Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
498 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 154 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 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1713 | C2 | semidirect |
| 4. #12448 | C2 | semidirect |
| 5. #12447 | C2 | semidirect |
| 6. #12004 | C2 | semidirect |
| 7. #11941 | C2 | semidirect |
| 79. #135 | Q16 | semidirect |
| 153. Q16 | C2×Q16 | semidirect |
| 156. D8 | C2×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33629 | 2 |
| C2×#1713 | 2 |
| C8×#135 | 2 |
| #12448 | 2 |
| #12447 | 2 |
| #12004 | 2 |
| #11941 | 2 |