Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
249 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 249 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #102 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×M32 | C2 | semidirect |
| 4. C4×#108 | C2 | semidirect |
| 5. #1387 | C2 | semidirect |
| 18. #361 | C4 | direct (N × H) |
| 39. M32 | C2×C4 | semidirect |
| 42. #108 | C2×C4 | semidirect |
| 127. C4 | #361 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D8 | 2 |
| C2×#361 | 2 |
| C4×M32 | 2 |
| C4×#108 | 2 |
| #1387 | 2 |