Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
558 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 558 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #21461744 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21461327 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #21461033 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #493×D61 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16840780 | C22 | 1 | 2 | yes | — |
| 6 | #16840767 | C22 | 1 | 2 | yes | — |
| 7 | #16840638 | — | 2 | 2 | no | — |
| 8 | #16840620 | — | 2 | 8 | no | — |
| 9 | #92420⋊C8 | — | 2 | 2 | no | — |
| 10 | C2×D8×D61 | — | 2 | 2 | no | — |
| 11 | #127×D61 | C22 | 1 | 2 | yes | — |
| 12 | #16840139 | C4 | 1 | 1 | yes | characteristic |
| 13 | #16840138 | C4 | 1 | 2 | yes | — |
| 14 | #16840137 | — | 2 | 2 | no | — |
| 15 | C4×C8×D61 | C22 | 1 | 1 | yes | characteristic |
| 16 | C61×#493 | C4 | 1 | 1 | yes | characteristic |
| 17 | C2×#11758213 | — | 2 | 2 | no | — |
| 18 | #14519844 | — | 2 | 4 | no | — |
| 19 | #14519844 | D4 | 1 | 4 | yes | — |
| 20 | #14519834 | — | 2 | 2 | no | — |
| 21 | C4×#92420 | — | 2 | 2 | no | — |
| 22 | C2×D4×D61 | — | 2 | 2 | no | — |
| 23 | C2×D4×D61 | — | 2 | 4 | no | — |
| 24 | C2×D488 | — | 2 | 2 | no | — |
| 25 | C2×#11758199 | — | 2 | 4 | no | — |
| 26 | D8×D61 | — | 4 | 8 | no | — |
| 27 | C2×C8×D61 | D4 | 1 | 2 | yes | — |
| 28 | #14519774 | C2×C4 | 1 | 2 | yes | — |
| 29 | #14519773 | — | 2 | 4 | no | — |
| 30 | #14519773 | — | 2 | 2 | no | — |
| 31 | C42×D61 | C23 | 1 | 1 | yes | characteristic |
| 32 | C4×#92418 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C8×Dic61 | D4 | 1 | 2 | yes | — |
| 34 | C2×D8×C61 | — | 2 | 2 | no | — |
| 35 | #127×C61 | C2×C4 | 1 | 2 | yes | — |
| 36 | C4×C8×C61 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C23×D61 | — | 4 | 2 | no | — |
| 38 | C2×D4×C61 | — | 2 | 2 | no | — |
| 39 | C2×D4×C61 | — | 2 | 4 | no | — |
| 40 | C2×C22⋊D61 | — | 4 | 4 | no | — |
| 41 | C2×C22⋊D61 | — | 4 | 2 | no | — |
| 42 | D4×D61 | — | 4 | 8 | no | — |
| 43 | D4×D61 | — | 8 | 8 | no | — |
| 44 | C2×D244 | — | 2 | 2 | no | — |
| 45 | C2×D244 | — | 2 | 4 | no | — |
| 46 | C2×C4×D61 | C2×D4 | 1 | 1 | yes | characteristic |
| 47 | C2×C4×D61 | #41 | 1 | 2 | yes | — |
| 48 | C2×#92420 | — | 2 | 4 | no | — |
| 49 | C2×#92420 | — | 4 | 2 | no | — |
| 50 | #11758213 | — | 4 | 4 | no | — |
| 51 | D8×C61 | — | 4 | 4 | no | — |
| 52 | C2×C8×C61 | C22⋊C4 | 1 | 2 | yes | — |
| 53 | C42×C61 | C22×C4 | 1 | 1 | yes | characteristic |
| 54 | #11758199 | — | 4 | 8 | no | — |
| 55 | C4×Dic61 | C2×D4 | 1 | 1 | yes | characteristic |
| 56 | C4×Dic61 | C2×D4 | 1 | 2 | yes | — |
| 57 | C2×#92418 | C22⋊C4 | 1 | 2 | yes | — |
| 58 | D488 | — | 4 | 4 | no | — |
| 59 | C8×D61 | — | 2 | 4 | no | — |
| 60 | C23×C61 | — | 4 | 2 | no | — |
| 61 | C22×D61 | — | 4 | 2 | no | — |
| 62 | C22×D61 | — | 8 | 8 | no | — |
| 63 | C22×D61 | — | 8 | 4 | no | — |
| 64 | C22×D61 | #124 | 1 | 1 | yes | characteristic |
| 65 | D4×C61 | — | 4 | 4 | no | — |
| 66 | D4×C61 | — | 8 | 4 | no | — |
| 67 | C2×C4×C61 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C2×C4×C61 | C4×D4 | 1 | 2 | yes | — |
| 69 | C22⋊D61 | — | 8 | 8 | no | — |
| 70 | C22⋊D61 | — | 8 | 4 | no | — |
| 71 | C2×Dic61 | #121 | 1 | 2 | yes | — |
| 72 | C2×Dic61 | #127 | 1 | 1 | yes | characteristic |
| 73 | D244 | — | 4 | 4 | no | — |
| 74 | D244 | — | 8 | 4 | no | — |
| 75 | C4×D61 | — | 2 | 4 | no | — |
| 76 | C4×D61 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 77 | #92420 | — | 4 | 4 | no | — |
| 78 | #92420 | — | 8 | 2 | no | — |
| 79 | C8×C61 | — | 2 | 2 | no | — |
| 80 | #92418 | — | 2 | 2 | no | — |
| 81 | #9780 | — | 61 | 1 | no | maximal, Sylow |
| 82 | C22×C61 | — | 8 | 2 | no | — |
| 83 | C22×C61 | — | 8 | 4 | no | — |
| 84 | C22×C61 | #390 | 1 | 1 | yes | characteristic |
| 85 | C2×D61 | — | 2 | 2 | no | — |
| 86 | C2×D61 | — | 8 | 2 | no | — |
| 87 | C2×D61 | — | 8 | 4 | no | — |
| 88 | C2×D61 | — | 16 | 8 | no | — |
| 89 | C2×D61 | #490 | 1 | 2 | yes | — |
| 90 | C4×C61 | — | 2 | 2 | no | — |
| 91 | C4×C61 | #418 | 1 | 2 | yes | — |
| 92 | Dic61 | #472 | 1 | 4 | yes | — |
| 93 | Dic61 | #496 | 1 | 2 | yes | — |
| 94 | C2×#493 | — | 61 | 1 | no | — |
| 95 | #1837 | — | 61 | 1 | no | — |
| 96 | #1545 | — | 61 | 4 | no | — |
| 97 | #1452 | — | 61 | 1 | no | — |
| 98 | C2×C61 | — | 16 | 2 | no | — |
| 99 | C2×C61 | #1961 | 1 | 1 | yes | characteristic |
| 100 | C2×C61 | #1978 | 1 | 2 | yes | — |
| 101 | D61 | — | 2 | 2 | no | — |
| 102 | D61 | — | 16 | 2 | no | — |
| 103 | C22×D8 | — | 122 | 2 | no | — |
| 104 | C2×#127 | — | 61 | 2 | no | — |
| 105 | #493 | — | 61 | 2 | no | — |
| 106 | #493 | — | 61 | 1 | no | — |
| 107 | #493 | — | 122 | 2 | no | — |
| 108 | #493 | #6923 | 1 | 1 | yes | characteristic |
| 109 | #431 | — | 61 | 2 | no | — |
| 110 | #423 | — | 61 | 2 | no | — |
| 111 | C2×C4×C8 | — | 61 | 1 | no | — |
| 112 | #341 | — | 122 | 2 | no | — |
| 113 | #331 | — | 122 | 8 | no | — |
| 114 | #322 | — | 122 | 2 | no | — |
| 115 | C61 | #9780 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 116 | C22×D4 | — | 122 | 4 | no | — |
| 117 | C22×D4 | — | 122 | 2 | no | — |
| 118 | C2×D8 | — | 2 | 2 | no | — |
| 119 | C2×D8 | — | 122 | 4 | no | — |
| 120 | C2×D8 | — | 122 | 2 | no | — |
| 121 | C2×D8 | — | 244 | 8 | no | — |
| 122 | C2×M16 | — | 122 | 2 | no | — |
| 123 | C22×C8 | — | 61 | 2 | no | — |
| 124 | #127 | — | 61 | 2 | no | — |
| 125 | #127 | — | 122 | 4 | no | — |
| 126 | #127 | — | 122 | 2 | no | — |
| 127 | #127 | C2×#6923 | 1 | 2 | yes | — |
| 128 | C2×C42 | — | 61 | 1 | no | — |
| 129 | C4⋊C8 | — | 61 | 4 | no | — |
| 130 | C4⋊C8 | — | 122 | 4 | no | — |
| 131 | C22⋊C8 | — | 122 | 2 | no | — |
| 132 | C4×C8 | — | 61 | 2 | no | — |
| 133 | C4×C8 | — | 61 | 1 | no | — |
| 134 | C4×C8 | — | 122 | 2 | no | — |
| 135 | C4×C8 | C2×#6923 | 1 | 1 | yes | characteristic |
| 136 | C24 | — | 244 | 2 | no | — |
| 137 | C2×D4 | — | 2 | 2 | no | — |
| 138 | C2×D4 | — | 2 | 4 | no | — |
| 139 | C2×D4 | — | 122 | 2 | no | — |
| 140 | C2×D4 | — | 122 | 4 | no | — |
| 141 | C2×D4 | — | 488 | 8 | no | — |
| 142 | C2×D4 | — | 244 | 8 | no | — |
| 143 | C2×D4 | — | 244 | 4 | no | — |
| 144 | C2×D4 | — | 244 | 2 | no | — |
| 145 | C22×C4 | — | 61 | 1 | no | — |
| 146 | C22×C4 | — | 61 | 2 | no | — |
| 147 | D8 | — | 4 | 4 | no | — |
| 148 | D8 | — | 244 | 4 | no | — |
| 149 | D8 | — | 244 | 8 | no | — |
| 150 | M16 | — | 244 | 4 | no | — |
| 151 | C2×C8 | — | 61 | 2 | no | — |
| 152 | C2×C8 | — | 122 | 4 | no | — |
| 153 | C2×C8 | — | 122 | 4 | no | — |
| 154 | C2×C8 | — | 244 | 2 | no | — |
| 155 | C2×C8 | C22⋊#6923 | 1 | 2 | yes | — |
| 156 | C42 | — | 61 | 1 | no | — |
| 157 | C42 | — | 61 | 2 | no | — |
| 158 | C42 | C22×#6923 | 1 | 1 | yes | characteristic |
| 159 | C23 | — | 4 | 2 | no | — |
| 160 | C23 | — | 61 | 1 | no | — |
| 161 | C23 | — | 488 | 4 | no | — |
| 162 | C23 | — | 488 | 8 | no | — |
| 163 | C23 | — | 244 | 2 | no | — |
| 164 | D4 | — | 4 | 4 | no | — |
| 165 | D4 | — | 8 | 4 | no | — |
| 166 | D4 | — | 488 | 4 | no | — |
| 167 | D4 | — | 488 | 8 | no | — |
| 168 | D4 | — | 488 | 4 | no | — |
| 169 | D4 | — | 244 | 4 | no | — |
| 170 | C2×C4 | — | 61 | 2 | no | — |
| 171 | C2×C4 | — | 61 | 1 | no | — |
| 172 | C2×C4 | — | 61 | 4 | no | — |
| 173 | C2×C4 | — | 122 | 4 | no | — |
| 174 | C2×C4 | C2×C22⋊#6923 | 1 | 1 | yes | characteristic |
| 175 | C2×C4 | D4×#6923 | 1 | 2 | yes | — |
| 176 | C8 | — | 2 | 2 | no | — |
| 177 | C8 | — | 122 | 2 | no | — |
| 178 | C8 | — | 488 | 2 | no | — |
| 179 | C8 | — | 244 | 4 | no | — |
| 180 | C22 | — | 8 | 2 | no | — |
| 181 | C22 | — | 8 | 4 | no | — |
| 182 | C22 | — | 61 | 2 | no | — |
| 183 | C22 | — | 122 | 2 | no | — |
| 184 | C22 | — | 976 | 8 | no | — |
| 185 | C22 | — | 488 | 2 | no | — |
| 186 | C22 | — | 488 | 4 | no | — |
| 187 | C22 | #16840714 | 1 | 1 | yes | characteristic |
| 188 | C4 | — | 2 | 2 | no | — |
| 189 | C4 | — | 61 | 2 | no | — |
| 190 | C4 | — | 61 | 4 | no | — |
| 191 | C4 | #16840754 | 1 | 2 | yes | — |
| 192 | C2 | — | 16 | 2 | no | — |
| 193 | C2 | — | 122 | 2 | no | — |
| 194 | C2 | — | 976 | 2 | no | — |
| 195 | C2 | #21461951 | 1 | 1 | yes | characteristic |
| 196 | C2 | #21461980 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21461744 | C2 | semidirect |
| 2. #21461327 | C2 | semidirect |
| 3. #21461033 | C2 | semidirect |
| 115. C61 | #9780 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21461744 | 2 |
| #21461327 | 2 |
| #21461033 | 2 |
| #493×D61 | 2 |
| #9780 | 61 |