Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
8671445 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C26×Q8 | C2 | 1 | 508 | yes | maximal, direct product factor |
| 2 | C27×C4 | C2 | 1 | 3 | yes | maximal |
| 3 | C28 | C22 | 1 | 1 | yes | characteristic |
| 4 | C25×Q8 | C22 | 1 | 42672 | yes | direct product factor |
| 5 | C26×C4 | C22 | 1 | 762 | yes | — |
| 6 | C27 | Q8 | 1 | 128 | yes | direct product factor |
| 7 | C27 | C23 | 1 | 127 | yes | — |
| 8 | C24×Q8 | C23 | 1 | 755904 | yes | direct product factor |
| 9 | C25×C4 | C23 | 1 | 32004 | yes | — |
| 10 | C26 | C2×Q8 | 1 | 8128 | yes | direct product factor |
| 11 | C26 | C24 | 1 | 2667 | yes | — |
| 12 | C23×Q8 | C24 | 1 | 3023616 | yes | direct product factor |
| 13 | C24×C4 | C24 | 1 | 283464 | yes | — |
| 14 | C25 | C22×Q8 | 1 | 85344 | yes | direct product factor |
| 15 | C25 | C25 | 1 | 11811 | yes | — |
| 16 | C22×Q8 | C25 | 1 | 2731008 | yes | direct product factor |
| 17 | C23×C4 | C25 | 1 | 566928 | yes | — |
| 18 | C24 | C23×Q8 | 1 | 188976 | yes | direct product factor |
| 19 | C24 | C26 | 1 | 11811 | yes | — |
| 20 | C2×Q8 | C26 | 1 | 520192 | yes | direct product factor |
| 21 | C22×C4 | C26 | 1 | 256032 | yes | — |
| 22 | C23 | C24×Q8 | 1 | 94488 | yes | direct product factor |
| 23 | C23 | C27 | 1 | 2667 | yes | — |
| 24 | Q8 | C27 | 1 | 16384 | yes | direct product factor |
| 25 | C2×C4 | C27 | 1 | 24384 | yes | — |
| 26 | C22 | C25×Q8 | 1 | 10668 | yes | direct product factor |
| 27 | C22 | C28 | 1 | 127 | yes | — |
| 28 | C4 | C28 | 1 | 384 | yes | — |
| 29 | C2 | C26×Q8 | 1 | 254 | yes | direct product factor |
| 30 | C2 | C29 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C28 | C22 |
| commutator, Frattini, cosocle | C2 | C29 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C28 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C26×Q8 | C2 | direct (N × H) |
| 4. C25×Q8 | C22 | direct (N × H) |
| 6. C27 | Q8 | direct (N × H) |
| 8. C24×Q8 | C23 | direct (N × H) |
| 10. C26 | C2×Q8 | direct (N × H) |
| 12. C23×Q8 | C24 | direct (N × H) |
| 14. C25 | C22×Q8 | direct (N × H) |
| 16. C22×Q8 | C25 | direct (N × H) |
| 18. C24 | C23×Q8 | direct (N × H) |
| 20. C2×Q8 | C26 | direct (N × H) |
| 22. C23 | C24×Q8 | direct (N × H) |
| 24. Q8 | C27 | direct (N × H) |
| 26. C22 | C25×Q8 | direct (N × H) |
| 29. C2 | C26×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C26×Q8 | 2 |
| C27×C4 | 2 |