Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
423 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 423 conjugacy classes of subgroups — too many to draw. Only its 129 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×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#473 | C2 | semidirect |
| 3. C2×C4×Q16 | C2 | semidirect |
| 5. #2052 | C2 | semidirect |
| 6. #1848 | C2 | semidirect |
| 7. #1816 | C2 | semidirect |
| 8. #1801 | C2 | semidirect |
| 45. C2×Q16 | D4 | semidirect |
| 198. C4 | #473 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#473 | 2 |
| #3037 | 2 |
| C2×C4×Q16 | 2 |
| #2931 | 2 |
| #2052 | 2 |
| #1848 | 2 |
| #1816 | 2 |
| #1801 | 2 |