Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3566 conjugacy classes of subgroups; only the 882 normal subgroups are shown.
This group has 3566 conjugacy classes of subgroups — too many to draw. Only its 882 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #14635 | C2 | semidirect |
| 4. C4×#393 | C2 | semidirect |
| 7. C22×C4×Q8 | C22 | semidirect |
| 8. #2840 | C4 | direct (N × H) |
| 13. #1446 | C22 | semidirect |
| 14. #1442 | C22 | semidirect |
| 17. C4×#95 | C22 | semidirect |
| 21. C24×C4 | Q8 | semidirect |
| 29. #393 | C2×C4 | semidirect |
| 44. C22×Q8 | C22×C4 | semidirect |
| 66. #95 | C22×C4 | semidirect |
| 75. C24 | C4×Q8 | semidirect |
| 148. C4 | #2840 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2840 | 2 |
| #14673 | 2 |
| #14635 | 2 |
| C4×#393 | 2 |
| C4×#379 | 2 |