Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
441 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 243 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C24 |
| commutator | C22 | C24×C9 |
| Frattini, socle, cosocle | C22×C3 | C24×C3 |
| prime radical | C3×#537 | C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9×#41 | C2 | semidirect |
| 2. C9×C22⋊Q8 | C2 | semidirect |
| 3. C9×#121 | C2 | semidirect |
| 4. C9×#121 | C2 | semidirect |
| 5. C2×C9×C4⋊C4 | C2 | semidirect |
| 31. C2×C4×C9 | D4 | semidirect |
| 35. #537 | C9 | direct (N × H) |
| 57. C2×#41 | C2×C9 | semidirect |
| 58. C22⋊Q8 | C2×C9 | semidirect |
| 59. #121 | C2×C9 | semidirect |
| 60. #121 | C2×C9 | semidirect |
| 61. C2×C4⋊C4 | C2×C9 | semidirect |
| 100. C9 | #537 | direct (N × H) |
| 109. C2×C4 | D4×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×#41 | 2 |
| C9×C22⋊Q8 | 2 |
| C9×#121 | 2 |
| C2×C9×C4⋊C4 | 2 |
| C3×#537 | 3 |