Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
3694 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#65930 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C5×#2656 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10614411 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #10609788 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #10609750 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #10608437 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10608398 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #10605112 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#64937 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C22×#711 | C22 | 1 | 4 | yes | — |
| 11 | C2×C5×#121 | C22 | 1 | 6 | yes | — |
| 12 | C2×#4086 | C22 | 1 | 6 | yes | — |
| 13 | C2×#4085 | C22 | 1 | 12 | yes | — |
| 14 | C2×D4×Dic5 | C22 | 1 | 6 | yes | — |
| 15 | C2×#4076 | C22 | 1 | 6 | yes | — |
| 16 | C22×C4×Dic5 | C22 | 1 | 3 | yes | — |
| 17 | #65930 | — | 2 | 48 | no | — |
| 18 | C2×#4039 | C22 | 1 | 6 | yes | — |
| 19 | C2×#4034 | C22 | 1 | 6 | yes | — |
| 20 | C2×#4031 | C22 | 1 | 12 | yes | — |
| 21 | C2×#4027 | C22 | 1 | 12 | yes | — |
| 22 | C5×#399 | C22 | 1 | 4 | yes | — |
| 23 | C5×#392 | C22 | 1 | 4 | yes | — |
| 24 | C2×C5×#95 | C22 | 1 | 1 | yes | characteristic |
| 25 | #65512 | C22 | 1 | 12 | yes | — |
| 26 | C2×#3963 | C22 | 1 | 3 | yes | — |
| 27 | #65281 | C22 | 1 | 12 | yes | — |
| 28 | #65249 | C22 | 1 | 12 | yes | — |
| 29 | #65238 | C22 | 1 | 12 | yes | — |
| 30 | #64966 | C22 | 1 | 4 | yes | — |
| 31 | #64960 | C22 | 1 | 4 | yes | — |
| 32 | #64937 | C22 | 1 | 4 | yes | — |
| 33 | #64937 | C22 | 1 | 4 | yes | — |
| 34 | #34641 | — | 5 | 1 | no | maximal, Sylow |
| 35 | C22×C5×D4 | C23 | 1 | 4 | yes | — |
| 36 | C22×C5×D4 | C23 | 1 | 6 | yes | — |
| 37 | C23×C4×C5 | C23 | 1 | 3 | yes | — |
| 38 | C22×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 39 | C22×C22⋊D5 | C23 | 1 | 4 | yes | — |
| 40 | C22×C22⋊D5 | C23 | 1 | 12 | yes | — |
| 41 | C23×Dic5 | C23 | 1 | 4 | yes | — |
| 42 | C23×Dic5 | C23 | 1 | 3 | yes | — |
| 43 | C23×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×#711 | — | 2 | 48 | no | — |
| 45 | C22×D20 | C23 | 1 | 6 | yes | — |
| 46 | C22×C4×D5 | C23 | 1 | 4 | yes | — |
| 47 | C22×Dic10 | C23 | 1 | 4 | yes | — |
| 48 | C5×#121 | — | 2 | 24 | no | — |
| 49 | C2×C5×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 50 | C2×C5×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 51 | C2×C22⋊Dic5 | C23 | 1 | 12 | yes | — |
| 52 | #4086 | — | 2 | 24 | no | — |
| 53 | #4085 | — | 2 | 48 | no | — |
| 54 | D4×Dic5 | — | 2 | 24 | no | — |
| 55 | #4076 | — | 2 | 24 | no | — |
| 56 | C2×#685 | C23 | 1 | 12 | yes | — |
| 57 | C2×#685 | C23 | 1 | 12 | yes | — |
| 58 | C2×C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 59 | C2×#683 | C23 | 1 | 12 | yes | — |
| 60 | C2×C4×Dic5 | D4 | 1 | 24 | yes | complemented |
| 61 | C2×C4×Dic5 | C23 | 1 | 6 | yes | — |
| 62 | C2×C4×Dic5 | C23 | 1 | 6 | yes | — |
| 63 | #4039 | — | 2 | 24 | no | — |
| 64 | #4034 | — | 2 | 24 | no | — |
| 65 | #4031 | — | 2 | 48 | no | — |
| 66 | #4027 | — | 2 | 48 | no | — |
| 67 | C5×#95 | C23 | 1 | 4 | yes | — |
| 68 | #3963 | C23 | 1 | 12 | yes | — |
| 69 | C2×#532 | — | 5 | 6 | no | — |
| 70 | #2658 | — | 5 | 4 | no | — |
| 71 | #2657 | — | 5 | 4 | no | — |
| 72 | #2656 | — | 5 | 1 | no | — |
| 73 | #2656 | — | 5 | 6 | no | — |
| 74 | #2656 | D5 | 1 | 1 | yes | characteristic, complemented |
| 75 | #2397 | — | 5 | 4 | no | — |
| 76 | #2397 | — | 5 | 4 | no | — |
| 77 | C2×#376 | — | 5 | 1 | no | — |
| 78 | C24×C5 | C24 | 1 | 4 | yes | — |
| 79 | C24×C5 | C24 | 1 | 1 | yes | characteristic |
| 80 | C23×D5 | C24 | 1 | 4 | yes | — |
| 81 | C2×C5×D4 | — | 2 | 24 | no | — |
| 82 | C2×C5×D4 | — | 2 | 12 | no | — |
| 83 | C2×C5×D4 | — | 4 | 12 | no | — |
| 84 | C22×C4×C5 | — | 4 | 6 | no | — |
| 85 | C22×C4×C5 | C2×D4 | 1 | 12 | yes | — |
| 86 | C22×C4×C5 | C24 | 1 | 4 | yes | — |
| 87 | C22×C4×C5 | C24 | 1 | 6 | yes | — |
| 88 | C2×C22⋊D5 | — | 2 | 24 | no | — |
| 89 | C2×C22⋊D5 | — | 2 | 24 | no | — |
| 90 | C2×C22⋊D5 | — | 2 | 24 | no | — |
| 91 | C2×C22⋊D5 | — | 4 | 12 | no | — |
| 92 | C2×C22⋊D5 | — | 4 | 12 | no | — |
| 93 | C22×Dic5 | — | 2 | 24 | no | — |
| 94 | C22×Dic5 | — | 4 | 3 | no | — |
| 95 | C22×Dic5 | — | 4 | 3 | no | — |
| 96 | C22×Dic5 | C2×D4 | 1 | 12 | yes | — |
| 97 | C22×Dic5 | C2×D4 | 1 | 12 | yes | — |
| 98 | C22×Dic5 | C24 | 1 | 4 | yes | — |
| 99 | C22×Dic5 | C24 | 1 | 6 | yes | — |
| 100 | C22×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 101 | C22×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 102 | #711 | — | 4 | 64 | no | — |
| 103 | C2×D20 | — | 2 | 12 | no | — |
| 104 | C2×D20 | — | 4 | 12 | no | — |
| 105 | C2×C4×D5 | — | 2 | 24 | no | — |
| 106 | C2×Dic10 | — | 2 | 24 | no | — |
| 107 | C5×C4⋊C4 | — | 2 | 12 | no | — |
| 108 | C5×C22⋊C4 | — | 2 | 24 | no | — |
| 109 | C22⋊Dic5 | — | 2 | 24 | no | — |
| 110 | #685 | — | 2 | 24 | no | — |
| 111 | #685 | — | 2 | 24 | no | — |
| 112 | C4⋊Dic5 | — | 2 | 12 | no | — |
| 113 | #683 | — | 2 | 24 | no | — |
| 114 | C4×Dic5 | C2×D4 | 1 | 24 | yes | — |
| 115 | C4×Dic5 | C2×D4 | 1 | 24 | yes | — |
| 116 | C22×#41 | — | 5 | 4 | no | — |
| 117 | #532 | — | 10 | 48 | no | — |
| 118 | C2×C4⋊Q8 | — | 5 | 6 | no | — |
| 119 | C2×#127 | — | 5 | 6 | no | — |
| 120 | C2×#124 | — | 5 | 12 | no | — |
| 121 | C2×#121 | — | 5 | 12 | no | — |
| 122 | C2×#121 | — | 5 | 6 | no | — |
| 123 | C2×#121 | C2×D5 | 1 | 6 | yes | — |
| 124 | C2×C4×D4 | — | 5 | 6 | no | — |
| 125 | C2×C4×D4 | — | 5 | 6 | no | — |
| 126 | C2×C4×D4 | — | 5 | 12 | no | — |
| 127 | C22×C42 | — | 5 | 3 | no | — |
| 128 | #399 | — | 5 | 12 | no | — |
| 129 | #399 | — | 5 | 12 | no | — |
| 130 | #399 | — | 5 | 4 | no | — |
| 131 | #399 | C2×D5 | 1 | 4 | yes | — |
| 132 | #392 | — | 5 | 12 | no | — |
| 133 | #392 | — | 5 | 12 | no | — |
| 134 | #392 | — | 5 | 4 | no | — |
| 135 | #392 | C2×D5 | 1 | 4 | yes | — |
| 136 | #376 | — | 5 | 4 | no | — |
| 137 | #376 | — | 5 | 4 | no | — |
| 138 | C2×#95 | — | 5 | 3 | no | — |
| 139 | C2×#95 | C2×D5 | 1 | 1 | yes | characteristic |
| 140 | C23×C5 | — | 2 | 1 | no | — |
| 141 | C23×C5 | — | 2 | 3 | no | — |
| 142 | C23×C5 | — | 2 | 12 | no | — |
| 143 | C23×C5 | — | 2 | 4 | no | — |
| 144 | C23×C5 | — | 2 | 12 | no | — |
| 145 | C23×C5 | C22×D4 | 1 | 6 | yes | — |
| 146 | C23×C5 | C25 | 1 | 1 | yes | characteristic |
| 147 | C22×D5 | — | 2 | 12 | no | — |
| 148 | C22×D5 | — | 2 | 12 | no | — |
| 149 | C22×D5 | — | 2 | 4 | no | — |
| 150 | C5×D4 | — | 4 | 24 | no | — |
| 151 | C5×D4 | — | 4 | 16 | no | — |
| 152 | C2×C4×C5 | — | 2 | 12 | no | — |
| 153 | C2×C4×C5 | — | 2 | 12 | no | — |
| 154 | C2×C4×C5 | — | 4 | 12 | no | — |
| 155 | C2×C4×C5 | C22×D4 | 1 | 12 | yes | — |
| 156 | C22⋊D5 | — | 4 | 48 | no | — |
| 157 | C22⋊D5 | — | 4 | 16 | no | — |
| 158 | C22⋊D5 | — | 4 | 16 | no | — |
| 159 | C2×Dic5 | — | 2 | 6 | no | — |
| 160 | C2×Dic5 | — | 2 | 6 | no | — |
| 161 | C2×Dic5 | — | 2 | 12 | no | — |
| 162 | C2×Dic5 | — | 4 | 12 | no | — |
| 163 | C2×Dic5 | — | 4 | 16 | no | — |
| 164 | C2×Dic5 | #127 | 1 | 16 | yes | — |
| 165 | C2×Dic5 | C22×D4 | 1 | 12 | yes | — |
| 166 | C2×Dic5 | C22×D4 | 1 | 6 | yes | — |
| 167 | C2×Dic5 | C22×D4 | 1 | 6 | yes | — |
| 168 | D20 | — | 4 | 24 | no | — |
| 169 | C4×D5 | — | 4 | 16 | no | — |
| 170 | Dic10 | — | 4 | 16 | no | — |
| 171 | C2×#41 | — | 10 | 48 | no | — |
| 172 | C22×Q8 | — | 5 | 4 | no | — |
| 173 | C22×D4 | — | 5 | 4 | no | — |
| 174 | C22×D4 | — | 5 | 4 | no | — |
| 175 | C22×D4 | — | 5 | 12 | no | — |
| 176 | C22×D4 | — | 5 | 6 | no | — |
| 177 | C22×D4 | C22×D5 | 1 | 4 | yes | — |
| 178 | C22×D4 | C22×D5 | 1 | 6 | yes | — |
| 179 | C23×C4 | — | 5 | 1 | no | — |
| 180 | C23×C4 | — | 5 | 3 | no | — |
| 181 | C23×C4 | — | 5 | 4 | no | — |
| 182 | C23×C4 | — | 5 | 4 | no | — |
| 183 | C23×C4 | C22×D5 | 1 | 3 | yes | — |
| 184 | C4⋊Q8 | — | 10 | 24 | no | — |
| 185 | #127 | — | 10 | 24 | no | — |
| 186 | #124 | — | 10 | 48 | no | — |
| 187 | #121 | — | 2 | 24 | no | — |
| 188 | #121 | — | 10 | 48 | no | — |
| 189 | #121 | — | 10 | 24 | no | — |
| 190 | C4×D4 | — | 10 | 48 | no | — |
| 191 | C4×D4 | — | 10 | 24 | no | — |
| 192 | C4×D4 | — | 10 | 24 | no | — |
| 193 | C2×C4⋊C4 | — | 5 | 12 | no | — |
| 194 | C2×C4⋊C4 | — | 5 | 6 | no | — |
| 195 | C2×C4⋊C4 | C22×D5 | 1 | 6 | yes | — |
| 196 | C2×C22⋊C4 | — | 5 | 12 | no | — |
| 197 | C2×C22⋊C4 | — | 5 | 12 | no | — |
| 198 | C2×C22⋊C4 | — | 5 | 12 | no | — |
| 199 | C2×C22⋊C4 | C22×D5 | 1 | 12 | yes | — |
| 200 | C2×C42 | — | 5 | 24 | no | — |
| 201 | C2×C42 | — | 5 | 6 | no | — |
| 202 | C2×C42 | — | 5 | 6 | no | — |
| 203 | #95 | — | 5 | 12 | no | — |
| 204 | #95 | C22×D5 | 1 | 4 | yes | — |
| 205 | C22×C5 | — | 2 | 6 | no | — |
| 206 | C22×C5 | — | 2 | 6 | no | — |
| 207 | C22×C5 | — | 4 | 4 | no | — |
| 208 | C22×C5 | — | 4 | 12 | no | — |
| 209 | C22×C5 | — | 4 | 12 | no | — |
| 210 | C22×C5 | C2×#127 | 1 | 4 | yes | — |
| 211 | C22×C5 | C23×D4 | 1 | 3 | yes | — |
| 212 | C22×C5 | #585 | 1 | 3 | yes | — |
| 213 | C22×C5 | #585 | 1 | 1 | yes | characteristic |
| 214 | C2×D5 | — | 4 | 4 | no | — |
| 215 | C2×D5 | — | 4 | 12 | no | — |
| 216 | C2×D5 | — | 4 | 12 | no | — |
| 217 | C4×C5 | — | 2 | 12 | no | — |
| 218 | C4×C5 | — | 4 | 4 | no | — |
| 219 | Dic5 | — | 2 | 12 | no | — |
| 220 | Dic5 | — | 4 | 4 | no | — |
| 221 | Dic5 | C2×#127 | 1 | 4 | yes | — |
| 222 | Dic5 | C2×#127 | 1 | 4 | yes | — |
| 223 | C24 | — | 5 | 4 | no | — |
| 224 | C24 | C23×D5 | 1 | 4 | yes | — |
| 225 | C24 | C23×D5 | 1 | 1 | yes | characteristic |
| 226 | #41 | — | 20 | 64 | no | — |
| 227 | C2×Q8 | — | 10 | 24 | no | — |
| 228 | C2×D4 | — | 2 | 24 | no | — |
| 229 | C2×D4 | — | 2 | 12 | no | — |
| 230 | C2×D4 | — | 4 | 12 | no | — |
| 231 | C2×D4 | — | 10 | 24 | no | — |
| 232 | C2×D4 | — | 10 | 24 | no | — |
| 233 | C2×D4 | — | 10 | 24 | no | — |
| 234 | C2×D4 | — | 10 | 12 | no | — |
| 235 | C2×D4 | — | 20 | 12 | no | — |
| 236 | C2×D4 | — | 20 | 12 | no | — |
| 237 | C2×D4 | — | 20 | 12 | no | — |
| 238 | C22×C4 | — | 4 | 6 | no | — |
| 239 | C22×C4 | — | 5 | 12 | no | — |
| 240 | C22×C4 | — | 5 | 1 | no | — |
| 241 | C22×C4 | — | 5 | 1 | no | — |
| 242 | C22×C4 | — | 5 | 6 | no | — |
| 243 | C22×C4 | — | 5 | 12 | no | — |
| 244 | C22×C4 | — | 5 | 4 | no | — |
| 245 | C22×C4 | — | 10 | 24 | no | — |
| 246 | C22×C4 | — | 10 | 24 | no | — |
| 247 | C22×C4 | — | 20 | 3 | no | — |
| 248 | C22×C4 | — | 20 | 3 | no | — |
| 249 | C22×C4 | D4×D5 | 1 | 12 | yes | — |
| 250 | C22×C4 | C23×D5 | 1 | 4 | yes | — |
| 251 | C22×C4 | C23×D5 | 1 | 6 | yes | — |
| 252 | C4⋊C4 | — | 2 | 12 | no | — |
| 253 | C4⋊C4 | — | 10 | 24 | no | — |
| 254 | C4⋊C4 | — | 10 | 12 | no | — |
| 255 | C22⋊C4 | — | 2 | 24 | no | — |
| 256 | C22⋊C4 | — | 10 | 24 | no | — |
| 257 | C22⋊C4 | — | 10 | 24 | no | — |
| 258 | C22⋊C4 | — | 10 | 24 | no | — |
| 259 | C42 | — | 5 | 24 | no | — |
| 260 | C42 | — | 5 | 24 | no | — |
| 261 | C2×C5 | — | 2 | 4 | no | — |
| 262 | C2×C5 | — | 8 | 4 | no | — |
| 263 | C2×C5 | C22×#127 | 1 | 1 | yes | characteristic |
| 264 | C2×C5 | #3449 | 1 | 3 | yes | — |
| 265 | C2×C5 | #3449 | 1 | 3 | yes | — |
| 266 | D5 | — | 8 | 4 | no | — |
| 267 | C23 | — | 2 | 3 | no | — |
| 268 | C23 | — | 2 | 1 | no | — |
| 269 | C23 | — | 2 | 12 | no | — |
| 270 | C23 | — | 2 | 12 | no | — |
| 271 | C23 | — | 2 | 4 | no | — |
| 272 | C23 | — | 10 | 12 | no | — |
| 273 | C23 | — | 10 | 12 | no | — |
| 274 | C23 | — | 10 | 4 | no | — |
| 275 | C23 | C2×D4×D5 | 1 | 6 | yes | — |
| 276 | C23 | C24×D5 | 1 | 1 | yes | characteristic |
| 277 | Q8 | — | 20 | 16 | no | — |
| 278 | D4 | — | 4 | 24 | no | — |
| 279 | D4 | — | 4 | 16 | no | — |
| 280 | D4 | — | 20 | 24 | no | — |
| 281 | D4 | — | 20 | 16 | no | — |
| 282 | D4 | — | 20 | 48 | no | — |
| 283 | D4 | — | 20 | 16 | no | — |
| 284 | C2×C4 | — | 2 | 12 | no | — |
| 285 | C2×C4 | — | 2 | 12 | no | — |
| 286 | C2×C4 | — | 4 | 12 | no | — |
| 287 | C2×C4 | — | 5 | 16 | no | — |
| 288 | C2×C4 | — | 5 | 6 | no | — |
| 289 | C2×C4 | — | 5 | 6 | no | — |
| 290 | C2×C4 | — | 5 | 12 | no | — |
| 291 | C2×C4 | — | 10 | 6 | no | — |
| 292 | C2×C4 | — | 10 | 6 | no | — |
| 293 | C2×C4 | — | 10 | 12 | no | — |
| 294 | C2×C4 | — | 20 | 16 | no | — |
| 295 | C2×C4 | — | 20 | 16 | no | — |
| 296 | C2×C4 | — | 20 | 12 | no | — |
| 297 | C2×C4 | C2×D4×D5 | 1 | 12 | yes | — |
| 298 | C5 | #34641 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 299 | C22 | — | 2 | 6 | no | — |
| 300 | C22 | — | 2 | 6 | no | — |
| 301 | C22 | — | 4 | 12 | no | — |
| 302 | C22 | — | 4 | 12 | no | — |
| 303 | C22 | — | 4 | 4 | no | — |
| 304 | C22 | — | 20 | 12 | no | — |
| 305 | C22 | — | 20 | 12 | no | — |
| 306 | C22 | — | 20 | 4 | no | — |
| 307 | C22 | D5×#127 | 1 | 4 | yes | — |
| 308 | C22 | C22×D4×D5 | 1 | 3 | yes | — |
| 309 | C22 | #66285 | 1 | 3 | yes | — |
| 310 | C22 | #66287 | 1 | 1 | yes | characteristic |
| 311 | C4 | — | 2 | 12 | no | — |
| 312 | C4 | — | 4 | 4 | no | — |
| 313 | C4 | — | 5 | 4 | no | — |
| 314 | C4 | — | 5 | 4 | no | — |
| 315 | C4 | — | 10 | 12 | no | — |
| 316 | C4 | — | 20 | 4 | no | — |
| 317 | C2 | — | 2 | 4 | no | — |
| 318 | C2 | — | 8 | 4 | no | — |
| 319 | C2 | — | 40 | 4 | no | — |
| 320 | C2 | C2×D5×#127 | 1 | 1 | yes | characteristic |
| 321 | C2 | #10618195 | 1 | 3 | yes | — |
| 322 | C2 | #10618264 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D5 |
| commutator, socle, cosocle | C23×C5 | C25 |
| Fitting | C5×#2656 | C2 |
| hypercenter | #2656 | D5 |
| hypocenter | C5 | #34641 |
| LCS | G ▷ C23×C5 ▷ C5 | |
| UCS | C1 ◁ C23 ◁ #2656 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#65930 | C2 | semidirect |
| 2. C5×#2656 | C2 | semidirect |
| 3. #10614411 | C2 | semidirect |
| 4. #10609788 | C2 | semidirect |
| 5. #10609750 | C2 | semidirect |
| 6. #10608437 | C2 | semidirect |
| 7. #10608398 | C2 | semidirect |
| 8. #10605112 | C2 | semidirect |
| 9. C2×#64937 | C2 | semidirect |
| 60. C2×C4×Dic5 | D4 | semidirect |
| 74. #2656 | D5 | semidirect |
| 298. C5 | #34641 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#65930 | 2 |
| C5×#2656 | 2 |
| #10614411 | 2 |
| #10609788 | 2 |
| #10609750 | 2 |
| #10608437 | 2 |
| #10608398 | 2 |
| #10605112 | 2 |
| C2×#64937 | 2 |
| #34641 | 5 |