Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
495 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 495 conjugacy classes of subgroups — too many to draw. Only its 183 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 | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C4⋊Q8 | C2 | semidirect |
| 4. #1990 | C2 | semidirect |
| 6. #1926 | C2 | semidirect |
| 7. #1864 | C2 | semidirect |
| 8. #1844 | C2 | semidirect |
| 9. #1744 | C2 | semidirect |
| 16. #486 | C4 | semidirect |
| 44. C4⋊Q8 | C2×C4 | semidirect |
| 73. #102 | C2×C4 | semidirect |
| 75. C4×C8 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#486 | 2 |
| C4×C4⋊Q8 | 2 |
| C4×#127 | 2 |
| #1990 | 2 |
| #1957 | 2 |
| #1926 | 2 |
| #1864 | 2 |
| #1844 | 2 |
| #1744 | 2 |