Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
291 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 291 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, socle | C22 | #1853 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#41 | C4⋊C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#131 ◁ 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. #12000 | C2 | semidirect |
| 4. C2×#1368 | C2 | semidirect |
| 6. #7158 | C2 | semidirect |
| 7. #7157 | C2 | semidirect |
| 103. M32 | M16 | semidirect |
| 111. C2×C16 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #17798 | 2 |
| #12604 | 2 |
| #12000 | 2 |
| C2×#1368 | 2 |
| #8604 | 2 |
| #7158 | 2 |
| #7157 | 2 |