Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1097 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1097 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #2584 | C2 | semidirect |
| 5. #2567 | C2 | semidirect |
| 6. #2546 | C2 | semidirect |
| 48. C2×C4⋊C4 | Q8 | semidirect |
| 127. C2×C4 | C4⋊Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#533 | 2 |
| #2857 | 2 |
| #2659 | 2 |
| #2584 | 2 |
| #2567 | 2 |
| #2546 | 2 |
| #2316 | 2 |
| C2×C4×C4⋊C4 | 2 |