Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
156 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C9 | #121 |
| commutator | C2×C4 | C23×C9 |
| Frattini, cosocle | C2×C3×C4 | C23×C3 |
| socle | C2×C3 | C3×#121 |
| prime radical | C3×AGL(1,ℤ/8) | C2×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C9 ◁ C2×C4×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#137 | C2 | semidirect |
| 3. C2×C9×SD16 | C2 | semidirect |
| 4. C9×#131 | C2 | semidirect |
| 5. C9×#108 | C2 | semidirect |
| 6. C9×#101 | C2 | semidirect |
| 7. C9×#100 | C2 | semidirect |
| 47. #471 | C9 | direct (N × H) |
| 71. #137 | C2×C9 | semidirect |
| 73. C2×SD16 | C2×C9 | semidirect |
| 74. #131 | C2×C9 | semidirect |
| 75. #108 | C2×C9 | semidirect |
| 76. #101 | C2×C9 | semidirect |
| 77. #100 | C2×C9 | semidirect |
| 121. C9 | #471 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#137 | 2 |
| C9×AGL(1,ℤ/8) | 2 |
| C2×C9×SD16 | 2 |
| C9×#131 | 2 |
| C9×#108 | 2 |
| C9×#101 | 2 |
| C9×#100 | 2 |
| C3×#471 | 3 |