Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
294 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #2233 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2218 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1426 | C2 | semidirect |
| 3. C2×#1419 | C2 | semidirect |
| 4. C2×#1419 | C2 | semidirect |
| 5. C2×#1400 | C2 | semidirect |
| 6. #12838 | C2 | semidirect |
| 7. #7474 | C2 | semidirect |
| 52. M64 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2218 | 2 |
| C2×#1426 | 2 |
| C2×#1419 | 2 |
| C2×#1400 | 2 |
| #12838 | 2 |
| #7474 | 2 |