Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
631 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×C9⋊S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C22×Q8×C9 | D3 |
| socle | C22×C3 | C2×C3⋊S4 |
| hypercenter | C23 | C9⋊S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ 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. #11722371 | C2 | direct (N × H) |
| 3. C2×C9⋊GL(2,𝔽3) | C2 | semidirect |
| 4. C2×C9⋊#226 | C2 | semidirect |
| 10. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 11. C9⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C9⋊#226 | C22 | semidirect |
| 17. C22×Q8×C9 | D3 | semidirect |
| 19. C22×Q8⋊C9 | D3 | semidirect |
| 45. C2×Q8×C9 | C2×D3 | semidirect |
| 55. C2×Q8⋊C9 | C2×D3 | semidirect |
| 75. C22×SL(2,𝔽3) | D9 | semidirect |
| 133. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 167. C22×Q8 | C3⋊D9 | semidirect |
| 208. C2×C9 | #983 | semidirect |
| 218. C2×Q8 | C2×C3⋊D9 | semidirect |
| 249. C9 | C2×#983 | semidirect |
| 290. C2 | #11722371 | 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 |
| #11722371 | 2 |
| C2×C9⋊GL(2,𝔽3) | 2 |
| C2×C9⋊#226 | 2 |
| C2×#64264 | 3 |
| C2×#63689 | 3 |
| C2×#63505 | 3 |
| C2×#6233 | 4 |