Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
408 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22⋊C4 |
| commutator | C22 | C2×C4×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C16×C22⋊C4 | C2 | semidirect |
| 5. #7199 | C2 | semidirect |
| 6. C23×C16 | C4 | semidirect |
| 11. C2×C4⋊C16 | C4 | semidirect |
| 12. C2×C22⋊C16 | C4 | semidirect |
| 16. C2×C4×C16 | C4 | semidirect |
| 21. #1322 | C4 | semidirect |
| 22. #1314 | C4 | semidirect |
| 101. #99 | C16 | direct (N × H) |
| 153. C22⋊C4 | C2×C16 | semidirect |
| 163. C16 | #99 | direct (N × H) |
| 170. C23 | C4×C16 | semidirect |
| 194. C2×C4 | C4×C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×C16 | 2 |
| C16×C22⋊C4 | 2 |
| #11465 | 2 |
| C8×#99 | 2 |
| #7199 | 2 |