Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
231 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C22×Q32 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×Q64 | C2 | 1 | 12 | yes | maximal, direct product factor |
| 3 | C22×C32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C22×Q16 | — | 2 | 2 | no | — |
| 5 | C2×Q32 | — | 2 | 6 | no | — |
| 6 | C2×Q32 | C22 | 1 | 12 | yes | — |
| 7 | C22×C16 | C22 | 1 | 1 | yes | characteristic |
| 8 | Q64 | C22 | 1 | 16 | yes | direct product factor |
| 9 | C2×C32 | C22 | 1 | 6 | yes | — |
| 10 | C22×Q8 | — | 4 | 2 | no | — |
| 11 | C2×Q16 | — | 2 | 12 | no | — |
| 12 | C2×Q16 | — | 4 | 6 | no | — |
| 13 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 14 | Q32 | — | 2 | 12 | no | — |
| 15 | Q32 | C23 | 1 | 8 | yes | — |
| 16 | C2×C16 | D4 | 1 | 3 | yes | — |
| 17 | C2×C16 | C23 | 1 | 3 | yes | — |
| 18 | C32 | C23 | 1 | 4 | yes | — |
| 19 | C2×Q8 | — | 4 | 12 | no | — |
| 20 | C2×Q8 | — | 8 | 6 | no | — |
| 21 | C22×C4 | — | 8 | 2 | no | — |
| 22 | C22×C4 | D8 | 1 | 1 | yes | characteristic |
| 23 | Q16 | — | 2 | 8 | no | — |
| 24 | Q16 | — | 4 | 12 | no | — |
| 25 | C2×C8 | D8 | 1 | 3 | yes | — |
| 26 | C2×C8 | C2×D4 | 1 | 3 | yes | — |
| 27 | C16 | C2×D4 | 1 | 3 | yes | — |
| 28 | C16 | C24 | 1 | 1 | yes | characteristic |
| 29 | C23 | D16 | 1 | 1 | yes | characteristic |
| 30 | Q8 | — | 4 | 8 | no | — |
| 31 | Q8 | — | 8 | 12 | no | — |
| 32 | C2×C4 | — | 8 | 12 | no | — |
| 33 | C2×C4 | D16 | 1 | 3 | yes | — |
| 34 | C2×C4 | C2×D8 | 1 | 3 | yes | — |
| 35 | C8 | C2×D8 | 1 | 3 | yes | — |
| 36 | C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 37 | C22 | Q64 | 1 | 4 | yes | direct product factor |
| 38 | C22 | C2×D16 | 1 | 3 | yes | — |
| 39 | C4 | — | 8 | 8 | no | — |
| 40 | C4 | C2×D16 | 1 | 3 | yes | — |
| 41 | C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 42 | C2 | C2×Q64 | 1 | 6 | yes | direct product factor |
| 43 | C2 | C22×D16 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | D16 |
| commutator, Frattini, cosocle | C16 | C24 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×Q64 | C2 | direct (N × H) |
| 8. Q64 | C22 | direct (N × H) |
| 37. C22 | Q64 | direct (N × H) |
| 42. C2 | C2×Q64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q32 | 2 |
| C2×Q64 | 2 |
| C22×C32 | 2 |