Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
701 conjugacy classes of subgroups; only the 235 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 235 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1818 | C2 | semidirect |
| 4. #1817 | C2 | semidirect |
| 5. #1729 | C2 | semidirect |
| 6. #1728 | C2 | semidirect |
| 7. C2×#336 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#41 | 2 |
| C2×#408 | 2 |
| #1818 | 2 |
| #1817 | 2 |
| #1729 | 2 |
| #1728 | 2 |
| C2×#336 | 2 |