Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
404 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 404 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×#5450 |
| commutator, cosocle | C24×C3 | C22×C32 |
| Frattini, socle | C22×C3 | C22×C3×A4 |
| Fitting | C32×H(3,𝔽4) | C3 |
| hypercenter | C32 | #5450 |
| hypocenter | C24 | C22×H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #120⋊H(3,𝔽3) | C2 | semidirect |
| 2. C32×H(3,𝔽4) | C3 | semidirect |
| 3. C3×#5450 | C3 | semidirect |
| 4. C32⋊(C22⋊A4) | C22 | semidirect |
| 8. C32×#120 | C2×C3 | semidirect |
| 9. C3×#120⋊C3 | C2×C3 | semidirect |
| 14. C24×C32 | A4 | semidirect |
| 15. C24×C32 | C22×C3 | semidirect |
| 16. C3×C22⋊A4 | C22×C3 | semidirect |
| 41. H(3,𝔽4) | H(3,𝔽3) | semidirect |
| 72. #120 | C2×H(3,𝔽3) | semidirect |
| 93. C24 | C22⋊H(3,𝔽3) | semidirect |
| 94. C24 | C22×H(3,𝔽3) | semidirect |
| 120. C32 | #5450 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #120⋊H(3,𝔽3) | 2 |
| C32×H(3,𝔽4) | 3 |
| C3×#5450 | 3 |
| C22×C22⋊H(3,𝔽3) | 4 |