Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
276 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | #3058 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×SD32 | C2 | semidirect |
| 2. C2×#473 | C2 | semidirect |
| 4. C2×#361 | C2 | semidirect |
| 5. #2141 | C2 | semidirect |
| 6. C2×#350 | C2 | semidirect |
| 7. #1382 | C2 | semidirect |
| 40. C2×Q16 | D4 | semidirect |
| 68. C2×C16 | D4 | semidirect |
| 70. #108 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD32 | 2 |
| C2×#473 | 2 |
| #3058 | 2 |
| C2×#361 | 2 |
| #2141 | 2 |
| C2×#350 | 2 |
| #1382 | 2 |