Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
473 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 473 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C2×#120 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #8164 | C2 | semidirect |
| 5. C8×#99 | C2 | semidirect |
| 6. #7204 | C2 | semidirect |
| 7. #7203 | C2 | semidirect |
| 9. C22×C22⋊C8 | C4 | semidirect |
| 17. #1850 | C4 | semidirect |
| 18. #1779 | C4 | semidirect |
| 21. #1752 | C4 | semidirect |
| 27. #1326 | C4 | semidirect |
| 28. #1316 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22030 | 2 |
| C4×AΣL(1,𝔽16) | 2 |
| #11471 | 2 |
| #8164 | 2 |
| C8×#99 | 2 |
| #7204 | 2 |
| #7203 | 2 |