C3×Q16 | semidirect |
| 88. C7 | C22×C3×Q16 | semidirect |
| 91. C22 | Q16×ΓL(1,𝔽8) | direct (N × H) |
| 97. C2 | C2×Q16×ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| C2×Q16×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| C22×C7×Q16 | 3 |
| C22×C3×Q16 | 7 |