Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
324 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 243 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. C4×C8×ΓL(1,𝔽8) | C2 | direct (N × H) |
| 4. C2×C4×C7×C8 | C3 | semidirect |
| 7. C2×C8×ΓL(1,𝔽8) | C4 | direct (N × H) |
| 13. C4×C7×C8 | C2×C3 | semidirect |
| 16. C2×C4×ΓL(1,𝔽8) | C8 | direct (N × H) |
| 21. C8×ΓL(1,𝔽8) | C2×C4 | direct (N × H) |
| 24. C2×C7×C8 | C3×C4 | semidirect |
| 35. C4×ΓL(1,𝔽8) | C2×C8 | direct (N × H) |
| 37. C2×C4×C8 | ΓL(1,𝔽8) | direct (N × H) |
| 39. C2×C4×C7 | C3×C8 | semidirect |
| 44. C7×C8 | C2×C3×C4 | semidirect |
| 51. C2×ΓL(1,𝔽8) | C4×C8 | direct (N × H) |
| 56. C4×C8 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 61. C4×C7 | C2×C3×C8 | semidirect |
| 70. ΓL(1,𝔽8) | C2×C4×C8 | direct (N × H) |
| 73. C2×C8 | C4×ΓL(1,𝔽8) | direct (N × H) |
| 77. C2×C7 | C3×C4×C8 | semidirect |
| 87. C2×C4 | C8×ΓL(1,𝔽8) | direct (N × H) |
| 92. C8 | C2×C4×ΓL(1,𝔽8) | direct (N × H) |
| 93. C7 | C2×C3×C4×C8 | semidirect |
| 101. C4 | C2×C8×ΓL(1,𝔽8) | direct (N × H) |
| 104. C2 | C4×C8×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×C8×ΓL(1,𝔽8) | 2 |
| C2×C42×ΓL(1,𝔽8) | 2 |
| C4×C8×ΓL(1,𝔽8) | 2 |
| C2×C4×C7×C8 | 3 |
| C2×C3×C4×C8 | 7 |