Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
424 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 240 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 | C22×#106 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#41 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #12030 | C2 | semidirect |
| 5. #12029 | C2 | semidirect |
| 6. #12020 | C2 | semidirect |
| 7. #12000 | C2 | semidirect |
| 125. C16 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33485 | 2 |
| #17835 | 2 |
| #17826 | 2 |
| #12030 | 2 |
| #12029 | 2 |
| #12020 | 2 |
| #12000 | 2 |