Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
366 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 95 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. SL(2,𝔽3)×Dic9 | C2 | direct (N × H) |
| 4. C2×Q8×Dic9 | C3 | semidirect |
| 6. C2×C9×SL(2,𝔽3) | C4 | semidirect |
| 13. Q8×Dic9 | C2×C3 | semidirect |
| 19. C9×SL(2,𝔽3) | C2×C4 | semidirect |
| 27. C2×Q8×C9 | C3×C4 | semidirect |
| 60. Q8×C9 | C2×C3×C4 | semidirect |
| 67. C2×Dic9 | SL(2,𝔽3) | direct (N × H) |
| 85. C2×SL(2,𝔽3) | Dic9 | direct (N × H) |
| 108. Dic9 | C2×SL(2,𝔽3) | direct (N × H) |
| 133. SL(2,𝔽3) | C2×Dic9 | direct (N × H) |
| 142. C2×C9 | C4×SL(2,𝔽3) | semidirect |
| 148. C2×Q8 | C3×Dic9 | semidirect |
| 177. C9 | C2×C4×SL(2,𝔽3) | semidirect |
| 182. Q8 | C2×C3×Dic9 | semidirect |
| 215. C2 | SL(2,𝔽3)×Dic9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×SL(2,𝔽3) | 2 |
| SL(2,𝔽3)×Dic9 | 2 |
| C2×Dic3×SL(2,𝔽3) | 3 |
| C2×Q8×Dic9 | 3 |
| C22×C3×Dic9 | 4 |