Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
486 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | #1113 |
| commutator, hypocenter | C33 | C22×C16 |
| Frattini | C8 | C2×#1113 |
| Fitting | C33×C16 | C22 |
| socle | C2×C33 | C22×C8 |
| prime radical | C2×#1113 | C8 |
| cosocle | C33×C8 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C16×C3⋊D3 | C2 | semidirect |
| 3. #11722065 | C2 | semidirect |
| 8. C33×C16 | C22 | semidirect |
| 9. C3×C3⋊(C3⋊C16) | C22 | semidirect |
| 11. C16×C3⋊D3 | D3 | semidirect |
| 30. C32×C16 | C2×D3 | semidirect |
| 31. C3⋊(C3⋊C16) | C2×D3 | semidirect |
| 35. #1113 | C16 | direct (N × H) |
| 58. C3×C3⋊D3 | C2×C16 | semidirect |
| 68. C3×C16 | D32 | semidirect |
| 84. C33 | C22×C16 | semidirect |
| 100. C3⋊D3 | D3×C16 | semidirect |
| 107. C16 | #1113 | direct (N × H) |
| 120. C32 | C2×D3×C16 | semidirect |
| 138. C3 | D32×C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#1113 | 2 |
| C3×C16×C3⋊D3 | 2 |
| #11722065 | 2 |
| D32×C16 | 3 |