Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
336 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #7980 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #2151 | C4 | semidirect |
| 10. #2151 | C4 | semidirect |
| 13. C4×M32 | C4 | semidirect |
| 14. C2×C4×C16 | C4 | semidirect |
| 15. #1841 | C4 | semidirect |
| 22. #1747 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33420 | 2 |
| #20787 | 2 |
| #7229 | 2 |
| #7228 | 2 |
| #7227 | 2 |