Finite Group DatabasebetaFinite Group DatabasebetaOrder 32768.
4073779669480638 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×2−1+12 | C2 | 1 | 8127 | yes | maximal, complemented |
| 2 | C22×2−1+10 | C22 | 1 | 5458635 | yes | complemented |
| 3 | C23×2−1+8 | C23 | 1 | 386003475 | yes | complemented |
| 4 | #72290098 | C2 | 1 | 8256 | yes | maximal, complemented |
| 5 | C2×#72274291 | C22 | 1 | 16904160 | yes | complemented |
| 6 | C22×#72258201 | C23 | 1 | 2882159280 | yes | complemented |
| 7 | C24×2−1+6 | C24 | 1 | 3062294235 | yes | complemented |
| 8 | C23×#63103 | C24 | 1 | 52496472600 | yes | complemented |
| 9 | H(13,𝔽2) | C22 | 1 | 5724160 | yes | complemented |
| 10 | 2−1+12 | C22 | 1 | 16644096 | yes | complemented |
| 11 | C2×H(11,𝔽2) | C23 | 1 | 2975132160 | yes | complemented |
| 12 | #72274291 | C23 | 1 | 11539906560 | yes | complemented |
| 13 | C2×2−1+10 | C23 | 1 | 8384463360 | yes | complemented |
| 14 | C2×#72258201 | C24 | 1 | 1475665551360 | yes | complemented |
| 15 | C22×H(9,𝔽2) | C24 | 1 | 130657887360 | yes | complemented |
| 16 | C22×2−1+8 | C24 | 1 | 345859113600 | yes | complemented |
| 17 | C22×#63103 | C25 | 1 | 15678946483200 | yes | complemented |
| 18 | C23×2−1+6 | C25 | 1 | 1469901232800 | yes | complemented |
| 19 | C25×2−1+4 | C25 | 1 | 2667159495 | yes | complemented |
| 20 | 2−1+10 | C24 | 1 | 953965608960 | yes | complemented |
| 21 | C24×#585 | C25 | 1 | 110242592460 | yes | complemented |
| 22 | H(11,𝔽2) | C24 | 1 | 609307066368 | yes | complemented |
| 23 | #72258201 | C25 | 1 | 50369384153088 | yes | complemented |
| 24 | C2×2−1+8 | C25 | 1 | 29513311027200 | yes | complemented |
| 25 | C2×H(9,𝔽2) | C25 | 1 | 20069051498496 | yes | complemented |
| 26 | C23×H(7,𝔽2) | C25 | 1 | 629957671200 | yes | complemented |
| 27 | 2−1+8 | C26 | 1 | 215868789227520 | yes | complemented |
| 28 | H(9,𝔽2) | C26 | 1 | 190284340133888 | yes | complemented |
| 29 | C2×#63103 | C26 | 1 | 401381029969920 | yes | complemented |
| 30 | C22×2−1+6 | C26 | 1 | 73168416921600 | yes | complemented |
| 31 | C22×H(7,𝔽2) | C26 | 1 | 56444207339520 | yes | complemented |
| 32 | C23×#585 | C26 | 1 | 17638814793600 | yes | complemented |
| 33 | C24×2−1+4 | C26 | 1 | 661455554760 | yes | complemented |
| 34 | C24×H(5,𝔽2) | C26 | 1 | 367475308200 | yes | complemented |
| 35 | C25×#41 | C26 | 1 | 26671594950 | yes | complemented |
| 36 | C26×Q8 | C26 | 1 | 211679325 | yes | complemented |
| 37 | #63103 | C27 | 1 | 815504314859520 | yes | — |
| 38 | C2×2−1+6 | C27 | 1 | 401381029969920 | yes | — |
| 39 | C2×H(7,𝔽2) | C27 | 1 | 401381029969920 | yes | — |
| 40 | C22×#585 | C27 | 1 | 263406300917760 | yes | — |
| 41 | C23×2−1+4 | C27 | 1 | 17638814793600 | yes | — |
| 42 | C23×H(5,𝔽2) | C27 | 1 | 17638814793600 | yes | — |
| 43 | C24×#41 | C27 | 1 | 2204851849200 | yes | — |
| 44 | C25×Q8 | C27 | 1 | 26671594950 | yes | — |
| 45 | C25×D4 | C27 | 1 | 26671594950 | yes | — |
| 46 | C26×C4 | C27 | 1 | 635037975 | yes | — |
| 47 | C27 | C28 | 1 | 211679325 | yes | — |
| 48 | 2−1+6 | C28 | 1 | 190284340133888 | yes | — |
| 49 | H(7,𝔽2) | C28 | 1 | 215868789227520 | yes | — |
| 50 | C2×#585 | C28 | 1 | 401381029969920 | yes | — |
| 51 | C22×2−1+4 | C28 | 1 | 56444207339520 | yes | — |
| 52 | C22×H(5,𝔽2) | C28 | 1 | 73168416921600 | yes | — |
| 53 | C23×#41 | C28 | 1 | 17638814793600 | yes | — |
| 54 | C24×Q8 | C28 | 1 | 367475308200 | yes | — |
| 55 | C24×D4 | C28 | 1 | 661455554760 | yes | — |
| 56 | C25×C4 | C28 | 1 | 26671594950 | yes | — |
| 57 | C26 | — | 64 | 211679325 | no | — |
| 58 | C26 | C29 | 1 | 2667159495 | yes | — |
| 59 | #585 | C29 | 1 | 50369384153088 | yes | — |
| 60 | C2×2−1+4 | C29 | 1 | 20069051498496 | yes | — |
| 61 | C2×H(5,𝔽2) | C29 | 1 | 29513311027200 | yes | — |
| 62 | C22×#41 | C29 | 1 | 15678946483200 | yes | — |
| 63 | C23×Q8 | C29 | 1 | 629957671200 | yes | — |
| 64 | C23×D4 | C29 | 1 | 1469901232800 | yes | — |
| 65 | C24×C4 | C29 | 1 | 110242592460 | yes | — |
| 66 | C25 | — | 32 | 2667159495 | no | — |
| 67 | C25 | C210 | 1 | 3062294235 | yes | — |
| 68 | 2−1+4 | C210 | 1 | 609307066368 | yes | — |
| 69 | H(5,𝔽2) | C210 | 1 | 953965608960 | yes | — |
| 70 | C2×#41 | C210 | 1 | 1475665551360 | yes | — |
| 71 | C22×Q8 | C210 | 1 | 130657887360 | yes | — |
| 72 | C22×D4 | C210 | 1 | 345859113600 | yes | — |
| 73 | C23×C4 | C210 | 1 | 52496472600 | yes | — |
| 74 | C24 | — | 16 | 3062294235 | no | — |
| 75 | C24 | C211 | 1 | 386003475 | yes | — |
| 76 | #41 | C211 | 1 | 11539906560 | yes | — |
| 77 | C2×Q8 | C211 | 1 | 2975132160 | yes | — |
| 78 | C2×D4 | C211 | 1 | 8384463360 | yes | — |
| 79 | C22×C4 | C211 | 1 | 2882159280 | yes | — |
| 80 | C23 | — | 8 | 386003475 | no | — |
| 81 | C23 | C212 | 1 | 5458635 | yes | — |
| 82 | Q8 | C212 | 1 | 5724160 | yes | — |
| 83 | D4 | C212 | 1 | 16644096 | yes | — |
| 84 | C2×C4 | C212 | 1 | 16904160 | yes | — |
| 85 | C22 | — | 4 | 5458635 | no | — |
| 86 | C22 | C213 | 1 | 8127 | yes | — |
| 87 | C4 | C213 | 1 | 8256 | yes | — |
| 88 | C2 | — | 2 | 8127 | no | — |
| 89 | C2 | C214 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C214 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×2−1+12 | C2 | semidirect |
| 2. C22×2−1+10 | C22 | semidirect |
| 3. C23×2−1+8 | C23 | semidirect |
| 4. #72290098 | C2 | semidirect |
| 5. C2×#72274291 | C22 | semidirect |
| 6. C22×#72258201 | C23 | semidirect |
| 7. C24×2−1+6 | C24 | semidirect |
| 8. C23×#63103 | C24 | semidirect |
| 9. H(13,𝔽2) | C22 | semidirect |
| 10. 2−1+12 | C22 | semidirect |
| 11. C2×H(11,𝔽2) | C23 | semidirect |
| 12. #72274291 | C23 | semidirect |
| 13. C2×2−1+10 | C23 | semidirect |
| 14. C2×#72258201 | C24 | semidirect |
| 15. C22×H(9,𝔽2) | C24 | semidirect |
| 16. C22×2−1+8 | C24 | semidirect |
| 17. C22×#63103 | C25 | semidirect |
| 18. C23×2−1+6 | C25 | semidirect |
| 19. C25×2−1+4 | C25 | semidirect |
| 20. 2−1+10 | C24 | semidirect |
| 21. C24×#585 | C25 | semidirect |
| 22. H(11,𝔽2) | C24 | semidirect |
| 23. #72258201 | C25 | semidirect |
| 24. C2×2−1+8 | C25 | semidirect |
| 25. C2×H(9,𝔽2) | C25 | semidirect |
| 26. C23×H(7,𝔽2) | C25 | semidirect |
| 27. 2−1+8 | C26 | semidirect |
| 28. H(9,𝔽2) | C26 | semidirect |
| 29. C2×#63103 | C26 | semidirect |
| 30. C22×2−1+6 | C26 | semidirect |
| 31. C22×H(7,𝔽2) | C26 | semidirect |
| 32. C23×#585 | C26 | semidirect |
| 33. C24×2−1+4 | C26 | semidirect |
| 34. C24×H(5,𝔽2) | C26 | semidirect |
| 35. C25×#41 | C26 | semidirect |
| 36. C26×Q8 | C26 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×2−1+12 | 2 |
| #72290098 | 2 |