Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
291 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C4×#97 | C2 | 1 | 3 | yes | maximal |
| 2 | C2×#322 | C2 | 1 | 3 | yes | maximal |
| 3 | C2×C82 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#97 | C4 | 1 | 12 | yes | — |
| 5 | C2×C4×C8 | C22 | 1 | 3 | yes | — |
| 6 | C2×C4×C8 | C22 | 1 | 3 | yes | — |
| 7 | C43 | C22 | 1 | 1 | yes | characteristic |
| 8 | #322 | C4 | 1 | 12 | yes | complemented |
| 9 | C82 | C4 | 1 | 4 | yes | complemented |
| 10 | C22×C8 | C2×C4 | 1 | 6 | yes | — |
| 11 | C22×C8 | C2×C4 | 1 | 6 | yes | — |
| 12 | C2×C42 | C2×C4 | 1 | 6 | yes | — |
| 13 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 14 | #97 | — | 2 | 24 | no | — |
| 15 | C4×C8 | C2×C4 | 1 | 6 | yes | — |
| 16 | C4×C8 | C2×C4 | 1 | 12 | yes | — |
| 17 | C4×C8 | C2×C4 | 1 | 6 | yes | — |
| 18 | C22×C4 | C42 | 1 | 4 | yes | — |
| 19 | C22×C4 | C22×C4 | 1 | 3 | yes | — |
| 20 | C2×C8 | — | 2 | 12 | no | — |
| 21 | C2×C8 | C42 | 1 | 12 | yes | — |
| 22 | C2×C8 | C42 | 1 | 12 | yes | — |
| 23 | C2×C8 | M16 | 1 | 24 | yes | — |
| 24 | C42 | — | 2 | 12 | no | — |
| 25 | C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 26 | C42 | C22×C4 | 1 | 3 | yes | — |
| 27 | C23 | C2×C42 | 1 | 1 | yes | characteristic |
| 28 | C2×C4 | — | 2 | 12 | no | — |
| 29 | C2×C4 | C2×C42 | 1 | 3 | yes | — |
| 30 | C2×C4 | C2×C42 | 1 | 3 | yes | — |
| 31 | C2×C4 | C2×M16 | 1 | 12 | yes | — |
| 32 | C8 | — | 2 | 12 | no | — |
| 33 | C8 | #97 | 1 | 24 | yes | complemented |
| 34 | C22 | C43 | 1 | 1 | yes | characteristic |
| 35 | C22 | C4×M16 | 1 | 6 | yes | — |
| 36 | C4 | — | 4 | 4 | no | — |
| 37 | C4 | C2×#97 | 1 | 6 | yes | — |
| 38 | C4 | C4×M16 | 1 | 6 | yes | — |
| 39 | C2 | #1455 | 1 | 4 | yes | — |
| 40 | C2 | C4×#97 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C23 |
| commutator | C22 | C43 |
| socle, prime radical | C23 | C2×C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #322 | C4 | semidirect |
| 9. C82 | C4 | semidirect |
| 33. C8 | #97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#97 | 2 |
| C2×#322 | 2 |
| C2×C82 | 2 |