Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
784 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C33×C8 | D4 |
| commutator | C4 | C2×C33×C8 |
| Frattini, cosocle | C2×C8 | C22×C33 |
| socle | C2×C33 | C22⋊C8 |
| prime radical | C33×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C8 ◁ C2×C33×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C33×M32 | C2 | semidirect |
| 3. C2×C33×C16 | C2 | semidirect |
| 4. C32×#350 | C3 | direct (N × H) |
| 13. C32×M32 | C2×C3 | semidirect |
| 14. C2×C32×C16 | C2×C3 | semidirect |
| 23. C3×#350 | C32 | direct (N × H) |
| 37. C3×M32 | C2×C32 | semidirect |
| 38. C2×C3×C16 | C2×C32 | semidirect |
| 47. #350 | C33 | direct (N × H) |
| 64. M32 | C2×C33 | semidirect |
| 65. C2×C16 | C2×C33 | semidirect |
| 66. C33 | #350 | direct (N × H) |
| 90. C32 | C3×#350 | direct (N × H) |
| 107. C3 | C32×#350 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#131 | 2 |
| C33×M32 | 2 |
| C2×C33×C16 | 2 |
| C32×#350 | 3 |