Finite Group DatabasebetaFinite Group DatabasebetaOrder 16384.
37559364027177 conjugacy classes of subgroups; only the 37558989808527 normal subgroups are shown.
This group has 37559364027177 conjugacy classes of subgroups — too many to draw. Only its 37558989808527 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#72274291 | C2 | 1 | 4095 | yes | maximal, complemented |
| 2 | C22×#72258201 | C22 | 1 | 1396395 | yes | complemented |
| 3 | C23×#63103 | C23 | 1 | 50868675 | yes | complemented |
| 4 | H(13,𝔽2) | C2 | 1 | 2080 | yes | maximal, complemented |
| 5 | 2−1+12 | C2 | 1 | 2016 | yes | maximal, complemented |
| 6 | C2×H(11,𝔽2) | C22 | 1 | 2162160 | yes | complemented |
| 7 | #72274291 | C22 | 1 | 1397760 | yes | complemented |
| 8 | #72274291 | C22 | 1 | 4193280 | yes | complemented |
| 9 | C2×2−1+10 | C22 | 1 | 2031120 | yes | complemented |
| 10 | C2×#72258201 | C23 | 1 | 357477120 | yes | complemented |
| 11 | C2×#72258201 | C23 | 1 | 1072431360 | yes | complemented |
| 12 | C22×H(9,𝔽2) | C23 | 1 | 189909720 | yes | complemented |
| 13 | C22×2−1+8 | C23 | 1 | 167567400 | yes | complemented |
| 14 | C22×#63103 | C24 | 1 | 22789166400 | yes | complemented |
| 15 | C22×#63103 | C24 | 1 | 7596388800 | yes | complemented |
| 16 | C23×2−1+6 | C24 | 1 | 1424322900 | yes | complemented |
| 17 | 2−1+10 | C23 | 1 | 693288960 | yes | complemented |
| 18 | C24×#585 | C24 | 1 | 213648435 | yes | complemented |
| 19 | H(11,𝔽2) | C23 | 1 | 738017280 | yes | complemented |
| 20 | #72258201 | C24 | 1 | 6100942848 | yes | complemented |
| 21 | #72258201 | C24 | 1 | 91514142720 | yes | complemented |
| 22 | C2×2−1+8 | C24 | 1 | 42897254400 | yes | complemented |
| 23 | C2×H(9,𝔽2) | C24 | 1 | 48616888320 | yes | complemented |
| 24 | C23×H(7,𝔽2) | C24 | 1 | 1831272300 | yes | complemented |
| 25 | 2−1+8 | C25 | 1 | 732113141760 | yes | complemented |
| 26 | H(9,𝔽2) | C25 | 1 | 829728227328 | yes | complemented |
| 27 | C2×#63103 | C25 | 1 | 97233776640 | yes | complemented |
| 28 | C2×#63103 | C25 | 1 | 1458506649600 | yes | complemented |
| 29 | C22×2−1+6 | C25 | 1 | 212698886400 | yes | complemented |
| 30 | C22×H(7,𝔽2) | C25 | 1 | 273469996800 | yes | complemented |
| 31 | C23×#585 | C25 | 1 | 51275624400 | yes | complemented |
| 32 | C23×#585 | C25 | 1 | 17091874800 | yes | complemented |
| 33 | C24×2−1+4 | C25 | 1 | 1281890610 | yes | complemented |
| 34 | C24×H(5,𝔽2) | C25 | 1 | 2136484350 | yes | complemented |
| 35 | C25×#41 | C25 | 1 | 103378275 | yes | complemented |
| 36 | #63103 | C26 | 1 | 98777169920 | yes | complemented |
| 37 | #63103 | C26 | 1 | 6222961704960 | yes | complemented |
| 38 | C2×2−1+6 | C26 | 1 | 2722545745920 | yes | complemented |
| 39 | C2×H(7,𝔽2) | C26 | 1 | 3500415959040 | yes | complemented |
| 40 | C22×#585 | C26 | 1 | 127619331840 | yes | complemented |
| 41 | C22×#585 | C26 | 1 | 1914289977600 | yes | complemented |
| 42 | C23×2−1+4 | C26 | 1 | 102551248800 | yes | complemented |
| 43 | C23×H(5,𝔽2) | C26 | 1 | 170918748000 | yes | complemented |
| 44 | C24×#41 | C26 | 1 | 4272968700 | yes | complemented |
| 45 | C24×#41 | C26 | 1 | 12818906100 | yes | complemented |
| 46 | C25×Q8 | C26 | 1 | 103378275 | yes | complemented |
| 47 | C25×D4 | C26 | 1 | 310134825 | yes | complemented |
| 48 | C26×C4 | C26 | 1 | 4922775 | yes | complemented |
| 49 | C27 | C27 | 1 | 4922775 | yes | — |
| 50 | 2−1+6 | C27 | 1 | 2765760757760 | yes | — |
| 51 | H(7,𝔽2) | C27 | 1 | 3555978117120 | yes | — |
| 52 | C2×#585 | C27 | 1 | 97233776640 | yes | — |
| 53 | C2×#585 | C27 | 1 | 6125727928320 | yes | — |
| 54 | C22×2−1+4 | C27 | 1 | 765715991040 | yes | — |
| 55 | C22×H(5,𝔽2) | C27 | 1 | 1276193318400 | yes | — |
| 56 | C23×#41 | C27 | 1 | 17091874800 | yes | — |
| 57 | C23×#41 | C27 | 1 | 256378122000 | yes | — |
| 58 | C24×Q8 | C27 | 1 | 4272968700 | yes | — |
| 59 | C24×D4 | C27 | 1 | 12818906100 | yes | — |
| 60 | C25×C4 | C27 | 1 | 103378275 | yes | — |
| 61 | C25×C4 | C27 | 1 | 310134825 | yes | — |
| 62 | C26 | — | 64 | 4922775 | no | — |
| 63 | C26 | C28 | 1 | 103378275 | yes | — |
| 64 | #585 | C28 | 1 | 6100942848 | yes | — |
| 65 | #585 | C28 | 1 | 1555740426240 | yes | — |
| 66 | C2×2−1+4 | C28 | 1 | 583402659840 | yes | — |
| 67 | C2×H(5,𝔽2) | C28 | 1 | 972337766400 | yes | — |
| 68 | C22×#41 | C28 | 1 | 7596388800 | yes | — |
| 69 | C22×#41 | C28 | 1 | 478572494400 | yes | — |
| 70 | C23×Q8 | C28 | 1 | 17091874800 | yes | — |
| 71 | C23×D4 | C28 | 1 | 51275624400 | yes | — |
| 72 | C24×C4 | C28 | 1 | 213648435 | yes | — |
| 73 | C24×C4 | C28 | 1 | 3204726525 | yes | — |
| 74 | C25 | — | 32 | 103378275 | no | — |
| 75 | C25 | C29 | 1 | 213648435 | yes | — |
| 76 | 2−1+4 | C29 | 1 | 36605657088 | yes | — |
| 77 | H(5,𝔽2) | C29 | 1 | 61009428480 | yes | — |
| 78 | C2×#41 | C29 | 1 | 357477120 | yes | — |
| 79 | C2×#41 | C29 | 1 | 91156665600 | yes | — |
| 80 | C22×Q8 | C29 | 1 | 7596388800 | yes | — |
| 81 | C22×D4 | C29 | 1 | 22789166400 | yes | — |
| 82 | C23×C4 | C29 | 1 | 50868675 | yes | — |
| 83 | C23×C4 | C29 | 1 | 3204726525 | yes | — |
| 84 | C24 | — | 16 | 213648435 | no | — |
| 85 | C24 | C210 | 1 | 50868675 | yes | — |
| 86 | #41 | C210 | 1 | 1397760 | yes | — |
| 87 | #41 | C210 | 1 | 1429908480 | yes | — |
| 88 | C2×Q8 | C210 | 1 | 357477120 | yes | — |
| 89 | C2×D4 | C210 | 1 | 1072431360 | yes | — |
| 90 | C22×C4 | C210 | 1 | 1396395 | yes | — |
| 91 | C22×C4 | C210 | 1 | 356080725 | yes | — |
| 92 | C23 | — | 8 | 50868675 | no | — |
| 93 | C23 | C211 | 1 | 1396395 | yes | — |
| 94 | Q8 | C211 | 1 | 1397760 | yes | — |
| 95 | D4 | C211 | 1 | 4193280 | yes | — |
| 96 | C2×C4 | C211 | 1 | 4189185 | yes | — |
| 97 | C2×C4 | C211 | 1 | 4095 | yes | — |
| 98 | C22 | — | 4 | 1396395 | no | — |
| 99 | C22 | C212 | 1 | 4095 | yes | — |
| 100 | C4 | C212 | 1 | 4095 | yes | — |
| 101 | C4 | C212 | 1 | 1 | yes | characteristic |
| 102 | C2 | — | 2 | 4095 | no | — |
| 103 | C2 | C213 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C4 | C212 |
| commutator, Frattini, cosocle | C2 | C213 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#72274291 | C2 | semidirect |
| 2. C22×#72258201 | C22 | semidirect |
| 3. C23×#63103 | C23 | semidirect |
| 4. H(13,𝔽2) | C2 | semidirect |
| 5. 2−1+12 | C2 | semidirect |
| 6. C2×H(11,𝔽2) | C22 | semidirect |
| 7. #72274291 | C22 | semidirect |
| 8. #72274291 | C22 | semidirect |
| 9. C2×2−1+10 | C22 | semidirect |
| 10. C2×#72258201 | C23 | semidirect |
| 11. C2×#72258201 | C23 | semidirect |
| 12. C22×H(9,𝔽2) | C23 | semidirect |
| 13. C22×2−1+8 | C23 | semidirect |
| 14. C22×#63103 | C24 | semidirect |
| 15. C22×#63103 | C24 | semidirect |
| 16. C23×2−1+6 | C24 | semidirect |
| 17. 2−1+10 | C23 | semidirect |
| 18. C24×#585 | C24 | semidirect |
| 19. H(11,𝔽2) | C23 | semidirect |
| 20. #72258201 | C24 | semidirect |
| 21. #72258201 | C24 | semidirect |
| 22. C2×2−1+8 | C24 | semidirect |
| 23. C2×H(9,𝔽2) | C24 | semidirect |
| 24. C23×H(7,𝔽2) | C24 | semidirect |
| 25. 2−1+8 | C25 | semidirect |
| 26. H(9,𝔽2) | C25 | semidirect |
| 27. C2×#63103 | C25 | semidirect |
| 28. C2×#63103 | C25 | semidirect |
| 29. C22×2−1+6 | C25 | semidirect |
| 30. C22×H(7,𝔽2) | C25 | semidirect |
| 31. C23×#585 | C25 | semidirect |
| 32. C23×#585 | C25 | semidirect |
| 33. C24×2−1+4 | C25 | semidirect |
| 34. C24×H(5,𝔽2) | C25 | semidirect |
| 35. C25×#41 | C25 | semidirect |
| 36. #63103 | C26 | semidirect |
| 37. #63103 | C26 | semidirect |
| 38. C2×2−1+6 | C26 | semidirect |
| 39. C2×H(7,𝔽2) | C26 | semidirect |
| 40. C22×#585 | C26 | semidirect |
| 41. C22×#585 | C26 | semidirect |
| 42. C23×2−1+4 | C26 | semidirect |
| 43. C23×H(5,𝔽2) | C26 | semidirect |
| 44. C24×#41 | C26 | semidirect |
| 45. C24×#41 | C26 | semidirect |
| 46. C25×Q8 | C26 | semidirect |
| 47. C25×D4 | C26 | semidirect |
| 48. C26×C4 | C26 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#72274291 | 2 |
| H(13,𝔽2) | 2 |
| 2−1+12 | 2 |