Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1134 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×C3×A4 |
| commutator, cosocle | C3×Q8 | C23×C32 |
| Frattini, socle | C2×C3 | C23×C3×A4 |
| Fitting | C32×#585 | C3 |
| hypercenter | C32×#41 | A4 |
| hypocenter | Q8 | C23×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C3×C4 ◁ C32×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. 2−1+4⋊H(3,𝔽3) | C2 | semidirect |
| 2. #11722577 | C2 | semidirect |
| 3. C2×#41⋊H(3,𝔽3) | C2 | semidirect |
| 4. C32×#585 | C3 | semidirect |
| 5. C3×#5945 | C3 | semidirect |
| 7. #41⋊H(3,𝔽3) | C22 | semidirect |
| 8. #41⋊H(3,𝔽3) | C22 | semidirect |
| 9. C2×Q8⋊H(3,𝔽3) | C22 | semidirect |
| 11. C32×2−1+4 | C2×C3 | semidirect |
| 13. C32×H(5,𝔽2) | C2×C3 | semidirect |
| 16. C2×C32×#41 | C2×C3 | semidirect |
| 17. C3×2−1+4⋊C3 | C2×C3 | semidirect |
| 18. C3×#994 | C2×C3 | semidirect |
| 19. C2×C3×#41⋊C3 | C2×C3 | semidirect |
| 47. C3×#41⋊C3 | C22×C3 | semidirect |
| 48. C3×#41⋊C3 | C22×C3 | semidirect |
| 49. C2×C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 93. #585 | H(3,𝔽3) | semidirect |
| 151. 2−1+4 | C2×H(3,𝔽3) | semidirect |
| 153. H(5,𝔽2) | C2×H(3,𝔽3) | semidirect |
| 156. C2×#41 | C2×H(3,𝔽3) | semidirect |
| 247. C32 | #5945 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| 2−1+4⋊H(3,𝔽3) | 2 |
| #11722577 | 2 |
| C2×#41⋊H(3,𝔽3) | 2 |
| C32×#585 | 3 |
| C3×#5945 | 3 |
| #41×H(3,𝔽3) | 4 |