Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
268 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 102 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 | C4×C22⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1752 |
| prime radical | #121 | M16 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12593 | C2 | semidirect |
| 4. #11414 | C2 | semidirect |
| 6. #7166 | C2 | semidirect |
| 7. #7150 | C2 | semidirect |
| 9. #2183 | C4 | semidirect |
| 10. Q8×C16 | C4 | semidirect |
| 12. #2169 | C4 | semidirect |
| 14. D4×C16 | C4 | semidirect |
| 27. #1338 | C4 | semidirect |
| 28. #1330 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33483 | 2 |
| #12593 | 2 |
| #11620 | 2 |
| #11414 | 2 |
| #8249 | 2 |
| #7166 | 2 |
| #7150 | 2 |