Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
124 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12078 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | #346 | C23 |
| prime radical | #508 | C2×C4 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1313 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12088 | C2 | semidirect |
| 4. #7457 | C2 | semidirect |
| 5. #7391 | C2 | semidirect |
| 7. #7333 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13657 | 2 |
| #12088 | 2 |
| #12068 | 2 |
| #7457 | 2 |
| #7391 | 2 |
| #7389 | 2 |
| #7333 | 2 |