Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
413 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 111 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 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#344 | C2 | semidirect |
| 5. C2×#343 | C2 | semidirect |
| 6. #1738 | C2 | semidirect |
| 7. #1736 | C2 | semidirect |
| 23. #413 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#421 | 2 |
| C2×#417 | 2 |
| C2×#413 | 2 |
| C2×#344 | 2 |
| C2×#343 | 2 |
| #1738 | 2 |
| #1736 | 2 |