Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1597 conjugacy classes of subgroups; only the 313 normal subgroups are shown.
This group has 1597 conjugacy classes of subgroups — too many to draw. Only its 313 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C3 | C3×A4 |
| commutator, cosocle | C3×Q8 | C23×C32 |
| Frattini | C2×C3 | C23×C3×A4 |
| Fitting | C23×C32×Q8 | C3 |
| hypercenter | C24×C32 | A4 |
| hypocenter | Q8 | C23×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C24×C3 ◁ C24×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8⋊H(3,𝔽3) | C2 | direct (N × H) |
| 2. C23×C32×Q8 | C3 | semidirect |
| 3. C23×C3×SL(2,𝔽3) | C3 | semidirect |
| 5. C2×Q8⋊H(3,𝔽3) | C22 | direct (N × H) |
| 7. C22×C32×Q8 | C2×C3 | semidirect |
| 9. C22×C3×SL(2,𝔽3) | C2×C3 | semidirect |
| 12. Q8⋊H(3,𝔽3) | C23 | direct (N × H) |
| 20. C2×C32×Q8 | C22×C3 | semidirect |
| 23. C2×C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 35. C23×C32 | SL(2,𝔽3) | semidirect |
| 39. C32×Q8 | C23×C3 | semidirect |
| 42. C3×SL(2,𝔽3) | C23×C3 | semidirect |
| 43. C23×Q8 | H(3,𝔽3) | semidirect |
| 62. C22×C32 | C2×SL(2,𝔽3) | semidirect |
| 67. C22×Q8 | C2×H(3,𝔽3) | semidirect |
| 89. C2×C32 | C22×SL(2,𝔽3) | semidirect |
| 94. C2×Q8 | C22×H(3,𝔽3) | semidirect |
| 108. C32 | C23×SL(2,𝔽3) | semidirect |
| 109. C23 | Q8⋊H(3,𝔽3) | direct (N × H) |
| 113. Q8 | C23×H(3,𝔽3) | semidirect |
| 122. C22 | C2×Q8⋊H(3,𝔽3) | direct (N × H) |
| 129. C2 | C22×Q8⋊H(3,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8⋊H(3,𝔽3) | 2 |
| C23×C32×Q8 | 3 |
| C23×C3×SL(2,𝔽3) | 3 |
| C24×H(3,𝔽3) | 4 |