Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
346 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 72 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 | #2054 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33771 | C2 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ 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. #13252 | C2 | semidirect |
| 3. #12015 | C2 | semidirect |
| 4. #11981 | C2 | semidirect |
| 5. #9802 | C2 | semidirect |
| 6. #9719 | C2 | semidirect |
| 7. #7296 | C2 | semidirect |
| 174. M16 | D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33771 | 2 |
| #13252 | 2 |
| #12015 | 2 |
| #11981 | 2 |
| #9802 | 2 |
| #9719 | 2 |
| #7296 | 2 |