Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
1278 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 287 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×#121 |
| 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. C2×#3757 | C2 | semidirect |
| 5. #64147 | C2 | direct (N × H) |
| 6. C2×C4⋊(C3⋊Dic3) | C2 | semidirect |
| 11. C2×C32×D4 | C22 | semidirect |
| 14. C2×#638 | C22 | semidirect |
| 18. C2×C4×C3⋊D3 | C22 | semidirect |
| 19. #3757 | C22 | semidirect |
| 20. C4⋊(C3⋊Dic3) | C22 | semidirect |
| 21. C22×C3×D4 | D3 | semidirect |
| 34. C22×C3⋊D3 | D4 | semidirect |
| 37. C2×C32×C4 | D4 | semidirect |
| 49. C2×C3×D4 | C2×D3 | semidirect |
| 70. C2×C3⋊D3 | C2×D4 | semidirect |
| 71. C32×C4 | C2×D4 | semidirect |
| 75. C22×D4 | C3⋊D3 | semidirect |
| 90. C2×C3×C4 | #67 | semidirect |
| 99. C2×C32 | #121 | semidirect |
| 109. C2×D4 | C2×C3⋊D3 | semidirect |
| 129. C3×C4 | C2×#67 | semidirect |
| 132. C32 | C2×#121 | semidirect |
| 149. C2×C4 | #638 | semidirect |
| 152. C2×C3 | #938 | semidirect |
| 171. C4 | C2×#638 | semidirect |
| 172. C3 | C2×#938 | semidirect |
| 175. C2 | #64147 | 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 |
| C2×#3757 | 2 |
| #64147 | 2 |
| C2×C4⋊(C3⋊Dic3) | 2 |
| C2×#938 | 3 |