Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
869 conjugacy classes of subgroups; only the 273 normal subgroups are shown.
This group has 869 conjugacy classes of subgroups — too many to draw. Only its 273 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 | C32×C22⋊C4 |
| commutator | C22×C3 | C22×C32×C4 |
| Frattini, cosocle | C23×C3 | C23×C32 |
| socle | C2×C3 | C2×C32×C22⋊C4 |
| prime radical | C2×#41×H(3,𝔽3) | C2 |
| LCS | G ▷ C22×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C4×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽3)×#117 | C2 | semidirect |
| 3. H(3,𝔽3)×#99 | C2 | semidirect |
| 4. C32×#410 | C3 | semidirect |
| 7. C2×Q8×H(3,𝔽3) | C4 | semidirect |
| 8. C2×D4×H(3,𝔽3) | C4 | semidirect |
| 10. C22×C4×H(3,𝔽3) | C4 | semidirect |
| 16. C32×#117 | C2×C3 | semidirect |
| 17. C32×#99 | C2×C3 | semidirect |
| 35. C2×C32×Q8 | C3×C4 | semidirect |
| 36. C2×C32×D4 | C3×C4 | semidirect |
| 38. C22×C32×C4 | C3×C4 | semidirect |
| 72. #410 | H(3,𝔽3) | direct (N × H) |
| 108. #117 | C2×H(3,𝔽3) | semidirect |
| 109. #99 | C2×H(3,𝔽3) | semidirect |
| 110. H(3,𝔽3) | #410 | direct (N × H) |
| 143. C2×Q8 | C4×H(3,𝔽3) | semidirect |
| 144. C2×D4 | C4×H(3,𝔽3) | semidirect |
| 146. C22×C4 | C4×H(3,𝔽3) | semidirect |
| 171. C32 | C3×#410 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×H(3,𝔽3) | 2 |
| H(3,𝔽3)×#117 | 2 |
| H(3,𝔽3)×#99 | 2 |
| C32×#410 | 3 |