Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
31663 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C24×Q8 | C2 | 1 | 124 | yes | maximal, direct product factor |
| 2 | C25×C4 | C2 | 1 | 3 | yes | maximal |
| 3 | C26 | C22 | 1 | 1 | yes | characteristic |
| 4 | C23×Q8 | C22 | 1 | 2480 | yes | direct product factor |
| 5 | C24×C4 | C22 | 1 | 186 | yes | — |
| 6 | C25 | Q8 | 1 | 32 | yes | direct product factor |
| 7 | C25 | C23 | 1 | 31 | yes | — |
| 8 | C22×Q8 | C23 | 1 | 9920 | yes | direct product factor |
| 9 | C23×C4 | C23 | 1 | 1860 | yes | — |
| 10 | C24 | C2×Q8 | 1 | 496 | yes | direct product factor |
| 11 | C24 | C24 | 1 | 155 | yes | — |
| 12 | C2×Q8 | C24 | 1 | 7936 | yes | direct product factor |
| 13 | C22×C4 | C24 | 1 | 3720 | yes | — |
| 14 | C23 | C22×Q8 | 1 | 1240 | yes | direct product factor |
| 15 | C23 | C25 | 1 | 155 | yes | — |
| 16 | Q8 | C25 | 1 | 1024 | yes | direct product factor |
| 17 | C2×C4 | C25 | 1 | 1488 | yes | — |
| 18 | C22 | C23×Q8 | 1 | 620 | yes | direct product factor |
| 19 | C22 | C26 | 1 | 31 | yes | — |
| 20 | C4 | C26 | 1 | 96 | yes | — |
| 21 | C2 | C24×Q8 | 1 | 62 | yes | direct product factor |
| 22 | C2 | C27 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C22 |
| commutator, Frattini, cosocle | C2 | C27 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8 | C2 | direct (N × H) |
| 4. C23×Q8 | C22 | direct (N × H) |
| 6. C25 | Q8 | direct (N × H) |
| 8. C22×Q8 | C23 | direct (N × H) |
| 10. C24 | C2×Q8 | direct (N × H) |
| 12. C2×Q8 | C24 | direct (N × H) |
| 14. C23 | C22×Q8 | direct (N × H) |
| 16. Q8 | C25 | direct (N × H) |
| 18. C22 | C23×Q8 | direct (N × H) |
| 21. C2 | C24×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C25×C4 | 2 |