Finite Group DatabasebetaFinite Group DatabasebetaOrder 15616.
1426 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 1426 conjugacy classes of subgroups — too many to draw. Only its 243 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 | #21444077 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #21444062 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #21444062 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #21444060 | C2 | 1 | 2 | yes | maximal |
| 5 | C61×#1438 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C22×#11758196 | C22 | 1 | 4 | yes | — |
| 7 | C22×C22⋊Dic61 | C22 | 1 | 2 | yes | — |
| 8 | C22×C22⋊Dic61 | C22 | 1 | 4 | yes | — |
| 9 | C22×C22⋊Dic61 | C22 | 1 | 2 | yes | — |
| 10 | C2×#14519684 | C22 | 1 | 4 | yes | — |
| 11 | C2×#14519684 | C22 | 1 | 4 | yes | — |
| 12 | C2×#14519684 | C22 | 1 | 4 | yes | — |
| 13 | C2×#14519684 | C22 | 1 | 4 | yes | — |
| 14 | C23×D4×C61 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×C22⋊C4×C61 | C22 | 1 | 4 | yes | — |
| 16 | C2×#95×C61 | C22 | 1 | 2 | yes | — |
| 17 | C23×Dic61 | C23 | 1 | 4 | yes | — |
| 18 | C23×Dic61 | C23 | 1 | 4 | yes | — |
| 19 | C2×#11758196 | — | 4 | 4 | no | — |
| 20 | C2×#11758196 | — | 4 | 4 | no | — |
| 21 | C2×#11758196 | — | 4 | 4 | no | — |
| 22 | C2×C22⋊Dic61 | — | 2 | 8 | no | — |
| 23 | C2×C22⋊Dic61 | — | 2 | 8 | no | — |
| 24 | C2×C22⋊Dic61 | — | 4 | 2 | no | — |
| 25 | C2×C22⋊Dic61 | — | 4 | 4 | no | — |
| 26 | C2×C22⋊Dic61 | — | 4 | 4 | no | — |
| 27 | C2×C22⋊Dic61 | — | 4 | 4 | no | — |
| 28 | C2×C22⋊Dic61 | — | 4 | 2 | no | — |
| 29 | #14519684 | — | 2 | 8 | no | — |
| 30 | #14519684 | — | 4 | 4 | no | — |
| 31 | #14519684 | — | 4 | 4 | no | — |
| 32 | #14519684 | — | 4 | 4 | no | — |
| 33 | C25×C61 | C23 | 1 | 2 | yes | — |
| 34 | C22×D4×C61 | — | 4 | 2 | no | — |
| 35 | C22×D4×C61 | — | 4 | 4 | no | — |
| 36 | C22×D4×C61 | Q8 | 1 | 4 | yes | — |
| 37 | C23×C4×C61 | C23 | 1 | 4 | yes | — |
| 38 | C23×C4×C61 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×C22⋊C4×C61 | — | 4 | 4 | no | — |
| 40 | C2×C22⋊C4×C61 | — | 4 | 4 | no | — |
| 41 | C2×C22⋊C4×C61 | — | 4 | 4 | no | — |
| 42 | #95×C61 | — | 4 | 2 | no | — |
| 43 | C24×C61 | — | 2 | 4 | no | — |
| 44 | C24×C61 | — | 2 | 4 | no | — |
| 45 | C24×C61 | — | 2 | 4 | no | — |
| 46 | C24×C61 | — | 2 | 4 | no | — |
| 47 | C24×C61 | — | 2 | 4 | no | — |
| 48 | C24×C61 | — | 2 | 4 | no | — |
| 49 | C24×C61 | — | 2 | 2 | no | — |
| 50 | C24×C61 | — | 2 | 2 | no | — |
| 51 | C24×C61 | C2×Q8 | 1 | 4 | yes | — |
| 52 | C24×C61 | C24 | 1 | 1 | yes | characteristic |
| 53 | C2×D4×C61 | — | 4 | 4 | no | — |
| 54 | C2×D4×C61 | — | 4 | 8 | no | — |
| 55 | C2×D4×C61 | — | 4 | 16 | no | — |
| 56 | C22×C4×C61 | — | 2 | 2 | no | — |
| 57 | C22×C4×C61 | — | 2 | 4 | no | — |
| 58 | C22×C4×C61 | — | 2 | 4 | no | — |
| 59 | C22×C4×C61 | — | 2 | 4 | no | — |
| 60 | C22×C4×C61 | — | 2 | 4 | no | — |
| 61 | C22×C4×C61 | — | 2 | 4 | no | — |
| 62 | C22×C4×C61 | — | 2 | 4 | no | — |
| 63 | C22×C4×C61 | — | 2 | 4 | no | — |
| 64 | C22×C4×C61 | — | 2 | 4 | no | — |
| 65 | C22×C4×C61 | C2×Q8 | 1 | 2 | yes | — |
| 66 | C22×Dic61 | — | 2 | 4 | no | — |
| 67 | C22×Dic61 | — | 2 | 4 | no | — |
| 68 | C22×Dic61 | — | 2 | 4 | no | — |
| 69 | C22×Dic61 | — | 2 | 4 | no | — |
| 70 | C22×Dic61 | — | 2 | 4 | no | — |
| 71 | C22×Dic61 | — | 2 | 4 | no | — |
| 72 | C22×Dic61 | — | 2 | 4 | no | — |
| 73 | C22×Dic61 | — | 2 | 4 | no | — |
| 74 | C22×Dic61 | — | 2 | 4 | no | — |
| 75 | C22×Dic61 | — | 2 | 4 | no | — |
| 76 | C22×Dic61 | — | 2 | 4 | no | — |
| 77 | C22×Dic61 | — | 2 | 4 | no | — |
| 78 | C22×Dic61 | — | 2 | 4 | no | — |
| 79 | C22×Dic61 | #41 | 1 | 8 | yes | — |
| 80 | C22⋊C4×C61 | — | 4 | 16 | no | — |
| 81 | C22⋊Dic61 | — | 4 | 8 | no | — |
| 82 | C22⋊Dic61 | — | 4 | 16 | no | — |
| 83 | C22⋊Dic61 | — | 4 | 8 | no | — |
| 84 | #11758196 | — | 4 | 16 | no | — |
| 85 | C23×C61 | — | 2 | 4 | no | — |
| 86 | C23×C61 | — | 2 | 4 | no | — |
| 87 | C23×C61 | — | 2 | 4 | no | — |
| 88 | C23×C61 | — | 2 | 8 | no | — |
| 89 | C23×C61 | — | 2 | 8 | no | — |
| 90 | C23×C61 | — | 4 | 4 | no | — |
| 91 | C23×C61 | — | 4 | 4 | no | — |
| 92 | C23×C61 | — | 4 | 4 | no | — |
| 93 | C23×C61 | — | 4 | 4 | no | — |
| 94 | C23×C61 | — | 4 | 4 | no | — |
| 95 | C23×C61 | — | 4 | 4 | no | — |
| 96 | C23×C61 | — | 4 | 2 | no | — |
| 97 | C23×C61 | — | 4 | 2 | no | — |
| 98 | C23×C61 | — | 4 | 4 | no | — |
| 99 | C23×C61 | — | 4 | 4 | no | — |
| 100 | C23×C61 | — | 4 | 2 | no | — |
| 101 | C23×C61 | — | 4 | 4 | no | — |
| 102 | C23×C61 | — | 4 | 2 | no | — |
| 103 | C23×C61 | — | 4 | 4 | no | — |
| 104 | C23×C61 | — | 4 | 4 | no | — |
| 105 | C23×C61 | — | 4 | 4 | no | — |
| 106 | C23×C61 | C22×Q8 | 1 | 1 | yes | characteristic |
| 107 | C23×C61 | C2×#41 | 1 | 4 | yes | — |
| 108 | C23×C61 | H(5,𝔽2) | 1 | 4 | yes | — |
| 109 | C23×C61 | H(5,𝔽2) | 1 | 4 | yes | — |
| 110 | C23×C61 | H(5,𝔽2) | 1 | 2 | yes | — |
| 111 | D4×C61 | — | 4 | 16 | no | — |
| 112 | C2×C4×C61 | — | 2 | 2 | no | — |
| 113 | C2×C4×C61 | — | 2 | 4 | no | — |
| 114 | C2×C4×C61 | — | 4 | 4 | no | — |
| 115 | C2×C4×C61 | — | 4 | 4 | no | — |
| 116 | C2×C4×C61 | — | 4 | 4 | no | — |
| 117 | C2×C4×C61 | — | 4 | 4 | no | — |
| 118 | C2×C4×C61 | — | 4 | 4 | no | — |
| 119 | C2×C4×C61 | — | 4 | 4 | no | — |
| 120 | C2×C4×C61 | — | 4 | 4 | no | — |
| 121 | C2×C4×C61 | — | 4 | 4 | no | — |
| 122 | C2×Dic61 | — | 2 | 8 | no | — |
| 123 | C2×Dic61 | — | 2 | 8 | no | — |
| 124 | C2×Dic61 | — | 2 | 8 | no | — |
| 125 | C2×Dic61 | — | 4 | 4 | no | — |
| 126 | C2×Dic61 | — | 4 | 4 | no | — |
| 127 | C2×Dic61 | — | 4 | 4 | no | — |
| 128 | C2×Dic61 | — | 4 | 4 | no | — |
| 129 | C2×Dic61 | — | 4 | 4 | no | — |
| 130 | C2×Dic61 | — | 4 | 4 | no | — |
| 131 | C2×Dic61 | — | 4 | 4 | no | — |
| 132 | C2×Dic61 | — | 4 | 4 | no | — |
| 133 | C2×Dic61 | — | 4 | 4 | no | — |
| 134 | C2×Dic61 | — | 4 | 4 | no | — |
| 135 | C2×Dic61 | — | 4 | 4 | no | — |
| 136 | #17271 | — | 61 | 1 | no | maximal, Sylow |
| 137 | C22×C61 | — | 4 | 4 | no | — |
| 138 | C22×C61 | — | 4 | 4 | no | — |
| 139 | C22×C61 | — | 4 | 4 | no | — |
| 140 | C22×C61 | — | 4 | 8 | no | — |
| 141 | C22×C61 | — | 4 | 8 | no | — |
| 142 | C22×C61 | — | 8 | 4 | no | — |
| 143 | C22×C61 | — | 8 | 4 | no | — |
| 144 | C22×C61 | — | 8 | 4 | no | — |
| 145 | C22×C61 | — | 8 | 4 | no | — |
| 146 | C22×C61 | #538 | 1 | 4 | yes | — |
| 147 | C22×C61 | #540 | 1 | 4 | yes | — |
| 148 | C22×C61 | #540 | 1 | 4 | yes | — |
| 149 | C22×C61 | #543 | 1 | 1 | yes | characteristic |
| 150 | C22×C61 | #543 | 1 | 2 | yes | — |
| 151 | C22×C61 | #551 | 1 | 2 | yes | — |
| 152 | C22×C61 | #553 | 1 | 4 | yes | — |
| 153 | C22×C61 | #554 | 1 | 4 | yes | — |
| 154 | C22×C61 | #560 | 1 | 2 | yes | — |
| 155 | C22×C61 | #560 | 1 | 4 | yes | — |
| 156 | C22×C61 | H(3,𝔽4) | 1 | 4 | yes | — |
| 157 | C4×C61 | — | 4 | 2 | no | — |
| 158 | C4×C61 | — | 8 | 4 | no | — |
| 159 | Dic61 | — | 4 | 8 | no | — |
| 160 | Dic61 | — | 8 | 4 | no | — |
| 161 | #1442 | — | 61 | 4 | no | — |
| 162 | #1439 | — | 61 | 4 | no | — |
| 163 | #1439 | — | 61 | 4 | no | — |
| 164 | #1438 | — | 61 | 2 | no | — |
| 165 | #1438 | D61 | 1 | 1 | yes | characteristic |
| 166 | C2×C61 | — | 8 | 4 | no | — |
| 167 | C2×C61 | #2748 | 1 | 2 | yes | — |
| 168 | C2×C61 | #2753 | 1 | 4 | yes | — |
| 169 | C2×C61 | #2780 | 1 | 4 | yes | — |
| 170 | C2×C61 | #2828 | 1 | 4 | yes | — |
| 171 | C2×C61 | #2840 | 1 | 1 | yes | characteristic |
| 172 | C23×D4 | C2×D61 | 1 | 1 | yes | characteristic |
| 173 | C22×C4⋊C4 | — | 61 | 4 | no | — |
| 174 | C22×C22⋊C4 | — | 61 | 2 | no | — |
| 175 | C22×C22⋊C4 | — | 61 | 4 | no | — |
| 176 | C22×C22⋊C4 | — | 61 | 2 | no | — |
| 177 | C22×C22⋊C4 | C2×D61 | 1 | 4 | yes | — |
| 178 | C2×#95 | — | 61 | 4 | no | — |
| 179 | C2×#95 | — | 61 | 4 | no | — |
| 180 | C2×#95 | — | 61 | 4 | no | — |
| 181 | C2×#95 | — | 61 | 4 | no | — |
| 182 | C2×#95 | C2×D61 | 1 | 2 | yes | — |
| 183 | C61 | #17271 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C25 | C22×D61 | 1 | 2 | yes | — |
| 185 | C22×D4 | — | 4 | 2 | no | — |
| 186 | C22×D4 | — | 4 | 4 | no | — |
| 187 | C22×D4 | Dic122 | 1 | 4 | yes | — |
| 188 | C23×C4 | — | 61 | 4 | no | — |
| 189 | C23×C4 | — | 61 | 4 | no | — |
| 190 | C23×C4 | C22×D61 | 1 | 4 | yes | — |
| 191 | C23×C4 | C22×D61 | 1 | 1 | yes | characteristic |
| 192 | C2×C4⋊C4 | — | 244 | 4 | no | — |
| 193 | C2×C4⋊C4 | — | 244 | 4 | no | — |
| 194 | C2×C4⋊C4 | — | 244 | 4 | no | — |
| 195 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 196 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 197 | C2×C22⋊C4 | — | 4 | 4 | no | — |
| 198 | C2×C22⋊C4 | — | 122 | 8 | no | — |
| 199 | C2×C22⋊C4 | — | 122 | 8 | no | — |
| 200 | C2×C22⋊C4 | — | 244 | 2 | no | — |
| 201 | C2×C22⋊C4 | — | 244 | 4 | no | — |
| 202 | C2×C22⋊C4 | — | 244 | 4 | no | — |
| 203 | C2×C22⋊C4 | — | 244 | 4 | no | — |
| 204 | C2×C22⋊C4 | — | 244 | 2 | no | — |
| 205 | #95 | — | 4 | 2 | no | — |
| 206 | #95 | — | 122 | 8 | no | — |
| 207 | #95 | — | 244 | 4 | no | — |
| 208 | #95 | — | 244 | 4 | no | — |
| 209 | #95 | — | 244 | 4 | no | — |
| 210 | C24 | — | 2 | 4 | no | — |
| 211 | C24 | — | 2 | 4 | no | — |
| 212 | C24 | — | 2 | 4 | no | — |
| 213 | C24 | — | 2 | 4 | no | — |
| 214 | C24 | — | 2 | 4 | no | — |
| 215 | C24 | — | 2 | 4 | no | — |
| 216 | C24 | — | 2 | 2 | no | — |
| 217 | C24 | — | 2 | 2 | no | — |
| 218 | C24 | C2×Dic122 | 1 | 4 | yes | — |
| 219 | C24 | C23×D61 | 1 | 1 | yes | characteristic |
| 220 | C2×D4 | — | 4 | 8 | no | — |
| 221 | C2×D4 | — | 4 | 4 | no | — |
| 222 | C2×D4 | — | 4 | 16 | no | — |
| 223 | C22×C4 | — | 2 | 2 | no | — |
| 224 | C22×C4 | — | 2 | 4 | no | — |
| 225 | C22×C4 | — | 2 | 4 | no | — |
| 226 | C22×C4 | — | 2 | 4 | no | — |
| 227 | C22×C4 | — | 2 | 4 | no | — |
| 228 | C22×C4 | — | 2 | 4 | no | — |
| 229 | C22×C4 | — | 2 | 4 | no | — |
| 230 | C22×C4 | — | 2 | 4 | no | — |
| 231 | C22×C4 | — | 2 | 4 | no | — |
| 232 | C22×C4 | — | 61 | 8 | no | — |
| 233 | C22×C4 | — | 122 | 4 | no | — |
| 234 | C22×C4 | — | 122 | 4 | no | — |
| 235 | C22×C4 | — | 122 | 4 | no | — |
| 236 | C22×C4 | — | 122 | 4 | no | — |
| 237 | C22×C4 | — | 122 | 4 | no | — |
| 238 | C22×C4 | — | 122 | 4 | no | — |
| 239 | C22×C4 | — | 122 | 4 | no | — |
| 240 | C22×C4 | — | 122 | 4 | no | — |
| 241 | C22×C4 | — | 122 | 4 | no | — |
| 242 | C22×C4 | — | 122 | 4 | no | — |
| 243 | C22×C4 | — | 122 | 4 | no | — |
| 244 | C22×C4 | — | 122 | 4 | no | — |
| 245 | C22×C4 | — | 122 | 4 | no | — |
| 246 | C22×C4 | C2×Dic122 | 1 | 2 | yes | — |
| 247 | C4⋊C4 | — | 244 | 16 | no | — |
| 248 | C22⋊C4 | — | 4 | 16 | no | — |
| 249 | C22⋊C4 | — | 244 | 8 | no | — |
| 250 | C22⋊C4 | — | 244 | 16 | no | — |
| 251 | C22⋊C4 | — | 244 | 8 | no | — |
| 252 | C23 | — | 2 | 8 | no | — |
| 253 | C23 | — | 2 | 8 | no | — |
| 254 | C23 | — | 2 | 4 | no | — |
| 255 | C23 | — | 2 | 4 | no | — |
| 256 | C23 | — | 2 | 4 | no | — |
| 257 | C23 | — | 4 | 4 | no | — |
| 258 | C23 | — | 4 | 4 | no | — |
| 259 | C23 | — | 4 | 2 | no | — |
| 260 | C23 | — | 4 | 2 | no | — |
| 261 | C23 | — | 4 | 4 | no | — |
| 262 | C23 | — | 4 | 4 | no | — |
| 263 | C23 | — | 4 | 4 | no | — |
| 264 | C23 | — | 4 | 4 | no | — |
| 265 | C23 | — | 4 | 4 | no | — |
| 266 | C23 | — | 4 | 4 | no | — |
| 267 | C23 | — | 4 | 4 | no | — |
| 268 | C23 | — | 4 | 4 | no | — |
| 269 | C23 | — | 4 | 4 | no | — |
| 270 | C23 | — | 4 | 4 | no | — |
| 271 | C23 | — | 4 | 2 | no | — |
| 272 | C23 | — | 4 | 2 | no | — |
| 273 | C23 | C22×Dic122 | 1 | 1 | yes | characteristic |
| 274 | C23 | #41×D61 | 1 | 4 | yes | — |
| 275 | C23 | #14519822 | 1 | 4 | yes | — |
| 276 | C23 | #14519822 | 1 | 4 | yes | — |
| 277 | C23 | #14519822 | 1 | 2 | yes | — |
| 278 | D4 | — | 4 | 16 | no | — |
| 279 | C2×C4 | — | 2 | 4 | no | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 4 | 4 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 4 | 4 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 4 | 4 | no | — |
| 287 | C2×C4 | — | 4 | 4 | no | — |
| 288 | C2×C4 | — | 4 | 4 | no | — |
| 289 | C2×C4 | — | 122 | 8 | no | — |
| 290 | C2×C4 | — | 122 | 8 | no | — |
| 291 | C2×C4 | — | 122 | 8 | no | — |
| 292 | C2×C4 | — | 244 | 4 | no | — |
| 293 | C2×C4 | — | 244 | 4 | no | — |
| 294 | C2×C4 | — | 244 | 4 | no | — |
| 295 | C2×C4 | — | 244 | 4 | no | — |
| 296 | C2×C4 | — | 244 | 4 | no | — |
| 297 | C2×C4 | — | 244 | 4 | no | — |
| 298 | C2×C4 | — | 244 | 4 | no | — |
| 299 | C2×C4 | — | 244 | 4 | no | — |
| 300 | C2×C4 | — | 244 | 4 | no | — |
| 301 | C2×C4 | — | 244 | 4 | no | — |
| 302 | C2×C4 | — | 244 | 4 | no | — |
| 303 | C22 | — | 4 | 8 | no | — |
| 304 | C22 | — | 4 | 8 | no | — |
| 305 | C22 | — | 4 | 4 | no | — |
| 306 | C22 | — | 4 | 4 | no | — |
| 307 | C22 | — | 4 | 4 | no | — |
| 308 | C22 | — | 8 | 4 | no | — |
| 309 | C22 | — | 8 | 4 | no | — |
| 310 | C22 | — | 8 | 4 | no | — |
| 311 | C22 | — | 8 | 4 | no | — |
| 312 | C22 | #16840344 | 1 | 4 | yes | — |
| 313 | C22 | #16840362 | 1 | 4 | yes | — |
| 314 | C22 | #16840362 | 1 | 4 | yes | — |
| 315 | C22 | #16840388 | 1 | 2 | yes | — |
| 316 | C22 | #16840388 | 1 | 1 | yes | characteristic |
| 317 | C22 | #16840442 | 1 | 2 | yes | — |
| 318 | C22 | #16840457 | 1 | 4 | yes | — |
| 319 | C22 | #16840471 | 1 | 4 | yes | — |
| 320 | C22 | #16840499 | 1 | 2 | yes | — |
| 321 | C22 | #16840501 | 1 | 4 | yes | — |
| 322 | C22 | #16840505 | 1 | 4 | yes | — |
| 323 | C4 | — | 4 | 2 | no | — |
| 324 | C4 | — | 8 | 4 | no | — |
| 325 | C4 | — | 488 | 4 | no | — |
| 326 | C4 | — | 244 | 8 | no | — |
| 327 | C2 | — | 8 | 4 | no | — |
| 328 | C2 | #21453032 | 1 | 2 | yes | — |
| 329 | C2 | #21453095 | 1 | 4 | yes | — |
| 330 | C2 | #21453464 | 1 | 4 | yes | — |
| 331 | C2 | #21453937 | 1 | 4 | yes | — |
| 332 | C2 | #21454072 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D61 |
| commutator, socle, cosocle | C24×C61 | C24 |
| Fitting | C61×#1438 | C2 |
| prime radical | C23×D4×C61 | C22 |
| hypercenter | #1438 | D61 |
| hypocenter | C61 | #17271 |
| LCS | G ▷ C24×C61 ▷ C61 | |
| UCS | C1 ◁ C24 ◁ #1438 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21444077 | C2 | semidirect |
| 2. #21444062 | C2 | semidirect |
| 3. #21444062 | C2 | semidirect |
| 183. C61 | #17271 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21444077 | 2 |
| #21444062 | 2 |
| #21444060 | 2 |
| C61×#1438 | 2 |
| #17271 | 61 |