Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
468 conjugacy classes of subgroups; only the 254 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 254 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator | C22 | C23×C16 |
| socle | C23 | C23×C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13159 | C2 | semidirect |
| 4. #13104 | C2 | semidirect |
| 6. #12898 | C2 | semidirect |
| 7. #12894 | C2 | semidirect |
| 59. #126 | C16 | semidirect |
| 108. C4⋊C4 | C2×C16 | semidirect |
| 116. C42 | C2×C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#126 | 2 |
| #13159 | 2 |
| #13120 | 2 |
| #13104 | 2 |
| #12912 | 2 |
| #12898 | 2 |
| #12894 | 2 |