Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
1014 conjugacy classes of subgroups; only the 359 normal subgroups are shown.
This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 359 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×C8 | C3⋊D3 |
| commutator, hypocenter | C32 | C23×C8 |
| Frattini | C4 | C23×C3⋊D3 |
| Fitting | C22×C32×C8 | C2 |
| socle | C23×C32 | C2×C4 |
| prime radical | C23×C3⋊D3 | C4 |
| cosocle | C32×C4 | C24 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C32×C8 | C2 | semidirect |
| 3. C22×C3⋊(C3⋊C8) | C2 | semidirect |
| 4. C2×C8×C3⋊D3 | C2 | direct (N × H) |
| 11. C2×C32×C8 | C22 | semidirect |
| 12. C2×C3⋊(C3⋊C8) | C22 | semidirect |
| 13. C8×C3⋊D3 | C22 | direct (N × H) |
| 15. C22×C3×C8 | D3 | semidirect |
| 19. C22×C3⋊D3 | C8 | direct (N × H) |
| 26. C32×C8 | C23 | semidirect |
| 27. C3⋊(C3⋊C8) | C23 | semidirect |
| 34. C2×C3×C8 | C2×D3 | semidirect |
| 37. C22×C32 | C2×C8 | semidirect |
| 39. C2×C3⋊D3 | C2×C8 | direct (N × H) |
| 47. C22×C8 | C3⋊D3 | direct (N × H) |
| 56. C3×C8 | C22×D3 | semidirect |
| 58. C2×C32 | C22×C8 | semidirect |
| 60. C3⋊D3 | C22×C8 | direct (N × H) |
| 68. C2×C8 | C2×C3⋊D3 | direct (N × H) |
| 69. C22×C3 | D3×C8 | semidirect |
| 76. C32 | C23×C8 | semidirect |
| 86. C8 | C22×C3⋊D3 | direct (N × H) |
| 87. C2×C3 | C2×D3×C8 | semidirect |
| 92. C22 | C8×C3⋊D3 | direct (N × H) |
| 97. C3 | C22×D3×C8 | semidirect |
| 99. C2 | C2×C8×C3⋊D3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C3⋊D3 | 2 |
| C22×C32×C8 | 2 |
| C22×C3⋊(C3⋊C8) | 2 |
| C2×C8×C3⋊D3 | 2 |
| C22×D3×C8 | 3 |