Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
22279 conjugacy classes of subgroups; only the 7728 normal subgroups are shown.
This group has 22279 conjugacy classes of subgroups — too many to draw. Only its 7728 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D4 |
| commutator | C4 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C2 | C24×C22⋊C4 |
| prime radical | #63103 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #63076 | C2 | semidirect |
| 3. C4×#585 | C2 | semidirect |
| 4. #60093 | C2 | semidirect |
| 5. C2×#2901 | C2 | semidirect |
| 7. 2−1+6 | C4 | semidirect |
| 9. H(7,𝔽2) | C4 | semidirect |
| 14. C2×#568 | C22 | semidirect |
| 17. #3428 | C22 | semidirect |
| 18. C2×C4×#41 | C22 | semidirect |
| 19. #2901 | C22 | semidirect |
| 20. #2901 | C22 | semidirect |
| 21. C2×#421 | C22 | semidirect |
| 22. C22×#104 | C22 | semidirect |
| 28. #585 | C2×C4 | semidirect |
| 33. C2×2−1+4 | C2×C4 | semidirect |
| 37. C2×H(5,𝔽2) | C2×C4 | semidirect |
| 43. #568 | C23 | semidirect |
| 45. C22×M16 | C23 | semidirect |
| 50. C4×#41 | C23 | semidirect |
| 51. C4×#41 | C23 | semidirect |
| 54. C2×#117 | C23 | semidirect |
| 55. C22×C42 | C23 | semidirect |
| 56. #421 | C23 | semidirect |
| 57. C2×#104 | C23 | semidirect |
| 58. C2×#104 | C23 | semidirect |
| 63. 2−1+4 | C22×C4 | semidirect |
| 68. H(5,𝔽2) | C22×C4 | semidirect |
| 76. C2×#41 | C22×C4 | semidirect |
| 82. C22×Q8 | C22×C4 | semidirect |
| 86. C22×D4 | C22×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63103 | 2 |
| #63076 | 2 |
| C4×#585 | 2 |
| #60093 | 2 |
| C2×#2901 | 2 |