Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
780 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16840151 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #21453865 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #21446642 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#16839501 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C61×#1295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#11758190 | — | 2 | 2 | no | — |
| 7 | C22×D488 | — | 2 | 2 | no | — |
| 8 | C2×#14519774 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14519765 | C22 | 1 | 2 | yes | — |
| 10 | #16840151 | — | 2 | 4 | no | — |
| 11 | #16840151 | — | 2 | 4 | no | — |
| 12 | C2×#14519697 | — | 2 | 2 | no | — |
| 13 | C2×#14519688 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16839679 | — | 2 | 2 | no | — |
| 15 | #16839675 | — | 2 | 2 | no | — |
| 16 | #16839672 | — | 2 | 2 | no | — |
| 17 | #16839501 | — | 2 | 4 | no | — |
| 18 | C2×#97×C61 | C22 | 1 | 2 | yes | — |
| 19 | C61×#382 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×Dic122 | D4 | 1 | 2 | yes | — |
| 21 | C22×D244 | — | 2 | 1 | no | — |
| 22 | C22×D244 | — | 2 | 2 | no | — |
| 23 | C22×D244 | — | 2 | 2 | no | — |
| 24 | C22×D244 | D4 | 1 | 2 | yes | — |
| 25 | C2×#11758190 | — | 2 | 8 | no | — |
| 26 | C2×#11758190 | — | 4 | 8 | no | — |
| 27 | C2×D488 | — | 2 | 8 | no | — |
| 28 | C2×D488 | — | 4 | 8 | no | — |
| 29 | C22×#92418 | — | 2 | 1 | no | — |
| 30 | #14519774 | — | 2 | 2 | no | — |
| 31 | #14519774 | — | 2 | 1 | no | — |
| 32 | #14519774 | D4 | 1 | 2 | yes | complemented |
| 33 | #14519765 | — | 2 | 2 | no | — |
| 34 | #14519765 | — | 2 | 2 | no | — |
| 35 | #14519765 | — | 2 | 2 | no | — |
| 36 | #14519765 | — | 2 | 2 | no | — |
| 37 | C2×#11758198 | — | 2 | 2 | no | — |
| 38 | C2×#11758198 | — | 2 | 2 | no | — |
| 39 | C2×#11758198 | — | 2 | 4 | no | — |
| 40 | #14519697 | — | 4 | 4 | no | — |
| 41 | #14519688 | — | 2 | 2 | no | — |
| 42 | #14519684 | — | 2 | 2 | no | — |
| 43 | C22×C8×C61 | — | 2 | 2 | no | — |
| 44 | C2×C4⋊C4×C61 | D4 | 1 | 2 | yes | — |
| 45 | C2×C42×C61 | C23 | 1 | 1 | yes | characteristic |
| 46 | #97×C61 | — | 2 | 2 | no | — |
| 47 | #97×C61 | D4 | 1 | 4 | yes | complemented |
| 48 | #95×C61 | — | 2 | 2 | no | — |
| 49 | C23×D61 | — | 2 | 2 | no | — |
| 50 | C23×D61 | — | 4 | 1 | no | — |
| 51 | C22×C4×C61 | — | 2 | 2 | no | — |
| 52 | C22×C4×C61 | C2×D4 | 1 | 2 | yes | — |
| 53 | C22×C4×C61 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C22×Dic61 | — | 2 | 2 | no | — |
| 55 | C2×D244 | — | 2 | 4 | no | — |
| 56 | C2×D244 | — | 2 | 2 | no | — |
| 57 | C2×D244 | — | 2 | 2 | no | — |
| 58 | C2×D244 | — | 2 | 2 | no | — |
| 59 | C2×D244 | — | 4 | 4 | no | — |
| 60 | C2×D244 | — | 4 | 2 | no | — |
| 61 | C2×D244 | — | 4 | 2 | no | — |
| 62 | C2×D244 | — | 4 | 4 | no | — |
| 63 | C2×D244 | — | 4 | 4 | no | — |
| 64 | C2×D244 | — | 4 | 2 | no | — |
| 65 | C2×D244 | — | 4 | 1 | no | — |
| 66 | C2×D244 | — | 8 | 4 | no | — |
| 67 | C2×D244 | — | 8 | 2 | no | — |
| 68 | C2×Dic122 | — | 2 | 2 | no | — |
| 69 | C2×Dic122 | — | 2 | 2 | no | — |
| 70 | C2×Dic122 | — | 4 | 4 | no | — |
| 71 | C2×C8×C61 | — | 2 | 8 | no | — |
| 72 | C2×C8×C61 | — | 2 | 4 | no | — |
| 73 | C4⋊C4×C61 | — | 4 | 2 | no | — |
| 74 | C42×C61 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C42×C61 | C2×D4 | 1 | 2 | yes | — |
| 76 | C42×C61 | #41 | 1 | 1 | yes | characteristic |
| 77 | #11758198 | — | 4 | 2 | no | — |
| 78 | #11758198 | — | 4 | 2 | no | — |
| 79 | #11758198 | — | 4 | 2 | no | — |
| 80 | #11758198 | — | 4 | 2 | no | — |
| 81 | #11758198 | — | 8 | 4 | no | — |
| 82 | C2×#92418 | — | 2 | 2 | no | — |
| 83 | C2×#92418 | — | 4 | 1 | no | — |
| 84 | C2×#92418 | — | 4 | 1 | no | — |
| 85 | D488 | — | 4 | 16 | no | — |
| 86 | #11758190 | — | 4 | 16 | no | — |
| 87 | C23×C61 | #120 | 1 | 1 | yes | characteristic |
| 88 | C22×D61 | — | 4 | 2 | no | — |
| 89 | C22×D61 | — | 4 | 2 | no | — |
| 90 | C22×D61 | — | 4 | 4 | no | — |
| 91 | C22×D61 | — | 4 | 4 | no | — |
| 92 | C22×D61 | — | 4 | 2 | no | — |
| 93 | C22×D61 | — | 4 | 2 | no | — |
| 94 | C22×D61 | — | 8 | 4 | no | — |
| 95 | C22×D61 | — | 8 | 1 | no | — |
| 96 | C22×D61 | — | 8 | 1 | no | — |
| 97 | C2×C4×C61 | — | 2 | 1 | no | — |
| 98 | C2×C4×C61 | — | 2 | 4 | no | — |
| 99 | C2×C4×C61 | — | 2 | 1 | no | — |
| 100 | C2×C4×C61 | — | 4 | 4 | no | — |
| 101 | C2×C4×C61 | — | 4 | 2 | no | — |
| 102 | C2×C4×C61 | #120 | 1 | 2 | yes | — |
| 103 | C2×C4×C61 | #121 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C61 | #121 | 1 | 2 | yes | — |
| 105 | C2×C4×C61 | #127 | 1 | 1 | yes | characteristic |
| 106 | C2×Dic61 | — | 4 | 2 | no | — |
| 107 | C2×Dic61 | — | 4 | 4 | no | — |
| 108 | D244 | — | 4 | 4 | no | — |
| 109 | D244 | — | 4 | 4 | no | — |
| 110 | D244 | — | 4 | 4 | no | — |
| 111 | D244 | — | 4 | 4 | no | — |
| 112 | D244 | — | 8 | 2 | no | — |
| 113 | D244 | — | 8 | 4 | no | — |
| 114 | D244 | — | 8 | 2 | no | — |
| 115 | D244 | — | 8 | 8 | no | — |
| 116 | Dic122 | — | 4 | 4 | no | — |
| 117 | Dic122 | — | 8 | 2 | no | — |
| 118 | C8×C61 | — | 2 | 8 | no | — |
| 119 | #92418 | — | 4 | 2 | no | — |
| 120 | #10078 | — | 61 | 1 | no | maximal, Sylow |
| 121 | C22×C61 | #392 | 1 | 1 | yes | characteristic |
| 122 | C22×C61 | #453 | 1 | 4 | yes | — |
| 123 | C22×C61 | #453 | 1 | 1 | yes | characteristic |
| 124 | C22×C61 | #454 | 1 | 1 | yes | characteristic |
| 125 | C2×D61 | — | 8 | 4 | no | — |
| 126 | C2×D61 | — | 8 | 4 | no | — |
| 127 | C2×D61 | — | 8 | 4 | no | — |
| 128 | C2×D61 | — | 8 | 2 | no | — |
| 129 | C2×D61 | — | 8 | 2 | no | — |
| 130 | C2×D61 | — | 8 | 2 | no | — |
| 131 | C2×D61 | — | 8 | 2 | no | — |
| 132 | C2×D61 | — | 16 | 2 | no | — |
| 133 | C2×D61 | — | 16 | 2 | no | — |
| 134 | C4×C61 | — | 2 | 2 | no | — |
| 135 | C4×C61 | — | 2 | 2 | no | — |
| 136 | C4×C61 | — | 2 | 1 | no | — |
| 137 | C4×C61 | — | 2 | 1 | no | — |
| 138 | C4×C61 | — | 8 | 2 | no | — |
| 139 | Dic61 | — | 8 | 2 | no | — |
| 140 | C2×#496 | — | 61 | 2 | no | — |
| 141 | #2819 | — | 61 | 1 | no | — |
| 142 | #1961 | — | 61 | 1 | no | — |
| 143 | C2×#329 | — | 61 | 2 | no | — |
| 144 | #1295 | D61 | 1 | 1 | yes | characteristic, complemented |
| 145 | C2×C61 | #1655 | 1 | 2 | yes | — |
| 146 | C2×C61 | #1656 | 1 | 2 | yes | — |
| 147 | C2×C61 | #2002 | 1 | 2 | yes | — |
| 148 | C2×C61 | #2006 | 1 | 1 | yes | characteristic |
| 149 | D61 | — | 16 | 2 | no | — |
| 150 | D61 | — | 16 | 2 | no | — |
| 151 | C22×SD16 | — | 122 | 2 | no | — |
| 152 | C22×D8 | — | 122 | 2 | no | — |
| 153 | C2×#127 | — | 61 | 1 | no | — |
| 154 | C2×#124 | — | 61 | 2 | no | — |
| 155 | #496 | — | 122 | 4 | no | — |
| 156 | #496 | — | 122 | 4 | no | — |
| 157 | C2×#102 | — | 122 | 2 | no | — |
| 158 | C2×#97 | — | 61 | 1 | no | — |
| 159 | C2×#97 | C2×D61 | 1 | 2 | yes | — |
| 160 | #394 | — | 122 | 2 | no | — |
| 161 | #393 | — | 122 | 2 | no | — |
| 162 | #392 | — | 122 | 2 | no | — |
| 163 | #382 | C2×D61 | 1 | 1 | yes | characteristic |
| 164 | #329 | — | 122 | 4 | no | — |
| 165 | C61 | #10078 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 166 | C22×Q8 | — | 61 | 2 | no | — |
| 167 | C22×D4 | — | 61 | 2 | no | — |
| 168 | C22×D4 | — | 122 | 2 | no | — |
| 169 | C22×D4 | — | 122 | 1 | no | — |
| 170 | C22×D4 | — | 122 | 2 | no | — |
| 171 | C2×SD16 | — | 122 | 8 | no | — |
| 172 | C2×SD16 | — | 244 | 8 | no | — |
| 173 | C2×D8 | — | 122 | 8 | no | — |
| 174 | C2×D8 | — | 244 | 8 | no | — |
| 175 | C22×C8 | — | 2 | 2 | no | — |
| 176 | C22×C8 | — | 122 | 1 | no | — |
| 177 | #127 | — | 61 | 2 | no | — |
| 178 | #127 | — | 122 | 2 | no | — |
| 179 | #127 | — | 122 | 1 | no | — |
| 180 | #124 | — | 122 | 2 | no | — |
| 181 | #124 | — | 122 | 2 | no | — |
| 182 | #124 | — | 122 | 2 | no | — |
| 183 | #124 | — | 122 | 2 | no | — |
| 184 | C2×C4⋊C4 | D244 | 1 | 2 | yes | — |
| 185 | C2×C22⋊C4 | — | 122 | 4 | no | — |
| 186 | C2×C22⋊C4 | — | 122 | 2 | no | — |
| 187 | C2×C22⋊C4 | — | 122 | 2 | no | — |
| 188 | C2×C42 | C22×D61 | 1 | 1 | yes | characteristic |
| 189 | #102 | — | 244 | 4 | no | — |
| 190 | #97 | — | 2 | 2 | no | — |
| 191 | #97 | — | 122 | 2 | no | — |
| 192 | #97 | D244 | 1 | 4 | yes | complemented |
| 193 | #95 | — | 2 | 2 | no | — |
| 194 | #95 | — | 122 | 2 | no | — |
| 195 | C24 | — | 122 | 2 | no | — |
| 196 | C24 | — | 244 | 1 | no | — |
| 197 | C2×Q8 | — | 122 | 2 | no | — |
| 198 | C2×Q8 | — | 122 | 2 | no | — |
| 199 | C2×Q8 | — | 244 | 4 | no | — |
| 200 | C2×D4 | — | 122 | 2 | no | — |
| 201 | C2×D4 | — | 122 | 4 | no | — |
| 202 | C2×D4 | — | 122 | 2 | no | — |
| 203 | C2×D4 | — | 122 | 2 | no | — |
| 204 | C2×D4 | — | 488 | 4 | no | — |
| 205 | C2×D4 | — | 488 | 2 | no | — |
| 206 | C2×D4 | — | 244 | 2 | no | — |
| 207 | C2×D4 | — | 244 | 4 | no | — |
| 208 | C2×D4 | — | 244 | 4 | no | — |
| 209 | C2×D4 | — | 244 | 2 | no | — |
| 210 | C2×D4 | — | 244 | 4 | no | — |
| 211 | C2×D4 | — | 244 | 2 | no | — |
| 212 | C2×D4 | — | 244 | 1 | no | — |
| 213 | C22×C4 | — | 2 | 2 | no | — |
| 214 | C22×C4 | — | 122 | 2 | no | — |
| 215 | C22×C4 | C2×D244 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | D4×D61 | 1 | 2 | yes | — |
| 217 | SD16 | — | 244 | 16 | no | — |
| 218 | D8 | — | 244 | 16 | no | — |
| 219 | C2×C8 | — | 2 | 8 | no | — |
| 220 | C2×C8 | — | 2 | 4 | no | — |
| 221 | C2×C8 | — | 122 | 2 | no | — |
| 222 | C2×C8 | — | 244 | 1 | no | — |
| 223 | C2×C8 | — | 244 | 1 | no | — |
| 224 | C4⋊C4 | — | 4 | 2 | no | — |
| 225 | C22⋊C4 | — | 488 | 4 | no | — |
| 226 | C22⋊C4 | — | 244 | 2 | no | — |
| 227 | C22⋊C4 | — | 244 | 2 | no | — |
| 228 | C22⋊C4 | — | 244 | 2 | no | — |
| 229 | C22⋊C4 | — | 244 | 2 | no | — |
| 230 | C42 | C2×D244 | 1 | 2 | yes | — |
| 231 | C42 | D4×D61 | 1 | 1 | yes | characteristic |
| 232 | C42 | #11758226 | 1 | 1 | yes | characteristic |
| 233 | C23 | — | 488 | 1 | no | — |
| 234 | C23 | — | 488 | 4 | no | — |
| 235 | C23 | — | 488 | 1 | no | — |
| 236 | C23 | — | 244 | 2 | no | — |
| 237 | C23 | — | 244 | 2 | no | — |
| 238 | C23 | — | 244 | 2 | no | — |
| 239 | C23 | — | 244 | 4 | no | — |
| 240 | C23 | — | 244 | 4 | no | — |
| 241 | C23 | — | 244 | 2 | no | — |
| 242 | C23 | #14519747 | 1 | 1 | yes | characteristic |
| 243 | Q8 | — | 488 | 2 | no | — |
| 244 | Q8 | — | 244 | 4 | no | — |
| 245 | D4 | — | 488 | 2 | no | — |
| 246 | D4 | — | 488 | 4 | no | — |
| 247 | D4 | — | 488 | 2 | no | — |
| 248 | D4 | — | 488 | 8 | no | — |
| 249 | D4 | — | 244 | 4 | no | — |
| 250 | D4 | — | 244 | 4 | no | — |
| 251 | D4 | — | 244 | 4 | no | — |
| 252 | D4 | — | 244 | 4 | no | — |
| 253 | C2×C4 | — | 2 | 1 | no | — |
| 254 | C2×C4 | — | 2 | 4 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 4 | 4 | no | — |
| 257 | C2×C4 | — | 4 | 2 | no | — |
| 258 | C2×C4 | — | 244 | 2 | no | — |
| 259 | C2×C4 | — | 244 | 4 | no | — |
| 260 | C2×C4 | #14519747 | 1 | 2 | yes | — |
| 261 | C2×C4 | #14519752 | 1 | 2 | yes | — |
| 262 | C2×C4 | #14519752 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | #14519774 | 1 | 1 | yes | characteristic |
| 264 | C8 | — | 2 | 8 | no | — |
| 265 | C8 | — | 244 | 2 | no | — |
| 266 | C22 | — | 976 | 2 | no | — |
| 267 | C22 | — | 976 | 2 | no | — |
| 268 | C22 | — | 488 | 4 | no | — |
| 269 | C22 | — | 488 | 2 | no | — |
| 270 | C22 | — | 488 | 2 | no | — |
| 271 | C22 | — | 488 | 2 | no | — |
| 272 | C22 | — | 488 | 2 | no | — |
| 273 | C22 | — | 488 | 4 | no | — |
| 274 | C22 | — | 488 | 4 | no | — |
| 275 | C22 | #16839672 | 1 | 1 | yes | characteristic |
| 276 | C22 | #16839931 | 1 | 4 | yes | — |
| 277 | C22 | #16839932 | 1 | 1 | yes | characteristic |
| 278 | C22 | #16839937 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 2 | no | — |
| 280 | C4 | — | 2 | 2 | no | — |
| 281 | C4 | — | 2 | 1 | no | — |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 8 | 2 | no | — |
| 284 | C4 | — | 488 | 2 | no | — |
| 285 | C2 | — | 976 | 2 | no | — |
| 286 | C2 | — | 976 | 2 | no | — |
| 287 | C2 | #21445109 | 1 | 2 | yes | — |
| 288 | C2 | #21445114 | 1 | 2 | yes | — |
| 289 | C2 | #21446837 | 1 | 2 | yes | — |
| 290 | C2 | #21446863 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16840151 | C2 | semidirect |
| 2. #21453865 | C2 | semidirect |
| 3. #21446642 | C2 | semidirect |
| 4. C2×#16839501 | C2 | semidirect |
| 5. C61×#1295 | C2 | semidirect |
| 32. #14519774 | D4 | semidirect |
| 47. #97×C61 | D4 | semidirect |
| 144. #1295 | D61 | semidirect |
| 165. C61 | #10078 | semidirect |
| 192. #97 | D244 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16840151 | 2 |
| #21453865 | 2 |
| #21446642 | 2 |
| C2×#16839501 | 2 |
| C61×#1295 | 2 |
| #10078 | 61 |