Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
303 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D4 |
| commutator | C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C8×C4⋊C4 | C2 | semidirect |
| 5. C2×C82 | C2 | semidirect |
| 6. #1279 | C2 | semidirect |
| 7. #1277 | C2 | semidirect |
| 11. C8×D4 | C4 | semidirect |
| 23. D4⋊C8 | C4 | semidirect |
| 49. #102 | C8 | direct (N × H) |
| 90. C2×C8 | C2×C8 | semidirect |
| 96. C4⋊C4 | C2×C8 | semidirect |
| 113. D4 | C4×C8 | semidirect |
| 142. C8 | #102 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D4 | 2 |
| C8×C4⋊C4 | 2 |
| C4×#102 | 2 |
| C2×D4⋊C8 | 2 |
| C2×C82 | 2 |
| #1279 | 2 |
| #1277 | 2 |