← C3×#63076 #10681683
Subgroups — C3×#63076 #10681683
Order 768.
10796 conjugacy classes of subgroups.
The subgroup lattice has 126 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×#585 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#3573 | C2 | 1 | 32 | yes | maximal, complemented |
| 3 | C2×C3×#568 | C2 | 1 | 30 | yes | maximal, complemented |
| 4 | #63076 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 5 | C3×#585 | C4 | 1 | 16 | yes | — |
| 6 | C3×#585 | C22 | 1 | 16 | yes | — |
| 7 | C2×C3×2−1+4 | C4 | 1 | 6 | yes | — |
| 8 | C2×C3×H(5,𝔽2) | C4 | 1 | 10 | yes | — |
| 9 | C22×C3×#41 | C22 | 1 | 15 | yes | — |
| 10 | C3×#568 | C22 | 1 | 80 | yes | complemented |
| 11 | C3×#568 | C22 | 1 | 240 | yes | complemented |
| 12 | C2×C3×#131 | C22 | 1 | 240 | yes | complemented |
| 13 | C22×C3×M16 | C22 | 1 | 60 | yes | complemented |
| 14 | C2×#585 | C2×C3 | 1 | 1 | yes | characteristic |
| 15 | #3573 | C2×C3 | 1 | 32 | yes | complemented |
| 16 | C2×#568 | C2×C3 | 1 | 30 | yes | complemented |
| 17 | C3×2−1+4 | C2×C4 | 1 | 96 | yes | — |
| 18 | C3×H(5,𝔽2) | C2×C4 | 1 | 160 | yes | — |
| 19 | C2×C3×#41 | C2×C4 | 1 | 60 | yes | — |
| 20 | C2×C3×#41 | C2×C4 | 1 | 120 | yes | — |
| 21 | C2×C3×#41 | C23 | 1 | 120 | yes | — |
| 22 | C2×C3×#41 | C23 | 1 | 20 | yes | — |
| 23 | C22×C3×Q8 | C2×C4 | 1 | 15 | yes | — |
| 24 | C22×C3×D4 | C2×C4 | 1 | 45 | yes | — |
| 25 | C23×C3×C4 | C23 | 1 | 15 | yes | — |
| 26 | C3×#131 | C23 | 1 | 640 | yes | complemented |
| 27 | C2×C3×M16 | C23 | 1 | 120 | yes | — |
| 28 | C2×C3×M16 | C23 | 1 | 360 | yes | complemented |
| 29 | C22×C3×C8 | C23 | 1 | 120 | yes | complemented |
| 30 | #585 | C3×C4 | 1 | 16 | yes | — |
| 31 | #585 | C22×C3 | 1 | 16 | yes | — |
| 32 | C2×2−1+4 | C3×C4 | 1 | 6 | yes | — |
| 33 | C2×H(5,𝔽2) | C3×C4 | 1 | 10 | yes | — |
| 34 | C22×#41 | C22×C3 | 1 | 15 | yes | — |
| 35 | #568 | C22×C3 | 1 | 80 | yes | complemented |
| 36 | #568 | C22×C3 | 1 | 240 | yes | complemented |
| 37 | C2×#131 | C22×C3 | 1 | 240 | yes | complemented |
| 38 | C22×M16 | C22×C3 | 1 | 60 | yes | complemented |
| 39 | C24×C3 | C22×C4 | 1 | 15 | yes | — |
| 40 | C3×#41 | C22×C4 | 1 | 80 | yes | — |
| 41 | C3×#41 | C22×C4 | 1 | 480 | yes | — |
| 42 | C3×#41 | C24 | 1 | 80 | yes | — |
| 43 | C2×C3×Q8 | C22×C4 | 1 | 20 | yes | — |
| 44 | C2×C3×Q8 | C22×C4 | 1 | 120 | yes | — |
| 45 | C2×C3×D4 | C22×C4 | 1 | 60 | yes | — |
| 46 | C2×C3×D4 | C22×C4 | 1 | 360 | yes | — |
| 47 | C22×C3×C4 | C22×C4 | 1 | 45 | yes | — |
| 48 | C22×C3×C4 | C22×C4 | 1 | 60 | yes | — |
| 49 | C22×C3×C4 | C24 | 1 | 60 | yes | — |
| 50 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 51 | C3×M16 | C24 | 1 | 16 | yes | — |
| 52 | C3×M16 | C24 | 1 | 240 | yes | — |
| 53 | C2×C3×C8 | C24 | 1 | 240 | yes | — |
| 54 | 2−1+4 | C2×C3×C4 | 1 | 96 | yes | — |
| 55 | H(5,𝔽2) | C2×C3×C4 | 1 | 160 | yes | — |
| 56 | C2×#41 | C2×C3×C4 | 1 | 60 | yes | — |
| 57 | C2×#41 | C2×C3×C4 | 1 | 120 | yes | — |
| 58 | C2×#41 | C23×C3 | 1 | 120 | yes | — |
| 59 | C2×#41 | C23×C3 | 1 | 20 | yes | — |
| 60 | C22×Q8 | C2×C3×C4 | 1 | 15 | yes | — |
| 61 | C22×D4 | C2×C3×C4 | 1 | 45 | yes | — |
| 62 | C23×C4 | C23×C3 | 1 | 15 | yes | — |
| 63 | #131 | C23×C3 | 1 | 640 | yes | complemented |
| 64 | C2×M16 | C23×C3 | 1 | 120 | yes | — |
| 65 | C2×M16 | C23×C3 | 1 | 360 | yes | complemented |
| 66 | C22×C8 | C23×C3 | 1 | 120 | yes | complemented |
| 67 | C23×C3 | — | 8 | 15 | no | — |
| 68 | C23×C3 | C23×C4 | 1 | 15 | yes | — |
| 69 | C23×C3 | C23×C4 | 1 | 60 | yes | — |
| 70 | C3×Q8 | C23×C4 | 1 | 80 | yes | — |
| 71 | C3×D4 | C23×C4 | 1 | 240 | yes | — |
| 72 | C2×C3×C4 | C23×C4 | 1 | 15 | yes | — |
| 73 | C2×C3×C4 | C23×C4 | 1 | 30 | yes | — |
| 74 | C2×C3×C4 | C23×C4 | 1 | 180 | yes | — |
| 75 | C2×C3×C4 | C25 | 1 | 30 | yes | — |
| 76 | C2×C3×C4 | C25 | 1 | 1 | yes | characteristic |
| 77 | C3×C8 | C25 | 1 | 32 | yes | — |
| 78 | C24 | C22×C3×C4 | 1 | 15 | yes | — |
| 79 | #41 | C22×C3×C4 | 1 | 80 | yes | — |
| 80 | #41 | C22×C3×C4 | 1 | 480 | yes | — |
| 81 | #41 | C24×C3 | 1 | 80 | yes | — |
| 82 | C2×Q8 | C22×C3×C4 | 1 | 20 | yes | — |
| 83 | C2×Q8 | C22×C3×C4 | 1 | 120 | yes | — |
| 84 | C2×D4 | C22×C3×C4 | 1 | 60 | yes | — |
| 85 | C2×D4 | C22×C3×C4 | 1 | 360 | yes | — |
| 86 | C22×C4 | C22×C3×C4 | 1 | 45 | yes | — |
| 87 | C22×C4 | C22×C3×C4 | 1 | 60 | yes | — |
| 88 | C22×C4 | C24×C3 | 1 | 60 | yes | — |
| 89 | C22×C4 | C24×C3 | 1 | 15 | yes | — |
| 90 | M16 | C24×C3 | 1 | 16 | yes | — |
| 91 | M16 | C24×C3 | 1 | 240 | yes | — |
| 92 | C2×C8 | C24×C3 | 1 | 240 | yes | — |
| 93 | C22×C3 | — | 4 | 15 | no | — |
| 94 | C22×C3 | — | 4 | 60 | no | — |
| 95 | C22×C3 | C24×C4 | 1 | 1 | yes | characteristic |
| 96 | C22×C3 | C24×C4 | 1 | 30 | yes | — |
| 97 | C3×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 98 | C3×C4 | C24×C4 | 1 | 30 | yes | — |
| 99 | C3×C4 | C26 | 1 | 1 | yes | characteristic |
| 100 | C23 | — | 8 | 15 | no | — |
| 101 | C23 | C23×C3×C4 | 1 | 15 | yes | — |
| 102 | C23 | C23×C3×C4 | 1 | 60 | yes | — |
| 103 | Q8 | C23×C3×C4 | 1 | 80 | yes | — |
| 104 | D4 | C23×C3×C4 | 1 | 240 | yes | — |
| 105 | C2×C4 | C23×C3×C4 | 1 | 15 | yes | — |
| 106 | C2×C4 | C23×C3×C4 | 1 | 30 | yes | — |
| 107 | C2×C4 | C23×C3×C4 | 1 | 180 | yes | — |
| 108 | C2×C4 | C25×C3 | 1 | 30 | yes | — |
| 109 | C2×C4 | C25×C3 | 1 | 1 | yes | characteristic |
| 110 | C8 | C25×C3 | 1 | 32 | yes | — |
| 111 | C2×C3 | — | 2 | 1 | no | — |
| 112 | C2×C3 | — | 2 | 30 | no | — |
| 113 | C2×C3 | C25×C4 | 1 | 1 | yes | characteristic |
| 114 | C22 | — | 4 | 15 | no | — |
| 115 | C22 | — | 4 | 60 | no | — |
| 116 | C22 | C24×C3×C4 | 1 | 1 | yes | characteristic |
| 117 | C22 | C24×C3×C4 | 1 | 30 | yes | — |
| 118 | C4 | C24×C3×C4 | 1 | 1 | yes | characteristic |
| 119 | C4 | C24×C3×C4 | 1 | 30 | yes | — |
| 120 | C4 | C26×C3 | 1 | 1 | yes | characteristic |
| 121 | C3 | #63076 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 122 | C2 | — | 2 | 1 | no | — |
| 123 | C2 | — | 2 | 30 | no | — |
| 124 | C2 | C25×C3×C4 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta