Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
369 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | C4×#99 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #8165 | C2 | semidirect |
| 5. C8×#101 | C2 | semidirect |
| 6. #7206 | C2 | semidirect |
| 7. #7206 | C2 | semidirect |
| 128. #101 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22210 | 2 |
| C4×#355 | 2 |
| #11476 | 2 |
| #8165 | 2 |
| C8×#101 | 2 |
| #7206 | 2 |