Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
428 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×D4 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q64 | C2 | semidirect |
| 3. #13726 | C2 | semidirect |
| 4. #13695 | C2 | semidirect |
| 5. #13688 | C2 | semidirect |
| 6. #13665 | C2 | semidirect |
| 7. C4×Q64 | C2 | semidirect |
| 47. Q64 | D4 | direct (N × H) |
| 132. D4 | Q64 | direct (N × H) |
| 149. C22 | C2×Q64 | semidirect |
| 156. C4 | C2×Q64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q64 | 2 |
| D4×Q32 | 2 |
| #13726 | 2 |
| #13695 | 2 |
| #13688 | 2 |
| #13665 | 2 |
| C4×Q64 | 2 |
| D4×C32 | 2 |