Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
434 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 434 conjugacy classes of subgroups — too many to draw. Only its 194 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 | D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle, prime radical | C23 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×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 |
|---|---|---|
| 3. C4×Q64 | C2 | direct (N × H) |
| 9. C2×Q64 | C4 | direct (N × H) |
| 34. Q64 | C2×C4 | direct (N × H) |
| 110. C2×C4 | Q64 | direct (N × H) |
| 129. C4 | C2×Q64 | direct (N × H) |
| 132. C2 | C4×Q64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q64 | 2 |
| C2×C4×Q32 | 2 |
| C4×Q64 | 2 |
| C2×#1423 | 2 |
| C2×#1416 | 2 |
| C2×C4×C32 | 2 |