Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
354 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1598 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C2 | #8869 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12924 | C2 | semidirect |
| 5. #11295 | C2 | semidirect |
| 6. #7257 | C2 | semidirect |
| 7. #7194 | C2 | semidirect |
| 27. #1342 | C4 | semidirect |
| 28. #1341 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22205 | 2 |
| #12998 | 2 |
| #12924 | 2 |
| #11688 | 2 |
| #11295 | 2 |
| #7257 | 2 |
| #7194 | 2 |