Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
186 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 87 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 | C2×D4 |
| commutator | C2×C4 | C23×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle | C22×C3 | C2×C3×D4 |
| prime radical | C3×#121 | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ C22×C4×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9×SD16 | C2 | semidirect |
| 2. C22×C8×C9 | C2 | semidirect |
| 3. C9×C22⋊Q8 | C2 | semidirect |
| 5. C9×#106 | C2 | semidirect |
| 6. C9×#103 | C2 | semidirect |
| 7. C9×#102 | C2 | semidirect |
| 44. C8×C9 | D4 | semidirect |
| 45. #465 | C9 | direct (N × H) |
| 64. C22×C9 | SD16 | semidirect |
| 72. C2×SD16 | C2×C9 | semidirect |
| 73. C22×C8 | C2×C9 | semidirect |
| 74. C22⋊Q8 | C2×C9 | semidirect |
| 76. #106 | C2×C9 | semidirect |
| 77. #103 | C2×C9 | semidirect |
| 78. #102 | C2×C9 | semidirect |
| 126. C9 | #465 | direct (N × H) |
| 142. C8 | D4×C9 | semidirect |
| 153. C22 | C9×SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×SD16 | 2 |
| C22×C8×C9 | 2 |
| C9×C22⋊Q8 | 2 |
| C9×#121 | 2 |
| C9×#106 | 2 |
| C9×#103 | 2 |
| C9×#102 | 2 |
| C3×#465 | 3 |