Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5634 conjugacy classes of subgroups; only the 2178 normal subgroups are shown.
This group has 5634 conjugacy classes of subgroups — too many to draw. Only its 2178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle, prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13856 | C2 | direct (N × H) |
| 10. C2×#376 | C4 | semidirect |
| 28. #376 | C2×C4 | semidirect |
| 52. C2×C42 | C4⋊C4 | semidirect |
| 85. C42 | C2×C4⋊C4 | semidirect |
| 122. C2 | #13856 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C4⋊C4 | 2 |
| C22×#376 | 2 |
| #13856 | 2 |
| C2×#1444 | 2 |