Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
971 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 971 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C9⋊S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×Q8×C9 | D3 |
| socle | C23×C3 | C3⋊S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×C9⋊GL(2,𝔽3) | C2 | direct (N × H) |
| 7. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 8. C9⋊GL(2,𝔽3) | C22 | direct (N × H) |
| 11. C22×Q8×C9 | D3 | semidirect |
| 13. C22×Q8⋊C9 | D3 | semidirect |
| 21. C9×SL(2,𝔽3) | C23 | semidirect |
| 30. C2×Q8×C9 | C2×D3 | semidirect |
| 34. C2×Q8⋊C9 | C2×D3 | semidirect |
| 45. C22×SL(2,𝔽3) | D9 | semidirect |
| 62. Q8×C9 | C22×D3 | semidirect |
| 67. Q8⋊C9 | C22×D3 | semidirect |
| 80. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 91. C22×C9 | GL(2,𝔽3) | semidirect |
| 99. C22×Q8 | C3⋊D9 | semidirect |
| 118. SL(2,𝔽3) | C22×D9 | semidirect |
| 125. C2×C9 | C2×GL(2,𝔽3) | semidirect |
| 131. C2×Q8 | C2×C3⋊D9 | semidirect |
| 150. C9 | C22×GL(2,𝔽3) | semidirect |
| 156. Q8 | C22×C3⋊D9 | semidirect |
| 170. C22 | C9⋊GL(2,𝔽3) | direct (N × H) |
| 177. C2 | C2×C9⋊GL(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 |
| C2×C9⋊GL(2,𝔽3) | 2 |
| C22×C3⋊GL(2,𝔽3) | 3 |
| C22×#3689 | 3 |
| C22×#3675 | 3 |
| C23×C3⋊D9 | 4 |