Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
1494 conjugacy classes of subgroups; only the 295 normal subgroups are shown.
This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 295 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×C3⋊D3 |
| commutator, cosocle | C22×C32 | C24 |
| Frattini | C22 | C23×C3⋊D3 |
| Fitting | C22×C32×D4 | C2 |
| socle | C23×C32 | C23 |
| hypercenter | C22×D4 | C3⋊D3 |
| hypocenter | C32 | C2×#127 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C32×D4 | C2 | semidirect |
| 2. C22×#638 | C2 | semidirect |
| 3. C22×C4⋊(C3⋊D3) | C2 | semidirect |
| 4. #64149 | C2 | direct (N × H) |
| 5. C2×C4×C3⋊Dic3 | C2 | semidirect |
| 10. C2×C32×D4 | C22 | semidirect |
| 13. C2×#638 | C22 | semidirect |
| 16. C2×C4⋊(C3⋊D3) | C22 | semidirect |
| 17. C4×C3⋊Dic3 | C22 | semidirect |
| 18. C22×C3×D4 | D3 | semidirect |
| 31. C2×C32×C4 | D4 | semidirect |
| 34. C2×C3⋊Dic3 | D4 | semidirect |
| 41. C2×C3×D4 | C2×D3 | semidirect |
| 58. C32×C4 | C2×D4 | semidirect |
| 59. C3⋊Dic3 | C2×D4 | semidirect |
| 62. C22×D4 | C3⋊D3 | semidirect |
| 73. C2×C3×C4 | #67 | semidirect |
| 80. C2×C32 | #127 | semidirect |
| 91. C2×D4 | C2×C3⋊D3 | semidirect |
| 104. C3×C4 | C2×#67 | semidirect |
| 106. C32 | C2×#127 | semidirect |
| 119. C2×C4 | #638 | semidirect |
| 122. C2×C3 | #940 | semidirect |
| 136. C4 | C2×#638 | semidirect |
| 137. C3 | C2×#940 | semidirect |
| 140. C2 | #64149 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×D4 | 2 |
| C22×#638 | 2 |
| C22×C4⋊(C3⋊D3) | 2 |
| #64149 | 2 |
| C2×C4×C3⋊Dic3 | 2 |
| C2×#940 | 3 |