Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
419 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C3×D4×A4 |
| commutator, cosocle | C3×#41 | C22×C32 |
| Frattini | C3×C4 | C22×C3×A4 |
| Fitting | C32×#578 | C3 |
| hypercenter | C32×D8 | A4 |
| hypocenter | Q8 | D4×H(3,𝔽3) |
| LCS | G ▷ C3×#41 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C32×C4 ◁ C32×D8 | |
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. #131⋊H(3,𝔽3) | C2 | semidirect |
| 3. C32×#578 | C3 | semidirect |
| 4. C3×#578⋊C3 | C3 | semidirect |
| 8. C32×2−1+4 | C2×C3 | semidirect |
| 9. C3×2−1+4⋊C3 | C2×C3 | semidirect |
| 13. C32×#131 | C2×C3 | semidirect |
| 14. C3×#131⋊C3 | C2×C3 | semidirect |
| 63. #578 | H(3,𝔽3) | semidirect |
| 98. 2−1+4 | C2×H(3,𝔽3) | semidirect |
| 102. #131 | C2×H(3,𝔽3) | semidirect |
| 158. C32 | #578⋊C3 | 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 |
| #131⋊H(3,𝔽3) | 2 |
| C32×#578 | 3 |
| C3×#578⋊C3 | 3 |
| D8×H(3,𝔽3) | 4 |