Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
563 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 563 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#107 | C2 | semidirect |
| 5. #2274 | C2 | semidirect |
| 6. #1928 | C2 | semidirect |
| 7. #1925 | C2 | semidirect |
| 8. #1736 | C2 | semidirect |
| 9. #1729 | C2 | semidirect |
| 20. #416 | C4 | semidirect |
| 77. #103 | C2×C4 | semidirect |
| 79. #102 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#107 | 2 |
| C2×#417 | 2 |
| C2×#416 | 2 |
| #2311 | 2 |
| #2274 | 2 |
| #1928 | 2 |
| #1925 | 2 |
| #1736 | 2 |
| #1729 | 2 |