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, hypercenter | C2×C4 | 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 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ 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. #11722373 | C2 | direct (N × H) |
| 3. C2×C9⋊GL(2,𝔽3) | C2 | semidirect |
| 4. C2×C9⋊#226 | C2 | semidirect |
| 9. C9×#41⋊C3 | C22 | semidirect |
| 11. C9⋊GL(2,𝔽3) | C22 | semidirect |
| 12. C9⋊#226 | C22 | semidirect |
| 17. C2×C9×#41 | D3 | semidirect |
| 19. C2×#41⋊C9 | D3 | semidirect |
| 43. C9×#41 | C2×D3 | semidirect |
| 54. #41⋊C9 | C2×D3 | semidirect |
| 74. C2×#41⋊C3 | D9 | semidirect |
| 130. #41⋊C3 | C2×D9 | semidirect |
| 163. C2×#41 | C3⋊D9 | semidirect |
| 202. C2×C9 | #985 | semidirect |
| 209. #41 | C2×C3⋊D9 | semidirect |
| 242. C9 | C2×#985 | semidirect |
| 282. C2 | #11722373 | 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 |
| #11722373 | 2 |
| C2×C9⋊GL(2,𝔽3) | 2 |
| C2×C9⋊#226 | 2 |
| C2×#64266 | 3 |
| C2×#63691 | 3 |
| C2×#63513 | 3 |
| C2×C4×C3⋊D9 | 4 |