Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
318 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#100 | C2 | semidirect |
| 4. C4×#99 | C2 | semidirect |
| 5. #1390 | C2 | semidirect |
| 8. C22×C22⋊C4 | C4 | semidirect |
| 17. #385 | C4 | semidirect |
| 18. #379 | C4 | semidirect |
| 20. C4×C22⋊C4 | C4 | semidirect |
| 21. AΣL(1,𝔽16) | C4 | semidirect |
| 69. C24 | C42 | semidirect |
| 114. C22⋊C4 | C42 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2321 | 2 |
| C2×AΣL(1,𝔽16) | 2 |
| C4×#100 | 2 |
| C4×#99 | 2 |
| #1390 | 2 |