Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
242 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 242 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 | C42 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1744 |
| prime radical | C23 | #431 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ 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 |
|---|---|---|
| 15. #1312 | C4 | semidirect |
| 16. #1312 | C4 | semidirect |
| 17. #1311 | C4 | semidirect |
| 18. C8×C16 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11411 | 2 |
| #11410 | 2 |
| #7844 | 2 |
| #7145 | 2 |
| #7142 | 2 |