Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
490 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C8×#97 | C2 | semidirect |
| 5. #7030 | C2 | semidirect |
| 53. #329 | C8 | direct (N × H) |
| 93. #97 | C2×C8 | semidirect |
| 202. C8 | #329 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#124 | 2 |
| C4×#329 | 2 |
| C8×#97 | 2 |
| #7567 | 2 |
| #7030 | 2 |