Finite Group DatabasebetaFinite Group DatabasebetaOrder 2624.
662 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 143 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×D41 |
| commutator, cosocle | C22×C41 | C24 |
| Frattini | C22 | C23×D41 |
| Fitting | C2×C22⋊C4×C41 | C2 |
| socle | C23×C41 | C23 |
| hypercenter | C2×C22⋊C4 | D41 |
| hypocenter | C41 | C2×#120 |
| LCS | G ▷ C22×C41 ▷ C41 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×D41 | C2 | semidirect |
| 2. C22×D164 | C2 | semidirect |
| 3. C22×C22⋊D41 | C2 | semidirect |
| 4. #13050134 | C2 | direct (N × H) |
| 5. C2×#10620281 | C2 | semidirect |
| 6. C2×C22⋊C4×C41 | C2 | semidirect |
| 11. C23×D41 | C22 | semidirect |
| 12. C23×D41 | C22 | semidirect |
| 13. C22×C4×C41 | C22 | semidirect |
| 15. C2×C22⋊D41 | C22 | semidirect |
| 16. C22×Dic41 | C22 | semidirect |
| 18. C2×D164 | C22 | semidirect |
| 19. C22⋊C4×C41 | C22 | semidirect |
| 20. #10620281 | C22 | semidirect |
| 22. C23×C41 | D4 | semidirect |
| 32. C22×D41 | D4 | semidirect |
| 34. C22×D41 | C23 | semidirect |
| 36. C2×C4×C41 | C23 | semidirect |
| 39. C2×Dic41 | C23 | semidirect |
| 45. C22×C41 | C2×D4 | semidirect |
| 54. C2×D41 | C2×D4 | semidirect |
| 58. C2×C41 | #120 | semidirect |
| 64. C41 | C2×#120 | semidirect |
| 70. C2×C22⋊C4 | D41 | semidirect |
| 82. C22×C4 | C2×D41 | semidirect |
| 84. C22⋊C4 | C2×D41 | semidirect |
| 96. C23 | D164 | semidirect |
| 104. C2×C4 | C22×D41 | semidirect |
| 116. C22 | C2×D164 | semidirect |
| 124. C2 | #13050134 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×D41 | 2 |
| C22×D164 | 2 |
| C22×C22⋊D41 | 2 |
| #13050134 | 2 |
| C2×#10620281 | 2 |
| C2×C22⋊C4×C41 | 2 |
| C2×#120 | 41 |