Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
393 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 116 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 | D4×D8 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×D8 | D8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C8×D8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33923 | C2 | semidirect |
| 3. #12522 | C2 | semidirect |
| 4. #12456 | C2 | semidirect |
| 5. #12450 | C2 | semidirect |
| 6. #12398 | C2 | semidirect |
| 7. #12393 | C2 | semidirect |
| 8. C8×Q32 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33923 | 2 |
| #20011 | 2 |
| #12522 | 2 |
| #12456 | 2 |
| #12450 | 2 |
| #12398 | 2 |
| #12393 | 2 |
| C8×Q32 | 2 |
| #12039 | 2 |