Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
679 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 679 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 | C2×C9×#41 | D3 |
| socle | C22×C3 | C2×C3⋊S4 |
| hypercenter | C2×C4 | C9⋊S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9×#41⋊C3 | C2 | semidirect |
| 2. #11722374 | C2 | direct (N × H) |
| 3. C2×C9⋊GL(2,𝔽3) | C2 | semidirect |
| 8. C9×#41⋊C3 | C22 | semidirect |
| 9. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 10. C9⋊GL(2,𝔽3) | C22 | semidirect |
| 14. C2×C9×#41 | D3 | semidirect |
| 16. C2×#41⋊C9 | D3 | semidirect |
| 27. C9×SL(2,𝔽3) | C23 | semidirect |
| 36. C9×#41 | C2×D3 | semidirect |
| 45. #41⋊C9 | C2×D3 | semidirect |
| 59. C2×#41⋊C3 | D9 | semidirect |
| 106. #41⋊C3 | C2×D9 | semidirect |
| 107. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 131. C2×#41 | C3⋊D9 | semidirect |
| 157. SL(2,𝔽3) | C22×D9 | semidirect |
| 165. C2×C9 | #986 | semidirect |
| 171. #41 | C2×C3⋊D9 | semidirect |
| 199. C9 | C2×#986 | semidirect |
| 236. C2 | #11722374 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×#41⋊C3 | 2 |
| #11722374 | 2 |
| C2×C9⋊GL(2,𝔽3) | 2 |
| C2×#64267 | 3 |
| C2×#63692 | 3 |
| C2×#63512 | 3 |
| C2×C4⋊(C3⋊D9) | 4 |