Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
167 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 167 conjugacy classes of subgroups — too many to draw. Only its 58 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 | #2246 |
| commutator, Frattini, cosocle | C4×C16 | C23 |
| prime radical | #2246 | C22 |
| LCS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13733 | C2 | semidirect |
| 2. #13731 | C2 | semidirect |
| 5. #12522 | C2 | semidirect |
| 7. #7331 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13733 | 2 |
| #13731 | 2 |
| #13719 | 2 |
| #12522 | 2 |
| #12503 | 2 |
| #7331 | 2 |