Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2998 conjugacy classes of subgroups; only the 1922 normal subgroups are shown.
This group has 2998 conjugacy classes of subgroups — too many to draw. Only its 1922 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C2×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1455 | C2 | direct (N × H) |
| 3. C22×#322 | C2 | semidirect |
| 4. C22×C82 | C2 | semidirect |
| 12. #1455 | C22 | direct (N × H) |
| 13. C2×#322 | C22 | semidirect |
| 14. C2×C82 | C22 | semidirect |
| 31. #322 | C23 | semidirect |
| 32. C82 | C23 | semidirect |
| 42. C22×C8 | M16 | semidirect |
| 71. C2×C8 | C2×M16 | semidirect |
| 93. C8 | C22×M16 | semidirect |
| 96. C22 | #1455 | direct (N × H) |
| 103. C2 | C2×#1455 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×M16 | 2 |
| C2×#1455 | 2 |
| C22×#322 | 2 |
| C22×C82 | 2 |