Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1199 conjugacy classes of subgroups; only the 343 normal subgroups are shown.
This group has 1199 conjugacy classes of subgroups — too many to draw. Only its 343 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| commutator | C22×C3 | C22×C32×C4 |
| prime radical | C24×H(3,𝔽3) | C22 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. H(3,𝔽3)×#95 | C2 | semidirect |
| 4. C32×#381 | C3 | semidirect |
| 7. C22×C4×H(3,𝔽3) | C4 | semidirect |
| 13. C32×#95 | C2×C3 | semidirect |
| 27. C22×C32×C4 | C3×C4 | semidirect |
| 54. #381 | H(3,𝔽3) | direct (N × H) |
| 81. #95 | C2×H(3,𝔽3) | semidirect |
| 82. H(3,𝔽3) | #381 | direct (N × H) |
| 107. C22×C4 | C4×H(3,𝔽3) | semidirect |
| 127. C32 | C3×#381 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×H(3,𝔽3) | 2 |
| C2×C22⋊C4×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#95 | 2 |
| C32×#381 | 3 |