Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
445 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 445 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2320 | C2 | semidirect |
| 4. #1928 | C2 | semidirect |
| 5. #1920 | C2 | semidirect |
| 8. C4×#107 | C2 | semidirect |
| 9. #1763 | C2 | semidirect |
| 11. #1746 | C2 | semidirect |
| 16. C4×SD16 | C4 | semidirect |
| 52. C4×Q8 | C2×C4 | semidirect |
| 72. #106 | C2×C4 | semidirect |
| 74. #103 | C2×C4 | semidirect |
| 78. 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×C4×SD16 | 2 |
| #2320 | 2 |
| #2275 | 2 |
| #1928 | 2 |
| #1920 | 2 |
| #1877 | 2 |
| #1862 | 2 |
| C4×#107 | 2 |
| #1763 | 2 |
| #1762 | 2 |
| #1746 | 2 |