Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
299 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8202 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M32 ◁ C4×M32 ◁ 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 |
|---|---|---|
| 2. #12667 | C2 | semidirect |
| 5. #11406 | C2 | semidirect |
| 6. #7180 | C2 | semidirect |
| 7. #7161 | C2 | semidirect |
| 11. #2170 | C4 | semidirect |
| 25. #1335 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33443 | 2 |
| #12667 | 2 |
| #12642 | 2 |
| #11602 | 2 |
| #11406 | 2 |
| #7180 | 2 |
| #7161 | 2 |