Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
357 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 96 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 | #1677 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ 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 |
|---|---|---|
| 3. #13207 | C2 | semidirect |
| 5. C2×#1358 | C2 | semidirect |
| 6. #7271 | C2 | semidirect |
| 7. #7161 | C2 | semidirect |
| 13. #2214 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2214 | 2 |
| C2×#2171 | 2 |
| #13207 | 2 |
| #12824 | 2 |
| C2×#1358 | 2 |
| #7271 | 2 |
| #7161 | 2 |