Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1172 conjugacy classes of subgroups; only the 361 normal subgroups are shown.
This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 361 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C9 | A4 |
| commutator, hypocenter | Q8 | C23×C3×C9 |
| Frattini | C2×C3 | C23×C3×A4 |
| Fitting | C23×Q8×C9 | C3 |
| socle | C24×C3 | C3×A4 |
| prime radical | C23×C3×SL(2,𝔽3) | C3 |
| cosocle | C3×Q8 | C23×C32 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C24×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×SL(2,𝔽3) | C2 | direct (N × H) |
| 3. C23×Q8×C9 | C3 | semidirect |
| 4. C23×Q8⋊C9 | C3 | semidirect |
| 6. C2×C9×SL(2,𝔽3) | C22 | direct (N × H) |
| 9. C22×Q8×C9 | C2×C3 | semidirect |
| 11. C22×Q8⋊C9 | C2×C3 | semidirect |
| 14. C9×SL(2,𝔽3) | C23 | direct (N × H) |
| 16. C23×SL(2,𝔽3) | C9 | direct (N × H) |
| 23. C2×Q8×C9 | C22×C3 | semidirect |
| 26. C2×Q8⋊C9 | C22×C3 | semidirect |
| 32. C22×SL(2,𝔽3) | C2×C9 | direct (N × H) |
| 38. C23×C9 | SL(2,𝔽3) | direct (N × H) |
| 42. Q8×C9 | C23×C3 | semidirect |
| 45. Q8⋊C9 | C23×C3 | semidirect |
| 46. C23×Q8 | C3×C9 | semidirect |
| 56. C2×SL(2,𝔽3) | C22×C9 | direct (N × H) |
| 61. C22×C9 | C2×SL(2,𝔽3) | direct (N × H) |
| 66. C22×Q8 | C2×C3×C9 | semidirect |
| 78. SL(2,𝔽3) | C23×C9 | direct (N × H) |
| 83. C2×C9 | C22×SL(2,𝔽3) | direct (N × H) |
| 88. C2×Q8 | C22×C3×C9 | semidirect |
| 99. C9 | C23×SL(2,𝔽3) | direct (N × H) |
| 100. C23 | C9×SL(2,𝔽3) | direct (N × H) |
| 104. Q8 | C23×C3×C9 | semidirect |
| 111. C22 | C2×C9×SL(2,𝔽3) | direct (N × H) |
| 117. C2 | C22×C9×SL(2,𝔽3) | 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 |
| C23×C3×SL(2,𝔽3) | 3 |
| C23×Q8×C9 | 3 |
| C23×Q8⋊C9 | 3 |
| C24×C3×C9 | 4 |