Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
596 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 596 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 |
|---|---|---|
| 1. C22×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×SD16×ΓL(1,𝔽8) | C2 | direct (N × H) |
| 4. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 5. C22×C7×SD16 | C3 | semidirect |
| 8. C2×Q8×ΓL(1,𝔽8) | C22 | semidirect |
| 13. SD16×ΓL(1,𝔽8) | C22 | direct (N × H) |
| 14. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 15. C22×C7×Q8 | C2×C3 | semidirect |
| 17. C2×C7×SD16 | C2×C3 | semidirect |
| 18. C22×C7×C8 | C2×C3 | semidirect |
| 24. Q8×ΓL(1,𝔽8) | C23 | semidirect |
| 30. C8×ΓL(1,𝔽8) | C23 | semidirect |
| 33. C2×C7×Q8 | C22×C3 | semidirect |
| 38. C7×SD16 | C22×C3 | semidirect |
| 39. C2×C7×C8 | C22×C3 | semidirect |
| 46. C22×ΓL(1,𝔽8) | SD16 | direct (N × H) |
| 51. C22×SD16 | ΓL(1,𝔽8) | direct (N × H) |
| 56. C7×Q8 | C23×C3 | semidirect |
| 62. C7×C8 | C23×C3 | semidirect |
| 73. C2×ΓL(1,𝔽8) | C2×SD16 | direct (N × H) |
| 75. C22×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 77. C2×SD16 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 78. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 81. C22×C7 | C3×SD16 | semidirect |
| 97. ΓL(1,𝔽8) | C22×SD16 | direct (N × H) |
| 100. C2×Q8 | C22×ΓL(1,𝔽8) | semidirect |
| 105. SD16 | C22×ΓL(1,𝔽8) | direct (N × H) |
| 106. C2×C8 | C22×ΓL(1,𝔽8) | semidirect |
| 108. C2×C7 | C2×C3×SD16 | semidirect |
| 121. Q8 | C23×ΓL(1,𝔽8) | semidirect |
| 127. C8 | C23×ΓL(1,𝔽8) | semidirect |
| 128. C7 | C22×C3×SD16 | semidirect |
| 134. C22 | SD16×ΓL(1,𝔽8) | direct (N × H) |
| 141. C2 | C2×SD16×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 |
| C22×D4×ΓL(1,𝔽8) | 2 |
| C2×SD16×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| C22×C7×SD16 | 3 |
| C22×C3×SD16 | 7 |