Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
952 conjugacy classes of subgroups; only the 364 normal subgroups are shown.
This group has 952 conjugacy classes of subgroups — too many to draw. Only its 364 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C33 | #99 |
| commutator | C23 | C2×C33×C4 |
| Frattini, cosocle | C2×D4 | C22×C33 |
| prime radical | C2×C33×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ C22×C33 ◁ C2×C33×D4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C32×#352 | C3 | direct (N × H) |
| 6. C22×C33×C4 | C4 | semidirect |
| 11. C33×C22⋊C4 | C4 | semidirect |
| 24. C3×#352 | C32 | direct (N × H) |
| 26. C22×C32×C4 | C3×C4 | semidirect |
| 31. C32×C22⋊C4 | C3×C4 | semidirect |
| 53. #352 | C33 | direct (N × H) |
| 59. C22×C3×C4 | C32×C4 | semidirect |
| 64. C3×C22⋊C4 | C32×C4 | semidirect |
| 77. C33 | #352 | direct (N × H) |
| 92. C22×C4 | C33×C4 | semidirect |
| 97. C22⋊C4 | C33×C4 | semidirect |
| 107. C32 | C3×#352 | direct (N × H) |
| 130. C3 | C32×#352 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#123 | 2 |
| C33×#100 | 2 |
| C33×#99 | 2 |
| C32×#352 | 3 |