Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
951 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 951 conjugacy classes of subgroups — too many to draw. Only its 567 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C23 | C22×D4 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #3162 | C2 | direct (N × H) |
| 3. C2×#501 | C2 | semidirect |
| 15. #501 | C22 | semidirect |
| 24. C22×C8 | Q8 | semidirect |
| 60. C2×C8 | C2×Q8 | semidirect |
| 101. C2 | #3162 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| #3162 | 2 |
| C2×#501 | 2 |
| C2×#427 | 2 |
| C22×#97 | 2 |