Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
411 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 411 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 52. #107 | Q8 | semidirect |
| 53. C4⋊C8 | Q8 | semidirect |
| 55. C4×C8 | Q8 | semidirect |
| 127. C8 | C4⋊Q8 | semidirect |
| 141. C4 | #500 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#500 | 2 |
| #2551 | 2 |
| #2057 | 2 |
| #1947 | 2 |
| #1942 | 2 |
| #1845 | 2 |
| C4×#107 | 2 |
| C8×C4⋊C4 | 2 |