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, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #3166 | C2 | direct (N × H) |
| 4. C2×#501 | C2 | semidirect |
| 5. C2×#499 | C2 | semidirect |
| 24. #501 | C22 | semidirect |
| 25. #499 | C22 | semidirect |
| 36. C2×M16 | Q8 | semidirect |
| 89. M16 | C2×Q8 | semidirect |
| 145. C2 | #3166 | 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 |
| C22×#125 | 2 |
| #3166 | 2 |
| C2×#501 | 2 |
| C2×#499 | 2 |
| C2×#428 | 2 |
| C2×C4×M16 | 2 |