Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
900 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#564 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2382 | C2 | 1 | 15 | yes | maximal |
| 3 | C4×#125 | C2 | 1 | 15 | yes | maximal |
| 4 | #564 | C4 | 1 | 16 | yes | direct product factor |
| 5 | C2×#125 | C22 | 1 | 15 | yes | — |
| 6 | #389 | C22 | 1 | 30 | yes | — |
| 7 | #387 | C22 | 1 | 60 | yes | — |
| 8 | #382 | C22 | 1 | 15 | yes | — |
| 9 | C4×C4⋊C4 | C22 | 1 | 30 | yes | — |
| 10 | C43 | C22 | 1 | 5 | yes | — |
| 11 | #125 | C2×C4 | 1 | 120 | yes | — |
| 12 | C2×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 13 | C2×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 14 | C2×C42 | C23 | 1 | 15 | yes | — |
| 15 | C2×C42 | C23 | 1 | 15 | yes | — |
| 16 | C2×C42 | C23 | 1 | 5 | yes | — |
| 17 | #95 | C23 | 1 | 60 | yes | — |
| 18 | C22×C4 | C24 | 1 | 15 | yes | — |
| 19 | C22×C4 | C24 | 1 | 15 | yes | — |
| 20 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 21 | C4⋊C4 | — | 4 | 30 | no | — |
| 22 | C4⋊C4 | C22×C4 | 1 | 120 | yes | — |
| 23 | C42 | — | 2 | 30 | no | — |
| 24 | C42 | — | 4 | 15 | no | — |
| 25 | C42 | C22×C4 | 1 | 20 | yes | — |
| 26 | C23 | C25 | 1 | 1 | yes | characteristic |
| 27 | C2×C4 | — | 2 | 15 | no | — |
| 28 | C2×C4 | — | 2 | 15 | no | — |
| 29 | C2×C4 | — | 2 | 15 | no | — |
| 30 | C2×C4 | — | 2 | 30 | no | — |
| 31 | C2×C4 | C23×C4 | 1 | 30 | yes | — |
| 32 | C2×C4 | 2−1+4 | 1 | 6 | yes | — |
| 33 | C22 | C24×C4 | 1 | 1 | yes | characteristic |
| 34 | C22 | C2×2−1+4 | 1 | 3 | yes | — |
| 35 | C22 | #585 | 1 | 3 | yes | — |
| 36 | C4 | — | 2 | 30 | no | — |
| 37 | C4 | — | 4 | 15 | no | — |
| 38 | C4 | #564 | 1 | 4 | yes | direct product factor |
| 39 | C2 | C4×2−1+4 | 1 | 3 | yes | — |
| 40 | C2 | C2×#564 | 1 | 1 | yes | characteristic |
| 41 | C2 | #3531 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #564 | C4 | direct (N × H) |
| 38. C4 | #564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#564 | 2 |
| #2382 | 2 |
| C4×#125 | 2 |