Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
175 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2110 | C2 | 1 | 3 | yes | maximal |
| 2 | #2108 | C2 | 1 | 3 | yes | maximal |
| 3 | #1747 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×M32 | C22 | 1 | 3 | yes | — |
| 5 | C22×C16 | C22 | 1 | 1 | yes | characteristic |
| 6 | #405 | C22 | 1 | 3 | yes | — |
| 7 | #347 | C4 | 1 | 12 | yes | complemented |
| 8 | #346 | C4 | 1 | 6 | yes | complemented |
| 9 | C4×C16 | C4 | 1 | 6 | yes | complemented |
| 10 | #338 | C4 | 1 | 1 | yes | characteristic |
| 11 | H(3,ℤ/4) | C4 | 1 | 3 | yes | — |
| 12 | C2×M16 | C2×C4 | 1 | 3 | yes | — |
| 13 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 14 | #117 | C2×C4 | 1 | 3 | yes | — |
| 15 | M32 | C2×C4 | 1 | 12 | yes | — |
| 16 | C2×C16 | C2×C4 | 1 | 6 | yes | — |
| 17 | C2×C16 | C2×C4 | 1 | 3 | yes | — |
| 18 | C2×C16 | C2×C4 | 1 | 3 | yes | — |
| 19 | #97 | C2×C4 | 1 | 6 | yes | — |
| 20 | C4×C8 | C2×C4 | 1 | 6 | yes | — |
| 21 | C22×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 22 | M16 | — | 2 | 6 | no | — |
| 23 | C2×C8 | C42 | 1 | 6 | yes | — |
| 24 | C2×C8 | C22×C4 | 1 | 3 | yes | — |
| 25 | C2×C8 | C22×C4 | 1 | 3 | yes | — |
| 26 | C4⋊C4 | — | 2 | 3 | no | — |
| 27 | C22⋊C4 | — | 2 | 3 | no | — |
| 28 | C42 | C42 | 1 | 6 | yes | — |
| 29 | C16 | C42 | 1 | 3 | yes | — |
| 30 | C16 | C42 | 1 | 12 | yes | — |
| 31 | C16 | C42 | 1 | 1 | yes | characteristic |
| 32 | C23 | #131 | 1 | 1 | yes | characteristic |
| 33 | C2×C4 | — | 2 | 6 | no | — |
| 34 | C2×C4 | C2×C42 | 1 | 3 | yes | — |
| 35 | C2×C4 | #131 | 1 | 3 | yes | — |
| 36 | C8 | — | 2 | 6 | no | — |
| 37 | C8 | C2×C42 | 1 | 3 | yes | — |
| 38 | C8 | C2×C42 | 1 | 1 | yes | characteristic |
| 39 | C22 | — | 4 | 1 | no | — |
| 40 | C22 | #405 | 1 | 3 | yes | — |
| 41 | C4 | — | 4 | 6 | no | — |
| 42 | C4 | C43 | 1 | 1 | yes | characteristic |
| 43 | C4 | #405 | 1 | 3 | yes | — |
| 44 | C2 | — | 2 | 3 | no | — |
| 45 | C2 | #1726 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C16 | C42 |
| commutator | C4 | C43 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1726 |
| prime radical | C23 | #131 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. #347 | C4 | semidirect |
| 8. #346 | C4 | semidirect |
| 9. C4×C16 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2110 | 2 |
| #2108 | 2 |
| #1747 | 2 |