Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1042 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3 | C22×D9 |
| commutator, cosocle | C23×C9 | C23×C3 |
| Frattini | C23×C3 | C22×C3×D3 |
| Fitting | C3×C9×#95 | C2 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C3×#95 | D9 |
| hypocenter | C9 | C3×#392 |
| LCS | G ▷ C23×C9 ▷ C9 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×D36 | C2 | semidirect |
| 2. C2×C3×(C2×D9)⋊C4 | C2 | semidirect |
| 3. C3×C9×#95 | C2 | semidirect |
| 5. #10588351 | C3 | direct (N × H) |
| 14. C22×D36 | C2×C3 | semidirect |
| 16. C2×(C2×D9)⋊C4 | C2×C3 | semidirect |
| 18. C9×#95 | C2×C3 | semidirect |
| 24. C22×C3×D9 | D4 | semidirect |
| 26. C2×C3×C4×C9 | D4 | semidirect |
| 59. C3×#95 | D9 | semidirect |
| 77. C22×D9 | C3×D4 | semidirect |
| 81. C2×C4×C9 | C3×D4 | semidirect |
| 128. #95 | C3×D9 | semidirect |
| 129. C3×C9 | #392 | semidirect |
| 149. C2×C3×C4 | D36 | semidirect |
| 196. C9 | C3×#392 | semidirect |
| 206. C2×C4 | C3×D36 | semidirect |
| 229. C3 | #10588351 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D36 | 2 |
| C2×C3×(C2×D9)⋊C4 | 2 |
| C3×C9×#95 | 2 |
| C3×#4671 | 3 |
| #10588351 | 3 |