Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
650 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 87 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×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 |
| hypercenter | C2×C4 | A4×D9 |
| LCS | G ▷ Q8×C9 | |
| 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. #11722412 | C2 | direct (N × H) |
| 3. C2×#88615 | C2 | semidirect |
| 5. C2×#63715 | C3 | semidirect |
| 7. C9×#41⋊C3 | C22 | semidirect |
| 8. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 9. #88615 | C22 | semidirect |
| 15. #63715 | C2×C3 | semidirect |
| 24. C9×SL(2,𝔽3) | C23 | semidirect |
| 65. C2×#41⋊C3 | D9 | semidirect |
| 117. #41⋊C3 | C2×D9 | semidirect |
| 119. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 168. SL(2,𝔽3) | C22×D9 | semidirect |
| 174. C2×C9 | #994 | semidirect |
| 213. C9 | C2×#994 | semidirect |
| 257. C2 | #11722412 | 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 |
| #11722412 | 2 |
| C2×#88615 | 2 |
| C2×C3⋊#994 | 3 |
| C2×#63715 | 3 |
| C2×C3×Dic18 | 4 |