Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
247 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #459 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9541 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C16 ◁ 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. #12664 | C2 | semidirect |
| 4. #12011 | C2 | semidirect |
| 6. #7181 | C2 | semidirect |
| 7. #7141 | C2 | semidirect |
| 48. #362 | C8 | semidirect |
| 49. #361 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13195 | 2 |
| #12664 | 2 |
| #12646 | 2 |
| #12011 | 2 |
| #11602 | 2 |
| #7181 | 2 |
| #7141 | 2 |