Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
648 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊#157 |
| commutator, socle | C22×C33 | C22×C4 |
| Fitting | C33×C42 | C4 |
| prime radical, cosocle | C22×C3×C3⋊D3 | C23 |
| hypercenter | C42 | C3⋊#157 |
| hypocenter | C33 | #384 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C33×C42 | C4 | semidirect |
| 8. C3×C4×C3⋊Dic3 | C4 | semidirect |
| 25. C32×C42 | Dic3 | semidirect |
| 26. C4×C3⋊Dic3 | Dic3 | semidirect |
| 32. C33×C4 | C4⋊C4 | semidirect |
| 33. C3×C3⋊Dic3 | C4⋊C4 | semidirect |
| 59. C3×C42 | #157 | semidirect |
| 72. C32×C4 | C4⋊Dic3 | semidirect |
| 73. C3⋊Dic3 | C4⋊Dic3 | semidirect |
| 77. C33 | #384 | semidirect |
| 101. C42 | C3⋊#157 | semidirect |
| 112. C3×C4 | C4⋊#157 | semidirect |
| 117. C32 | #4935 | semidirect |
| 136. C4 | C4⋊(C3⋊#157) | semidirect |
| 139. C3 | #10593735 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊(C3⋊#157) | 2 |
| C3×C42×C3⋊D3 | 2 |
| #10593735 | 3 |
| #4935 | 9 |