Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
550 conjugacy classes of subgroups; only the 250 normal subgroups are shown.
This group has 550 conjugacy classes of subgroups — too many to draw. Only its 250 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C9 | C2×D4 |
| commutator | C2×C4 | C23×C3×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C32 |
| socle | C22×C32 | C2×C3×D4 |
| prime radical | C32×#127 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C9 ◁ C3×C42×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C9×#125 | C2 | semidirect |
| 3. C3×C9×#102 | C2 | semidirect |
| 4. C3×C9×#97 | C2 | semidirect |
| 6. C9×#490 | C3 | direct (N × H) |
| 18. C9×#125 | C2×C3 | semidirect |
| 19. C9×#102 | C2×C3 | semidirect |
| 20. C9×#97 | C2×C3 | semidirect |
| 28. C3×#490 | C9 | direct (N × H) |
| 50. C3×#125 | C2×C9 | semidirect |
| 52. C3×#102 | C2×C9 | semidirect |
| 54. C3×#97 | C2×C9 | semidirect |
| 70. #490 | C3×C9 | direct (N × H) |
| 99. #125 | C2×C3×C9 | semidirect |
| 100. #102 | C2×C3×C9 | semidirect |
| 101. #97 | C2×C3×C9 | semidirect |
| 102. C3×C9 | #490 | direct (N × H) |
| 142. C9 | C3×#490 | direct (N × H) |
| 162. C3 | C9×#490 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C9×#127 | 2 |
| C3×C9×#125 | 2 |
| C3×C9×#102 | 2 |
| C3×C9×#97 | 2 |
| C32×#490 | 3 |
| C9×#490 | 3 |