Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
453 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 128 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 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33796 | C2 | semidirect |
| 4. #18086 | C2 | semidirect |
| 5. #12500 | C2 | semidirect |
| 6. #12486 | C2 | semidirect |
| 7. #12448 | C2 | semidirect |
| 8. #12419 | C2 | semidirect |
| 9. #12018 | C2 | semidirect |
| 11. #11941 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33820 | 2 |
| #33796 | 2 |
| #19630 | 2 |
| #18086 | 2 |
| #12500 | 2 |
| #12486 | 2 |
| #12448 | 2 |
| #12419 | 2 |
| #12018 | 2 |
| #11948 | 2 |
| #11941 | 2 |