Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
250 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 73 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 | #120 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #386 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×M16 | C2 | semidirect |
| 4. C4×#103 | C2 | semidirect |
| 6. #1303 | C2 | semidirect |
| 7. #1278 | C2 | semidirect |
| 11. #475 | C4 | semidirect |
| 12. #475 | C4 | semidirect |
| 13. #460 | C4 | semidirect |
| 14. #460 | C4 | semidirect |
| 112. M16 | SD16 | semidirect |
| 152. Q8 | #104 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3103 | 2 |
| Q8×M16 | 2 |
| #1980 | 2 |
| C4×#103 | 2 |
| #1492 | 2 |
| #1303 | 2 |
| #1278 | 2 |