Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
521 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C9⋊S4 |
| commutator, cosocle, hypocenter | C9×A4 | C23 |
| Frattini | C3 | C22×C3⋊S4 |
| Fitting | C24×C9 | D3 |
| socle | C24×C3 | C3⋊D3 |
| LCS | G ▷ C9×A4 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C9×A4 | C2 | semidirect |
| 2. C2×C9⋊S4 | C2 | direct (N × H) |
| 7. C2×C9×A4 | C22 | semidirect |
| 8. C9⋊S4 | C22 | direct (N × H) |
| 11. C24×C9 | D3 | semidirect |
| 13. C22×C22⋊C9 | D3 | semidirect |
| 20. C9×A4 | C23 | semidirect |
| 28. C23×C9 | C2×D3 | semidirect |
| 33. C2×C22⋊C9 | C2×D3 | semidirect |
| 42. C22×A4 | D9 | semidirect |
| 55. C22×C9 | S4 | semidirect |
| 56. C22×C9 | C22×D3 | semidirect |
| 61. C22⋊C9 | C22×D3 | semidirect |
| 72. C2×A4 | C2×D9 | semidirect |
| 82. C2×C9 | C2×S4 | semidirect |
| 86. C24 | C3⋊D9 | semidirect |
| 101. A4 | C22×D9 | semidirect |
| 105. C9 | C22×S4 | semidirect |
| 111. C23 | C2×C3⋊D9 | semidirect |
| 128. C22 | C9⋊S4 | direct (N × H) |
| 129. C22 | C22×C3⋊D9 | semidirect |
| 136. C2 | C2×C9⋊S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C9×A4 | 2 |
| C2×C9⋊S4 | 2 |
| C22×C3⋊S4 | 3 |
| C22×(C22⋊C9)⋊C2 | 3 |
| C22×#611 | 3 |
| C22×C3⋊D9 | 4 |