Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
509 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 183 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 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2921 | C2 | semidirect |
| 3. #2905 | C2 | semidirect |
| 4. #1997 | C2 | semidirect |
| 5. #1967 | C2 | semidirect |
| 6. #1901 | C2 | semidirect |
| 8. #1801 | C2 | semidirect |
| 9. #1800 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#542 | 2 |
| #2921 | 2 |
| #2905 | 2 |
| #1997 | 2 |
| #1967 | 2 |
| #1901 | 2 |
| #1900 | 2 |
| #1801 | 2 |
| #1800 | 2 |