Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1576 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 258 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. C32×#951 | C2 | semidirect |
| 3. C32×#936 | C2 | semidirect |
| 4. C32×#837 | C2 | semidirect |
| 5. C32×#833 | C2 | semidirect |
| 7. C3×#5242 | C3 | direct (N × H) |
| 13. C2×C32×Dic6 | C22 | semidirect |
| 17. C32×C4×Dic3 | C22 | semidirect |
| 18. C32×#208 | C22 | semidirect |
| 24. C3×#951 | C2×C3 | semidirect |
| 25. C3×#936 | C2×C3 | semidirect |
| 28. C3×#837 | C2×C3 | semidirect |
| 29. C3×#833 | C2×C3 | semidirect |
| 45. #5242 | C32 | direct (N × H) |
| 62. C2×C3×Dic6 | C22×C3 | semidirect |
| 71. C3×C4×Dic3 | C22×C3 | semidirect |
| 72. C3×#208 | C22×C3 | semidirect |
| 87. #951 | C2×C32 | semidirect |
| 88. #936 | C2×C32 | semidirect |
| 91. #837 | C2×C32 | semidirect |
| 92. #833 | C2×C32 | semidirect |
| 152. C2×Dic6 | C22×C32 | semidirect |
| 161. C4×Dic3 | C22×C32 | semidirect |
| 162. #208 | C22×C32 | semidirect |
| 193. C33 | #455 | semidirect |
| 281. C32 | #5242 | direct (N × H) |
| 282. C32 | C3×#455 | semidirect |
| 319. C3 | C3×#5242 | direct (N × H) |
| 320. C3 | C32×#455 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×H(5,𝔽2) | 2 |
| C32×#951 | 2 |
| C32×#936 | 2 |
| C32×#837 | 2 |
| C32×#833 | 2 |
| C32×#455 | 3 |
| C3×#5242 | 3 |