Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
686 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 686 conjugacy classes of subgroups — too many to draw. Only its 95 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 | A4×D9 |
| commutator, cosocle, hypocenter | Q8×C9 | C23×C3 |
| Frattini | C2×C3 | C22×D3×A4 |
| Fitting | C2×C9×#41 | C2×C3 |
| socle | C22×C3 | C2×D3×A4 |
| LCS | G ▷ Q8×C9 | |
| 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. D9×#41⋊C3 | C2 | direct (N × H) |
| 3. C2×D9×SL(2,𝔽3) | C2 | semidirect |
| 4. C2×#88615 | C2 | semidirect |
| 6. C2×#41×D9 | C3 | semidirect |
| 8. C9×#41⋊C3 | C22 | semidirect |
| 9. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 10. D9×SL(2,𝔽3) | C22 | semidirect |
| 11. #88615 | C22 | semidirect |
| 16. C2×C9×#41 | C2×C3 | semidirect |
| 18. #41×D9 | C2×C3 | semidirect |
| 19. C2×#3701 | C2×C3 | semidirect |
| 30. C9×SL(2,𝔽3) | C23 | semidirect |
| 40. C9×#41 | C22×C3 | semidirect |
| 47. #3701 | C22×C3 | semidirect |
| 83. C2×#41⋊C3 | D9 | direct (N × H) |
| 152. #41⋊C3 | C2×D9 | direct (N × H) |
| 155. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 168. C2×D9 | #41⋊C3 | direct (N × H) |
| 181. C2×#41 | C3×D9 | semidirect |
| 219. SL(2,𝔽3) | C22×D9 | semidirect |
| 226. C2×C9 | C2×#41⋊C3 | semidirect |
| 229. D9 | C2×#41⋊C3 | direct (N × H) |
| 238. #41 | C2×C3×D9 | semidirect |
| 278. C9 | C22×#41⋊C3 | semidirect |
| 335. C2 | D9×#41⋊C3 | 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 |
| D9×#41⋊C3 | 2 |
| C2×D9×SL(2,𝔽3) | 2 |
| C2×#88615 | 2 |
| C2×D3×#41⋊C3 | 3 |
| C2×#41×D9 | 3 |
| C2×C3×C4×D9 | 4 |