Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
497 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 227 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1493 | C2 | semidirect |
| 5. #1487 | C2 | semidirect |
| 6. #1481 | C2 | semidirect |
| 7. C2×Q8⋊C8 | C2 | semidirect |
| 131. Q8 | #131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×#41 | 2 |
| C8×#41 | 2 |
| #2906 | 2 |
| #1493 | 2 |
| #1487 | 2 |
| #1481 | 2 |
| C2×Q8⋊C8 | 2 |