Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
455 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 455 conjugacy classes of subgroups — too many to draw. Only its 117 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×C9×SL(2,𝔽3) | C2 | semidirect |
| 2. C4×C9×SL(2,𝔽3) | C2 | semidirect |
| 4. D4×Q8×C9 | C3 | semidirect |
| 5. D4×Q8⋊C9 | C3 | semidirect |
| 11. C22×Q8×C9 | C2×C3 | semidirect |
| 12. C22×Q8⋊C9 | C2×C3 | semidirect |
| 15. C4×Q8×C9 | C2×C3 | semidirect |
| 17. C4×Q8⋊C9 | C2×C3 | semidirect |
| 21. C9×SL(2,𝔽3) | D4 | direct (N × H) |
| 23. D4×SL(2,𝔽3) | C9 | direct (N × H) |
| 47. C22×SL(2,𝔽3) | C2×C9 | semidirect |
| 52. C4×SL(2,𝔽3) | C2×C9 | semidirect |
| 62. Q8×C9 | C3×D4 | semidirect |
| 64. D4×C9 | SL(2,𝔽3) | direct (N × H) |
| 72. Q8⋊C9 | C3×D4 | semidirect |
| 73. D4×Q8 | C3×C9 | semidirect |
| 103. C22×C9 | C2×SL(2,𝔽3) | semidirect |
| 110. C4×C9 | C2×SL(2,𝔽3) | semidirect |
| 111. C22×Q8 | C2×C3×C9 | semidirect |
| 114. C4×Q8 | C2×C3×C9 | semidirect |
| 132. SL(2,𝔽3) | D4×C9 | direct (N × H) |
| 172. C9 | D4×SL(2,𝔽3) | direct (N × H) |
| 177. Q8 | C3×D4×C9 | semidirect |
| 178. D4 | C9×SL(2,𝔽3) | direct (N × H) |
| 195. C22 | C2×C9×SL(2,𝔽3) | semidirect |
| 201. C4 | C2×C9×SL(2,𝔽3) | semidirect |
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 |
| C4×C9×SL(2,𝔽3) | 2 |
| C3×D4×SL(2,𝔽3) | 3 |
| D4×Q8×C9 | 3 |
| D4×Q8⋊C9 | 3 |
| C2×C3×D4×C9 | 4 |