Finite Group DatabasebetaFinite Group DatabasebetaOrder 32768.
4073782660872416 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72290098 | C2 | 1 | 8128 | yes | maximal, complemented |
| 2 | C2×H(13,𝔽2) | C2 | 1 | 8255 | yes | maximal, complemented |
| 3 | C2×#72274291 | C22 | 1 | 16642080 | yes | complemented |
| 4 | C22×H(11,𝔽2) | C22 | 1 | 5720715 | yes | complemented |
| 5 | C22×#72258201 | C23 | 1 | 2837474640 | yes | complemented |
| 6 | C23×H(9,𝔽2) | C23 | 1 | 430688115 | yes | complemented |
| 7 | C23×#63103 | C24 | 1 | 51682573800 | yes | complemented |
| 8 | C24×H(7,𝔽2) | C24 | 1 | 3876193035 | yes | complemented |
| 9 | H(13,𝔽2) | C22 | 1 | 16906240 | yes | complemented |
| 10 | 2−1+12 | C22 | 1 | 5462016 | yes | complemented |
| 11 | C2×H(11,𝔽2) | C23 | 1 | 8787018240 | yes | complemented |
| 12 | #72274291 | C23 | 1 | 11360993280 | yes | complemented |
| 13 | C2×2−1+10 | C23 | 1 | 2751490560 | yes | complemented |
| 14 | C2×#72258201 | C24 | 1 | 1452787015680 | yes | complemented |
| 15 | C22×H(9,𝔽2) | C24 | 1 | 385896551040 | yes | complemented |
| 16 | C22×2−1+8 | C24 | 1 | 113498985600 | yes | complemented |
| 17 | C22×#63103 | C25 | 1 | 15435862041600 | yes | complemented |
| 18 | C23×2−1+6 | C25 | 1 | 482370688800 | yes | complemented |
| 19 | 2−1+10 | C24 | 1 | 563505266688 | yes | complemented |
| 20 | C24×#585 | C25 | 1 | 108533404980 | yes | complemented |
| 21 | H(11,𝔽2) | C24 | 1 | 999767408640 | yes | complemented |
| 22 | #72258201 | C25 | 1 | 49588463468544 | yes | complemented |
| 23 | C2×2−1+8 | C25 | 1 | 17433444188160 | yes | complemented |
| 24 | C25×H(5,𝔽2) | C25 | 1 | 4376346975 | yes | complemented |
| 25 | C2×H(9,𝔽2) | C25 | 1 | 32929839022080 | yes | complemented |
| 26 | C23×H(7,𝔽2) | C25 | 1 | 1860572656800 | yes | complemented |
| 27 | 2−1+8 | C26 | 1 | 165294878228480 | yes | complemented |
| 28 | H(9,𝔽2) | C26 | 1 | 240858251132928 | yes | complemented |
| 29 | C2×#63103 | C26 | 1 | 395158068264960 | yes | complemented |
| 30 | C22×2−1+6 | C26 | 1 | 43220413716480 | yes | complemented |
| 31 | C22×H(7,𝔽2) | C26 | 1 | 92615172249600 | yes | complemented |
| 32 | C23×#585 | C26 | 1 | 17365344796800 | yes | complemented |
| 33 | C24×2−1+4 | C26 | 1 | 217066809960 | yes | complemented |
| 34 | C24×H(5,𝔽2) | C26 | 1 | 1085334049800 | yes | complemented |
| 35 | C25×#41 | C26 | 1 | 26258081850 | yes | complemented |
| 36 | C26×D4 | C26 | 1 | 625192425 | yes | complemented |
| 37 | C28 | C27 | 1 | 9845550 | yes | complemented |
| 38 | #63103 | C27 | 1 | 802860837109760 | yes | complemented |
| 39 | C2×2−1+6 | C27 | 1 | 307345164206080 | yes | complemented |
| 40 | C2×H(7,𝔽2) | C27 | 1 | 508060373483520 | yes | complemented |
| 41 | C22×#585 | C27 | 1 | 259322482298880 | yes | complemented |
| 42 | C23×2−1+4 | C27 | 1 | 10419206878080 | yes | complemented |
| 43 | C23×H(5,𝔽2) | C27 | 1 | 28942241328000 | yes | complemented |
| 44 | C24×#41 | C27 | 1 | 2170668099600 | yes | complemented |
| 45 | C25×Q8 | C27 | 1 | 8752693950 | yes | complemented |
| 46 | C25×D4 | C27 | 1 | 78774245550 | yes | complemented |
| 47 | C26×C4 | C27 | 1 | 625192425 | yes | complemented |
| 48 | C27 | — | 128 | 9845550 | no | — |
| 49 | C27 | C28 | 1 | 625192425 | yes | — |
| 50 | 2−1+6 | C28 | 1 | 165294878228480 | yes | — |
| 51 | H(7,𝔽2) | C28 | 1 | 240858251132928 | yes | — |
| 52 | C2×#585 | C28 | 1 | 395158068264960 | yes | — |
| 53 | C22×2−1+4 | C28 | 1 | 43220413716480 | yes | — |
| 54 | C22×H(5,𝔽2) | C28 | 1 | 92615172249600 | yes | — |
| 55 | C23×#41 | C28 | 1 | 17365344796800 | yes | — |
| 56 | C24×Q8 | C28 | 1 | 217066809960 | yes | — |
| 57 | C24×D4 | C28 | 1 | 1085334049800 | yes | — |
| 58 | C25×C4 | C28 | 1 | 26258081850 | yes | — |
| 59 | C26 | — | 64 | 625192425 | no | — |
| 60 | C26 | C29 | 1 | 4376346975 | yes | — |
| 61 | #585 | C29 | 1 | 49588463468544 | yes | — |
| 62 | C2×2−1+4 | C29 | 1 | 17433444188160 | yes | — |
| 63 | C2×H(5,𝔽2) | C29 | 1 | 32929839022080 | yes | — |
| 64 | C22×#41 | C29 | 1 | 15435862041600 | yes | — |
| 65 | C23×Q8 | C29 | 1 | 482370688800 | yes | — |
| 66 | C23×D4 | C29 | 1 | 1860572656800 | yes | — |
| 67 | C24×C4 | C29 | 1 | 108533404980 | yes | — |
| 68 | C25 | — | 32 | 4376346975 | no | — |
| 69 | C25 | C210 | 1 | 3876193035 | yes | — |
| 70 | 2−1+4 | C210 | 1 | 563505266688 | yes | — |
| 71 | H(5,𝔽2) | C210 | 1 | 999767408640 | yes | — |
| 72 | C2×#41 | C210 | 1 | 1452787015680 | yes | — |
| 73 | C22×Q8 | C210 | 1 | 113498985600 | yes | — |
| 74 | C22×D4 | C210 | 1 | 385896551040 | yes | — |
| 75 | C23×C4 | C210 | 1 | 51682573800 | yes | — |
| 76 | C24 | — | 16 | 3876193035 | no | — |
| 77 | C24 | C211 | 1 | 430688115 | yes | — |
| 78 | #41 | C211 | 1 | 11360993280 | yes | — |
| 79 | C2×Q8 | C211 | 1 | 2751490560 | yes | — |
| 80 | C2×D4 | C211 | 1 | 8787018240 | yes | — |
| 81 | C22×C4 | C211 | 1 | 2837474640 | yes | — |
| 82 | C23 | — | 8 | 430688115 | no | — |
| 83 | C23 | C212 | 1 | 5720715 | yes | — |
| 84 | Q8 | C212 | 1 | 5462016 | yes | — |
| 85 | D4 | C212 | 1 | 16906240 | yes | — |
| 86 | C2×C4 | C212 | 1 | 16642080 | yes | — |
| 87 | C22 | — | 4 | 5720715 | no | — |
| 88 | C22 | C213 | 1 | 8255 | yes | — |
| 89 | C4 | C213 | 1 | 8128 | yes | — |
| 90 | C2 | — | 2 | 8255 | no | — |
| 91 | 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. #72290098 | C2 | semidirect |
| 2. C2×H(13,𝔽2) | C2 | semidirect |
| 3. C2×#72274291 | C22 | semidirect |
| 4. C22×H(11,𝔽2) | C22 | semidirect |
| 5. C22×#72258201 | C23 | semidirect |
| 6. C23×H(9,𝔽2) | C23 | semidirect |
| 7. C23×#63103 | C24 | semidirect |
| 8. C24×H(7,𝔽2) | 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. 2−1+10 | C24 | semidirect |
| 20. C24×#585 | C25 | semidirect |
| 21. H(11,𝔽2) | C24 | semidirect |
| 22. #72258201 | C25 | semidirect |
| 23. C2×2−1+8 | C25 | semidirect |
| 24. C25×H(5,𝔽2) | 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×D4 | C26 | semidirect |
| 37. C28 | C27 | semidirect |
| 38. #63103 | C27 | semidirect |
| 39. C2×2−1+6 | C27 | semidirect |
| 40. C2×H(7,𝔽2) | C27 | semidirect |
| 41. C22×#585 | C27 | semidirect |
| 42. C23×2−1+4 | C27 | semidirect |
| 43. C23×H(5,𝔽2) | C27 | semidirect |
| 44. C24×#41 | C27 | semidirect |
| 45. C25×Q8 | C27 | semidirect |
| 46. C25×D4 | C27 | semidirect |
| 47. C26×C4 | C27 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72290098 | 2 |
| C2×H(13,𝔽2) | 2 |