Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1372 conjugacy classes of subgroups; only the 427 normal subgroups are shown.
This group has 1372 conjugacy classes of subgroups — too many to draw. Only its 427 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×C33×C16 | C2 | semidirect |
| 3. C2×C3⋊(C3⋊(C3⋊C16)) | C2 | semidirect |
| 4. C16×C3⋊(C3⋊D3) | C2 | direct (N × H) |
| 11. C33×C16 | C22 | semidirect |
| 12. C3⋊(C3⋊(C3⋊C16)) | C22 | semidirect |
| 14. C2×C32×C16 | D3 | semidirect |
| 31. C32×C16 | C2×D3 | semidirect |
| 34. C2×C3⋊(C3⋊D3) | C16 | direct (N × H) |
| 40. C2×C3×C16 | C3⋊D3 | semidirect |
| 52. C2×C33 | C2×C16 | semidirect |
| 54. C3⋊(C3⋊D3) | C2×C16 | direct (N × H) |
| 60. C3×C16 | C2×C3⋊D3 | semidirect |
| 71. C2×C16 | C3⋊(C3⋊D3) | direct (N × H) |
| 72. C33 | C22×C16 | semidirect |
| 81. C2×C32 | D3×C16 | semidirect |
| 90. C16 | C2×C3⋊(C3⋊D3) | direct (N × H) |
| 97. C32 | C2×D3×C16 | semidirect |
| 106. C2×C3 | C16×C3⋊D3 | semidirect |
| 115. C3 | C2×C16×C3⋊D3 | semidirect |
| 117. C2 | C16×C3⋊(C3⋊D3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×C3⋊(C3⋊D3) | 2 |
| C2×C33×C16 | 2 |
| C2×C3⋊(C3⋊(C3⋊C16)) | 2 |
| C16×C3⋊(C3⋊D3) | 2 |
| C2×C16×C3⋊D3 | 3 |