Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
290 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 101 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3334 | C2 | semidirect |
| 2. #3285 | C2 | semidirect |
| 3. #2243 | C2 | semidirect |
| 4. #2236 | C2 | semidirect |
| 5. #2232 | C2 | semidirect |
| 6. #2227 | C2 | semidirect |
| 7. #2220 | C2 | semidirect |
| 8. #2215 | C2 | semidirect |
| 9. #2210 | C2 | semidirect |
| 10. #2186 | C2 | semidirect |
| 11. #2185 | C2 | semidirect |
| 12. #2175 | C2 | semidirect |
| 13. #2168 | C2 | semidirect |
| 14. #2139 | C2 | semidirect |
| 15. C2×#358 | C2 | semidirect |
| 146. Q16 | #41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3334 | 2 |
| #3285 | 2 |
| #2243 | 2 |
| #2236 | 2 |
| #2232 | 2 |
| #2227 | 2 |
| #2220 | 2 |
| #2215 | 2 |
| #2210 | 2 |
| #2186 | 2 |
| #2185 | 2 |
| #2175 | 2 |
| #2168 | 2 |
| #2139 | 2 |
| C2×#358 | 2 |