Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
542 conjugacy classes of subgroups; only the 442 normal subgroups are shown.
This group has 542 conjugacy classes of subgroups — too many to draw. Only its 442 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C16 | C22 |
| commutator | C2 | C23×C32 |
| Frattini, cosocle | C2×C16 | C24 |
| socle, prime radical | C24 | C2×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×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 |
|---|---|---|
| 3. C2×C4⋊C32 | C2 | direct (N × H) |
| 11. C4⋊C32 | C22 | direct (N × H) |
| 50. C22×C4 | C32 | semidirect |
| 76. C2×C4 | C2×C32 | semidirect |
| 87. C22 | C4⋊C32 | direct (N × H) |
| 94. C4 | C22×C32 | semidirect |
| 97. C2 | C2×C4⋊C32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C32 | 2 |
| C22×C4×C16 | 2 |
| C2×C4⋊C32 | 2 |