Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
64 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2258 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | M128 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C64 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #504 | C4 | 1 | 1 | yes | characteristic |
| 5 | M64 | C4 | 1 | 3 | yes | — |
| 6 | C2×C32 | C22 | 1 | 3 | yes | — |
| 7 | C64 | C22 | 1 | 3 | yes | — |
| 8 | C64 | C22 | 1 | 1 | yes | characteristic |
| 9 | #131 | C8 | 1 | 1 | yes | characteristic |
| 10 | M32 | C8 | 1 | 3 | yes | — |
| 11 | C2×C16 | C2×C4 | 1 | 3 | yes | — |
| 12 | C32 | C2×C4 | 1 | 3 | yes | — |
| 13 | C32 | C23 | 1 | 1 | yes | characteristic |
| 14 | #41 | C16 | 1 | 1 | yes | characteristic |
| 15 | M16 | C16 | 1 | 3 | yes | — |
| 16 | C2×C8 | C2×C8 | 1 | 3 | yes | — |
| 17 | C16 | C2×C8 | 1 | 3 | yes | — |
| 18 | C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 19 | Q8 | C32 | 1 | 1 | yes | characteristic |
| 20 | D4 | C32 | 1 | 3 | yes | — |
| 21 | C2×C4 | C2×C16 | 1 | 3 | yes | — |
| 22 | C8 | C2×C16 | 1 | 3 | yes | — |
| 23 | C8 | C22×C8 | 1 | 1 | yes | characteristic |
| 24 | C22 | C2×C32 | 1 | 3 | yes | — |
| 25 | C4 | C2×C32 | 1 | 3 | yes | — |
| 26 | C4 | C22×C16 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 3 | no | — |
| 28 | C2 | C22×C32 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C64 | C22 |
| commutator, socle | C2 | C22×C32 |
| Frattini, cosocle | C32 | C23 |
| prime radical | #41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C64 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. M128 | C2 | semidirect |
| 3. C2×C64 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2258 | 2 |
| M128 | 2 |
| C2×C64 | 2 |