Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
1312 conjugacy classes of subgroups; only the 220 normal subgroups are shown.
This group has 1312 conjugacy classes of subgroups — too many to draw. Only its 220 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 | #21462430 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #21462118 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #21462110 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #21459593 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16840842 | — | 2 | 2 | no | — |
| 6 | #16840748 | C22 | 1 | 12 | yes | complemented |
| 7 | D61⋊#100 | — | 2 | 6 | no | — |
| 8 | #16840672 | C22 | 1 | 4 | yes | complemented |
| 9 | #16840670 | C22 | 1 | 12 | yes | complemented |
| 10 | C2×#14519823 | C22 | 1 | 1 | yes | characteristic |
| 11 | #16840431 | — | 2 | 2 | no | — |
| 12 | D4×C22⋊D61 | — | 2 | 6 | no | — |
| 13 | #16840329 | C22 | 1 | 3 | yes | — |
| 14 | #16840326 | C4 | 1 | 3 | yes | — |
| 15 | #16840308 | C22 | 1 | 3 | yes | — |
| 16 | C2×#14519773 | C4 | 1 | 3 | yes | — |
| 17 | #41×Dic61 | C4 | 1 | 1 | yes | characteristic |
| 18 | C2×H(5,𝔽2)×C61 | C4 | 1 | 1 | yes | characteristic |
| 19 | #14519865 | — | 4 | 8 | no | — |
| 20 | D61⋊M16 | — | 2 | 12 | no | — |
| 21 | #14519838 | C23 | 1 | 8 | yes | — |
| 22 | #14519837 | — | 2 | 12 | no | — |
| 23 | #14519823 | — | 4 | 4 | no | — |
| 24 | C2×#11758226 | D4 | 1 | 2 | yes | — |
| 25 | C2×#11758222 | C23 | 1 | 3 | yes | — |
| 26 | C22×D244 | C2×C4 | 1 | 3 | yes | — |
| 27 | C22×C22⋊D61 | — | 2 | 6 | no | — |
| 28 | C2×D4×D61 | D4 | 1 | 6 | yes | — |
| 29 | #14519775 | C23 | 1 | 1 | yes | characteristic |
| 30 | #14519773 | — | 2 | 6 | no | — |
| 31 | #14519773 | C2×C4 | 1 | 3 | yes | — |
| 32 | #14519773 | C2×C4 | 1 | 6 | yes | — |
| 33 | #14519766 | C2×C4 | 1 | 3 | yes | — |
| 34 | #14519764 | C23 | 1 | 3 | yes | — |
| 35 | #14519751 | — | 2 | 6 | no | — |
| 36 | #14519749 | — | 2 | 6 | no | — |
| 37 | #14519748 | — | 2 | 6 | no | — |
| 38 | #14519746 | — | 2 | 6 | no | — |
| 39 | #14519745 | — | 2 | 6 | no | — |
| 40 | Q8×Dic61 | — | 2 | 1 | no | — |
| 41 | D4×Dic61 | — | 2 | 3 | no | — |
| 42 | C4×C22⋊D61 | — | 2 | 6 | no | — |
| 43 | #14519736 | C2×C4 | 1 | 3 | yes | — |
| 44 | C2×C4×Dic61 | C2×C4 | 1 | 3 | yes | — |
| 45 | H(5,𝔽2)×C61 | — | 4 | 4 | no | — |
| 46 | C2×#41×C61 | — | 2 | 1 | no | — |
| 47 | C2×#41×C61 | C2×C4 | 1 | 3 | yes | — |
| 48 | C2×#41×C61 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C22×D4×C61 | — | 2 | 3 | no | — |
| 50 | C22×D4×C61 | C2×C4 | 1 | 3 | yes | — |
| 51 | C24×C61 | — | 2 | 3 | no | — |
| 52 | C23×D61 | C22⋊C4 | 1 | 6 | yes | — |
| 53 | #41×C61 | — | 2 | 4 | no | — |
| 54 | #41×C61 | — | 4 | 6 | no | — |
| 55 | #41×C61 | — | 4 | 4 | no | — |
| 56 | C2×Q8×C61 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 57 | C2×Q8×C61 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C2×D4×C61 | — | 2 | 3 | no | — |
| 59 | C2×D4×C61 | — | 2 | 3 | no | — |
| 60 | C2×D4×C61 | — | 4 | 6 | no | — |
| 61 | C2×D4×C61 | — | 4 | 12 | no | — |
| 62 | C2×D4×C61 | C22⋊C4 | 1 | 3 | yes | — |
| 63 | C2×D4×C61 | C22×C4 | 1 | 3 | yes | — |
| 64 | C22×C4×C61 | — | 2 | 3 | no | — |
| 65 | C22×C4×C61 | C22×C4 | 1 | 3 | yes | — |
| 66 | C2×C22⋊D61 | — | 2 | 6 | no | — |
| 67 | C2×C22⋊D61 | — | 2 | 12 | no | — |
| 68 | C2×C22⋊D61 | — | 8 | 12 | no | — |
| 69 | C2×C22⋊D61 | C2×D4 | 1 | 6 | yes | — |
| 70 | C22×Dic61 | — | 2 | 3 | no | — |
| 71 | #11758226 | — | 2 | 8 | no | — |
| 72 | D4×D61 | — | 4 | 12 | no | — |
| 73 | #11758222 | — | 4 | 6 | no | — |
| 74 | C2×D244 | — | 4 | 6 | no | — |
| 75 | C2×D244 | C22⋊C4 | 1 | 6 | yes | — |
| 76 | C2×D244 | C22×C4 | 1 | 3 | yes | — |
| 77 | C2×C4×D61 | C2×D4 | 1 | 6 | yes | — |
| 78 | C2×Dic122 | C24 | 1 | 1 | yes | characteristic |
| 79 | #11758217 | — | 2 | 8 | no | — |
| 80 | #11758213 | — | 4 | 12 | no | — |
| 81 | D61⋊C8 | — | 4 | 12 | no | — |
| 82 | C22⋊Dic61 | — | 2 | 3 | no | — |
| 83 | C22⋊Dic61 | — | 2 | 6 | no | — |
| 84 | #11758198 | — | 2 | 6 | no | — |
| 85 | C4⋊Dic61 | — | 2 | 3 | no | — |
| 86 | #11758196 | — | 2 | 2 | no | — |
| 87 | C4×Dic61 | — | 2 | 3 | no | — |
| 88 | C4×Dic61 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C4×Dic61 | C22×C4 | 1 | 3 | yes | — |
| 90 | C23×C61 | — | 2 | 3 | no | — |
| 91 | C23×C61 | — | 4 | 6 | no | — |
| 92 | C23×C61 | — | 8 | 6 | no | — |
| 93 | C23×C61 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 94 | C22×D61 | — | 4 | 6 | no | — |
| 95 | C22×D61 | — | 8 | 6 | no | — |
| 96 | C22×D61 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 97 | Q8×C61 | — | 4 | 1 | no | — |
| 98 | Q8×C61 | #99 | 1 | 4 | yes | — |
| 99 | D4×C61 | — | 2 | 6 | no | — |
| 100 | D4×C61 | — | 4 | 3 | no | — |
| 101 | D4×C61 | — | 4 | 6 | no | — |
| 102 | D4×C61 | — | 4 | 6 | no | — |
| 103 | C2×C4×C61 | — | 2 | 6 | no | — |
| 104 | C2×C4×C61 | — | 2 | 1 | no | — |
| 105 | C2×C4×C61 | — | 4 | 6 | no | — |
| 106 | C2×C4×C61 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 107 | C2×C4×C61 | C23×C4 | 1 | 1 | yes | characteristic |
| 108 | C22⋊D61 | — | 4 | 6 | no | — |
| 109 | C22⋊D61 | — | 4 | 12 | no | — |
| 110 | C22⋊D61 | — | 8 | 12 | no | — |
| 111 | C2×Dic61 | — | 2 | 1 | no | — |
| 112 | C2×Dic61 | — | 2 | 3 | no | — |
| 113 | C2×Dic61 | — | 4 | 6 | no | — |
| 114 | C2×Dic61 | C22×D4 | 1 | 2 | yes | — |
| 115 | D244 | — | 2 | 12 | no | — |
| 116 | D244 | — | 4 | 3 | no | — |
| 117 | C4×D61 | — | 2 | 12 | no | — |
| 118 | Dic122 | — | 4 | 1 | no | — |
| 119 | #92420 | — | 4 | 8 | no | — |
| 120 | #33373 | — | 61 | 1 | no | maximal, Sylow |
| 121 | C22×C61 | — | 2 | 3 | no | — |
| 122 | C22×C61 | — | 4 | 3 | no | — |
| 123 | C22×C61 | — | 4 | 6 | no | — |
| 124 | C22×C61 | — | 4 | 3 | no | — |
| 125 | C22×C61 | — | 16 | 6 | no | — |
| 126 | C22×C61 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 127 | C2×D61 | — | 2 | 6 | no | — |
| 128 | C2×D61 | — | 4 | 6 | no | — |
| 129 | C2×D61 | — | 16 | 6 | no | — |
| 130 | C4×C61 | — | 2 | 1 | no | — |
| 131 | C4×C61 | — | 4 | 1 | no | — |
| 132 | C4×C61 | C2×#99 | 1 | 6 | yes | — |
| 133 | Dic61 | — | 2 | 2 | no | — |
| 134 | Dic61 | — | 2 | 2 | no | — |
| 135 | Dic61 | — | 4 | 3 | no | — |
| 136 | #3498 | — | 61 | 1 | no | — |
| 137 | #2888 | — | 61 | 2 | no | — |
| 138 | #2134 | — | 61 | 6 | no | — |
| 139 | #2132 | — | 61 | 6 | no | — |
| 140 | C2×C61 | — | 2 | 1 | no | — |
| 141 | C2×C61 | — | 4 | 3 | no | — |
| 142 | C2×C61 | — | 8 | 3 | no | — |
| 143 | C2×C61 | C22×#99 | 1 | 1 | yes | characteristic |
| 144 | D61 | — | 4 | 6 | no | — |
| 145 | C2×H(5,𝔽2) | — | 61 | 1 | no | — |
| 146 | C2×H(5,𝔽2) | #6923 | 1 | 1 | yes | characteristic |
| 147 | #568 | — | 122 | 2 | no | — |
| 148 | #550 | — | 122 | 2 | no | — |
| 149 | D42 | — | 122 | 6 | no | — |
| 150 | #535 | — | 61 | 3 | no | — |
| 151 | #535 | — | 61 | 3 | no | — |
| 152 | #532 | — | 61 | 3 | no | — |
| 153 | C2×#127 | — | 61 | 3 | no | — |
| 154 | C4×#41 | — | 61 | 1 | no | — |
| 155 | #413 | — | 61 | 12 | no | — |
| 156 | C2×#100 | — | 122 | 6 | no | — |
| 157 | #356 | — | 61 | 4 | no | — |
| 158 | #355 | — | 61 | 12 | no | — |
| 159 | C61 | #33373 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 160 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 161 | H(5,𝔽2) | — | 244 | 4 | no | — |
| 162 | C2×#41 | — | 2 | 1 | no | — |
| 163 | C2×#41 | — | 61 | 2 | no | — |
| 164 | C2×#41 | — | 61 | 3 | no | — |
| 165 | C2×#41 | C2×#6923 | 1 | 3 | yes | — |
| 166 | C2×#41 | C2×#6923 | 1 | 1 | yes | characteristic |
| 167 | C22×D4 | — | 2 | 3 | no | — |
| 168 | C22×D4 | — | 61 | 6 | no | — |
| 169 | C22×D4 | — | 61 | 3 | no | — |
| 170 | C22×D4 | — | 122 | 6 | no | — |
| 171 | C22×D4 | C2×#6923 | 1 | 3 | yes | — |
| 172 | #131 | — | 244 | 8 | no | — |
| 173 | C2×M16 | — | 122 | 12 | no | — |
| 174 | C4⋊Q8 | — | 61 | 1 | no | — |
| 175 | #127 | — | 61 | 3 | no | — |
| 176 | #127 | — | 61 | 6 | no | — |
| 177 | #127 | — | 122 | 6 | no | — |
| 178 | #124 | — | 61 | 3 | no | — |
| 179 | #124 | — | 61 | 3 | no | — |
| 180 | #121 | — | 122 | 6 | no | — |
| 181 | #121 | — | 122 | 6 | no | — |
| 182 | #121 | — | 122 | 6 | no | — |
| 183 | #120 | — | 122 | 6 | no | — |
| 184 | #120 | — | 122 | 6 | no | — |
| 185 | C4×Q8 | — | 122 | 1 | no | — |
| 186 | C4×D4 | — | 122 | 6 | no | — |
| 187 | C4×D4 | — | 122 | 3 | no | — |
| 188 | #117 | — | 61 | 3 | no | — |
| 189 | C2×C42 | — | 61 | 3 | no | — |
| 190 | #101 | — | 61 | 8 | no | — |
| 191 | #100 | — | 122 | 12 | no | — |
| 192 | C24 | — | 2 | 3 | no | — |
| 193 | C24 | — | 61 | 6 | no | — |
| 194 | #41 | — | 2 | 4 | no | — |
| 195 | #41 | — | 4 | 6 | no | — |
| 196 | #41 | — | 4 | 4 | no | — |
| 197 | #41 | — | 122 | 8 | no | — |
| 198 | #41 | — | 244 | 6 | no | — |
| 199 | C2×Q8 | — | 61 | 1 | no | — |
| 200 | C2×Q8 | C22⋊#6923 | 1 | 1 | yes | characteristic |
| 201 | C2×Q8 | C22×#6923 | 1 | 1 | yes | characteristic |
| 202 | C2×D4 | — | 2 | 3 | no | — |
| 203 | C2×D4 | — | 2 | 3 | no | — |
| 204 | C2×D4 | — | 4 | 6 | no | — |
| 205 | C2×D4 | — | 4 | 12 | no | — |
| 206 | C2×D4 | — | 61 | 6 | no | — |
| 207 | C2×D4 | — | 61 | 6 | no | — |
| 208 | C2×D4 | — | 61 | 3 | no | — |
| 209 | C2×D4 | — | 122 | 12 | no | — |
| 210 | C2×D4 | — | 122 | 6 | no | — |
| 211 | C2×D4 | — | 488 | 12 | no | — |
| 212 | C2×D4 | — | 244 | 6 | no | — |
| 213 | C2×D4 | — | 244 | 12 | no | — |
| 214 | C2×D4 | C22⋊#6923 | 1 | 3 | yes | — |
| 215 | C2×D4 | C22×#6923 | 1 | 3 | yes | — |
| 216 | C22×C4 | — | 2 | 3 | no | — |
| 217 | C22×C4 | — | 61 | 6 | no | — |
| 218 | C22×C4 | — | 122 | 3 | no | — |
| 219 | C22×C4 | C22×#6923 | 1 | 3 | yes | — |
| 220 | M16 | — | 122 | 8 | no | — |
| 221 | M16 | — | 244 | 12 | no | — |
| 222 | C2×C8 | — | 244 | 12 | no | — |
| 223 | C4⋊C4 | — | 122 | 2 | no | — |
| 224 | C4⋊C4 | — | 122 | 3 | no | — |
| 225 | C22⋊C4 | — | 122 | 6 | no | — |
| 226 | C22⋊C4 | — | 122 | 6 | no | — |
| 227 | C22⋊C4 | — | 122 | 3 | no | — |
| 228 | C42 | — | 61 | 1 | no | — |
| 229 | C42 | — | 61 | 3 | no | — |
| 230 | C42 | — | 122 | 3 | no | — |
| 231 | C23 | — | 2 | 3 | no | — |
| 232 | C23 | — | 4 | 6 | no | — |
| 233 | C23 | — | 8 | 6 | no | — |
| 234 | C23 | — | 61 | 6 | no | — |
| 235 | C23 | — | 488 | 6 | no | — |
| 236 | C23 | — | 244 | 6 | no | — |
| 237 | C23 | C2×C22⋊#6923 | 1 | 3 | yes | — |
| 238 | Q8 | — | 4 | 1 | no | — |
| 239 | Q8 | — | 244 | 1 | no | — |
| 240 | Q8 | #14519835 | 1 | 4 | yes | — |
| 241 | D4 | — | 2 | 6 | no | — |
| 242 | D4 | — | 4 | 3 | no | — |
| 243 | D4 | — | 4 | 6 | no | — |
| 244 | D4 | — | 4 | 6 | no | — |
| 245 | D4 | — | 122 | 12 | no | — |
| 246 | D4 | — | 488 | 12 | no | — |
| 247 | D4 | — | 244 | 3 | no | — |
| 248 | D4 | — | 244 | 6 | no | — |
| 249 | D4 | — | 244 | 12 | no | — |
| 250 | C2×C4 | — | 2 | 6 | no | — |
| 251 | C2×C4 | — | 2 | 1 | no | — |
| 252 | C2×C4 | — | 4 | 6 | no | — |
| 253 | C2×C4 | — | 61 | 2 | no | — |
| 254 | C2×C4 | — | 122 | 1 | no | — |
| 255 | C2×C4 | — | 122 | 12 | no | — |
| 256 | C2×C4 | — | 122 | 3 | no | — |
| 257 | C2×C4 | — | 244 | 6 | no | — |
| 258 | C2×C4 | C2×C22⋊#6923 | 1 | 3 | yes | — |
| 259 | C2×C4 | C23×#6923 | 1 | 1 | yes | characteristic |
| 260 | C8 | — | 244 | 8 | no | — |
| 261 | C22 | — | 2 | 3 | no | — |
| 262 | C22 | — | 4 | 3 | no | — |
| 263 | C22 | — | 4 | 6 | no | — |
| 264 | C22 | — | 4 | 3 | no | — |
| 265 | C22 | — | 16 | 6 | no | — |
| 266 | C22 | — | 122 | 6 | no | — |
| 267 | C22 | — | 976 | 6 | no | — |
| 268 | C22 | — | 244 | 6 | no | — |
| 269 | C22 | C22×C22⋊#6923 | 1 | 1 | yes | characteristic |
| 270 | C4 | — | 2 | 1 | no | — |
| 271 | C4 | — | 4 | 1 | no | — |
| 272 | C4 | — | 122 | 2 | no | — |
| 273 | C4 | — | 122 | 2 | no | — |
| 274 | C4 | — | 244 | 3 | no | — |
| 275 | C4 | C2×#14519835 | 1 | 6 | yes | — |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 4 | 3 | no | — |
| 278 | C2 | — | 8 | 3 | no | — |
| 279 | C2 | — | 244 | 6 | no | — |
| 280 | C2 | C22×#14519835 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21462430 | C2 | semidirect |
| 2. #21462118 | C2 | semidirect |
| 3. #21462110 | C2 | semidirect |
| 6. #16840748 | C22 | semidirect |
| 8. #16840672 | C22 | semidirect |
| 9. #16840670 | C22 | semidirect |
| 159. C61 | #33373 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21462430 | 2 |
| #21462118 | 2 |
| #21462110 | 2 |
| #21459593 | 2 |
| #33373 | 61 |