Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
418 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 186 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 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×#97 |
| prime radical | C24 | #97 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ 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 |
|---|---|---|
| 3. C2×#1313 | C2 | semidirect |
| 5. #7137 | C2 | semidirect |
| 24. C22×C16 | C8 | semidirect |
| 123. C16 | C22⋊C8 | semidirect |
| 160. C22 | #1313 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#346 | 2 |
| #12599 | 2 |
| C2×#1313 | 2 |
| #8144 | 2 |
| #7137 | 2 |