Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
367 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #385 |
| prime radical | C22×D4 | C4⋊C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #8808 | C2 | semidirect |
| 4. #8786 | C2 | semidirect |
| 6. #7030 | C2 | semidirect |
| 7. #7030 | C2 | semidirect |
| 140. #97 | M16 | semidirect |
| 141. #97 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23353 | 2 |
| #9640 | 2 |
| #8808 | 2 |
| #8786 | 2 |
| C4×#329 | 2 |
| #7030 | 2 |