Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
304 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3169 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C4×#97 ◁ 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 |
|---|---|---|
| 2. #12594 | C2 | semidirect |
| 3. #12020 | C2 | semidirect |
| 4. #11681 | C2 | semidirect |
| 6. #9850 | C2 | semidirect |
| 10. #2252 | C4 | semidirect |
| 11. #2248 | C4 | semidirect |
| 12. #2241 | C4 | semidirect |
| 13. #2240 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33751 | 2 |
| #12594 | 2 |
| #12020 | 2 |
| #11681 | 2 |
| #11680 | 2 |
| #9850 | 2 |
| #9849 | 2 |