Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
21970 conjugacy classes of subgroups; only the 4418 normal subgroups are shown.
This group has 21970 conjugacy classes of subgroups — too many to draw. Only its 4418 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#1446 | C2 | direct (N × H) |
| 5. C24×Q8 | C4 | semidirect |
| 11. #1446 | C22 | direct (N × H) |
| 16. C23×Q8 | C2×C4 | semidirect |
| 23. C24×C4 | Q8 | semidirect |
| 37. C22×Q8 | C22×C4 | semidirect |
| 49. C23×C4 | C2×Q8 | semidirect |
| 80. C22×C4 | C22×Q8 | semidirect |
| 104. C22 | #1446 | direct (N × H) |
| 117. C2 | C2×#1446 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×Q8 | 2 |
| C23×#95 | 2 |
| C2×#1446 | 2 |