Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
269 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #106 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#106 |
| prime radical | C23 | C4×SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C2×C82 ◁ 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 |
|---|---|---|
| 19. #1368 | C4 | semidirect |
| 20. #1311 | C4 | semidirect |
| 21. C8×C16 | C4 | semidirect |
| 37. #365 | C8 | direct (N × H) |
| 132. C16 | C4×C8 | semidirect |
| 158. C8 | #365 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#365 | 2 |
| #12000 | 2 |
| C8×M32 | 2 |
| C8×#108 | 2 |
| C8×#106 | 2 |
| #7174 | 2 |
| #7173 | 2 |