Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
347 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C24 |
| commutator, socle | C22 | C23×C16 |
| prime radical | C2×#41 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×C32 | C2 | semidirect |
| 3. #13646 | C2 | semidirect |
| 4. #13640 | C2 | semidirect |
| 5. C4×M64 | C2 | semidirect |
| 61. C32 | #41 | semidirect |
| 87. Q8 | M64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C16×#41 | 2 |
| Q8×C32 | 2 |
| #13646 | 2 |
| #13640 | 2 |
| C4×M64 | 2 |