Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
298 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #88813 |
| commutator | C33×C4 | C42 |
| Frattini, hypercenter | C2×C8 | #1113 |
| Fitting, cosocle | C2×C33×C8 | C22 |
| socle | C2×C33 | #97 |
| prime radical | C22×C33 | M16 |
| hypocenter | C33 | #347 |
| LCS | G ▷ C33×C4 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C3×C3⋊(C3⋊C16) | C4 | semidirect |
| 33. C3⋊(C3⋊C16) | Dic3 | semidirect |
| 104. C33 | #347 | semidirect |
| 158. C32 | #4466 | semidirect |
| 159. C32 | #4512 | semidirect |
| 181. C3 | #10589105 | semidirect |
| 182. C3 | #10589106 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63642 | 2 |
| C3×#63639 | 2 |
| #10589106 | 3 |
| #10589105 | 3 |