Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
502 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 143 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, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2657 | C2 |
| LCS | G ▷ C22×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 |
|---|---|---|
| 1. C2×#490 | C2 | semidirect |
| 2. C2×#489 | C2 | semidirect |
| 3. C2×#488 | C2 | semidirect |
| 4. C2×#486 | C2 | semidirect |
| 5. #2660 | C2 | semidirect |
| 7. #1961 | C2 | semidirect |
| 8. #1959 | C2 | semidirect |
| 9. #1958 | C2 | semidirect |
| 10. #1957 | C2 | semidirect |
| 11. #1726 | C2 | semidirect |
| 88. #97 | D4 | semidirect |
| 90. C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#490 | 2 |
| C2×#489 | 2 |
| C2×#488 | 2 |
| C2×#486 | 2 |
| #2660 | 2 |
| #2657 | 2 |
| #1961 | 2 |
| #1959 | 2 |
| #1958 | 2 |
| #1957 | 2 |
| #1726 | 2 |