Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1184 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 167 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 | C5×#3483 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10615053 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #10602735 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10602723 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #10602613 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #66283 | C22 | 1 | 1 | yes | characteristic |
| 7 | C5×Q82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C5×#555 | — | 2 | 3 | no | — |
| 9 | C5×#555 | C22 | 1 | 3 | yes | complemented |
| 10 | C5×#550 | — | 2 | 3 | no | — |
| 11 | C5×#532 | — | 2 | 3 | no | — |
| 12 | C5×#532 | C22 | 1 | 3 | yes | — |
| 13 | C4×C5×#41 | — | 2 | 3 | no | — |
| 14 | #66156 | — | 2 | 3 | no | — |
| 15 | #65526 | C22 | 1 | 3 | yes | — |
| 16 | #65384 | C22 | 1 | 6 | yes | — |
| 17 | #65366 | — | 2 | 3 | no | — |
| 18 | #65351 | C22 | 1 | 6 | yes | — |
| 19 | #65350 | — | 2 | 3 | no | — |
| 20 | C2×#3937 | — | 2 | 3 | no | — |
| 21 | #64768 | C22 | 1 | 6 | yes | complemented |
| 22 | #64762 | C22 | 1 | 6 | yes | complemented |
| 23 | #33567 | — | 5 | 1 | no | maximal, Sylow |
| 24 | C2×C5×#41 | — | 2 | 3 | no | — |
| 25 | C2×C5×#41 | — | 2 | 1 | no | — |
| 26 | C2×C5×#41 | D4 | 1 | 3 | yes | — |
| 27 | C2×C5×#41 | C23 | 1 | 1 | yes | characteristic |
| 28 | #4150 | — | 4 | 1 | no | — |
| 29 | #4149 | — | 4 | 1 | no | — |
| 30 | D5×#41 | — | 4 | 2 | no | — |
| 31 | #4147 | C23 | 1 | 2 | yes | — |
| 32 | #4144 | — | 2 | 3 | no | — |
| 33 | C2×#709 | — | 2 | 3 | no | — |
| 34 | C5×C4⋊Q8 | — | 2 | 3 | no | — |
| 35 | C5×C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 36 | C5×#127 | — | 2 | 3 | no | — |
| 37 | C5×#127 | D4 | 1 | 3 | yes | — |
| 38 | C5×#124 | — | 2 | 3 | no | — |
| 39 | C5×#124 | — | 2 | 3 | no | — |
| 40 | C5×#124 | D4 | 1 | 3 | yes | — |
| 41 | C5×#124 | C23 | 1 | 3 | yes | — |
| 42 | C5×#121 | — | 2 | 6 | no | — |
| 43 | C5×#121 | — | 2 | 6 | no | — |
| 44 | C5×#121 | — | 2 | 6 | no | — |
| 45 | C4×C5×Q8 | — | 2 | 1 | no | — |
| 46 | C4×C5×Q8 | — | 2 | 3 | no | — |
| 47 | C4×C5×D4 | — | 2 | 3 | no | — |
| 48 | C4×C5×D4 | — | 2 | 6 | no | — |
| 49 | C4×C5×D4 | — | 2 | 6 | no | — |
| 50 | C4×C5×D4 | — | 2 | 3 | no | — |
| 51 | C5×#117 | — | 2 | 3 | no | — |
| 52 | C5×#117 | — | 2 | 6 | no | — |
| 53 | C2×C42×C5 | — | 2 | 3 | no | — |
| 54 | C2×C42×C5 | D4 | 1 | 3 | yes | — |
| 55 | #4097 | — | 4 | 3 | no | — |
| 56 | #4096 | — | 4 | 6 | no | — |
| 57 | #4095 | — | 4 | 3 | no | — |
| 58 | #4088 | — | 2 | 6 | no | — |
| 59 | #4078 | — | 2 | 6 | no | — |
| 60 | C2×#681 | — | 2 | 3 | no | — |
| 61 | #3968 | C23 | 1 | 6 | yes | — |
| 62 | #3965 | — | 2 | 3 | no | — |
| 63 | #3937 | — | 2 | 6 | no | — |
| 64 | #3937 | — | 2 | 6 | no | — |
| 65 | #3483 | D5 | 1 | 1 | yes | characteristic, complemented |
| 66 | #3018 | — | 5 | 3 | no | — |
| 67 | #2205 | — | 5 | 2 | no | — |
| 68 | #2204 | — | 5 | 6 | no | — |
| 69 | #2132 | — | 5 | 3 | no | — |
| 70 | C5×#41 | — | 4 | 1 | no | — |
| 71 | C5×#41 | — | 4 | 1 | no | — |
| 72 | C5×#41 | — | 4 | 3 | no | — |
| 73 | C5×#41 | — | 4 | 3 | no | — |
| 74 | C5×#41 | — | 4 | 12 | no | — |
| 75 | C5×#41 | — | 8 | 1 | no | — |
| 76 | C5×#41 | — | 8 | 1 | no | — |
| 77 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 78 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 79 | C2×C5×Q8 | C2×D4 | 1 | 3 | yes | — |
| 80 | C2×C5×Q8 | C24 | 1 | 1 | yes | characteristic |
| 81 | C2×C5×D4 | — | 2 | 3 | no | — |
| 82 | C2×C5×D4 | — | 2 | 6 | no | — |
| 83 | C2×C5×D4 | — | 2 | 3 | no | — |
| 84 | C2×C5×D4 | — | 2 | 3 | no | — |
| 85 | C2×C5×D4 | C2×D4 | 1 | 3 | yes | — |
| 86 | C2×C5×D4 | C2×D4 | 1 | 3 | yes | — |
| 87 | C22×C4×C5 | — | 2 | 6 | no | — |
| 88 | C22×C4×C5 | — | 2 | 3 | no | — |
| 89 | C22×C4×C5 | — | 2 | 3 | no | — |
| 90 | C22×C4×C5 | C2×D4 | 1 | 3 | yes | — |
| 91 | C2×C22⋊D5 | — | 4 | 3 | no | — |
| 92 | #713 | — | 4 | 2 | no | — |
| 93 | Q8×D5 | — | 4 | 2 | no | — |
| 94 | #711 | — | 4 | 6 | no | — |
| 95 | D4×D5 | — | 4 | 6 | no | — |
| 96 | #709 | — | 2 | 6 | no | — |
| 97 | #709 | — | 2 | 6 | no | — |
| 98 | #709 | — | 4 | 3 | no | — |
| 99 | #709 | — | 4 | 1 | no | — |
| 100 | C2×D20 | — | 2 | 3 | no | — |
| 101 | C2×C4×D5 | — | 4 | 3 | no | — |
| 102 | C2×Dic10 | — | 2 | 3 | no | — |
| 103 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 104 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 105 | C5×C4⋊C4 | — | 2 | 6 | no | — |
| 106 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 107 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 108 | C5×C22⋊C4 | — | 2 | 6 | no | — |
| 109 | C42×C5 | — | 2 | 3 | no | — |
| 110 | C42×C5 | — | 2 | 3 | no | — |
| 111 | C42×C5 | — | 2 | 3 | no | — |
| 112 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 113 | C42×C5 | C2×D4 | 1 | 3 | yes | — |
| 114 | #689 | — | 4 | 6 | no | — |
| 115 | #688 | — | 4 | 6 | no | — |
| 116 | #687 | — | 4 | 6 | no | — |
| 117 | #686 | — | 4 | 6 | no | — |
| 118 | #681 | — | 2 | 6 | no | — |
| 119 | #681 | — | 4 | 3 | no | — |
| 120 | C2×#168 | — | 4 | 3 | no | — |
| 121 | #585 | — | 5 | 1 | no | — |
| 122 | #575 | — | 10 | 3 | no | — |
| 123 | Q82 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 124 | #555 | — | 2 | 3 | no | — |
| 125 | #555 | C2×D5 | 1 | 3 | yes | complemented |
| 126 | #550 | — | 2 | 3 | no | — |
| 127 | #532 | — | 2 | 3 | no | — |
| 128 | #532 | C2×D5 | 1 | 3 | yes | — |
| 129 | C4×#41 | — | 2 | 3 | no | — |
| 130 | #456 | — | 5 | 6 | no | — |
| 131 | #455 | — | 10 | 3 | no | — |
| 132 | #454 | — | 5 | 6 | no | — |
| 133 | #453 | — | 10 | 3 | no | — |
| 134 | C2×#104 | — | 10 | 3 | no | — |
| 135 | #413 | — | 5 | 3 | no | — |
| 136 | #356 | — | 5 | 6 | no | — |
| 137 | #355 | — | 5 | 6 | no | — |
| 138 | C23×C5 | — | 2 | 3 | no | — |
| 139 | C23×C5 | #120 | 1 | 3 | yes | — |
| 140 | C22×D5 | — | 4 | 3 | no | — |
| 141 | C5×Q8 | — | 2 | 4 | no | — |
| 142 | C5×Q8 | — | 4 | 1 | no | — |
| 143 | C5×Q8 | — | 4 | 3 | no | — |
| 144 | C5×Q8 | — | 8 | 1 | no | — |
| 145 | C5×D4 | — | 4 | 3 | no | — |
| 146 | C5×D4 | — | 4 | 3 | no | — |
| 147 | C5×D4 | — | 4 | 12 | no | — |
| 148 | C5×D4 | — | 4 | 6 | no | — |
| 149 | C5×D4 | — | 8 | 3 | no | — |
| 150 | C5×D4 | — | 8 | 3 | no | — |
| 151 | C2×C4×C5 | — | 2 | 3 | no | — |
| 152 | C2×C4×C5 | — | 2 | 3 | no | — |
| 153 | C2×C4×C5 | — | 2 | 3 | no | — |
| 154 | C2×C4×C5 | — | 2 | 3 | no | — |
| 155 | C2×C4×C5 | — | 2 | 3 | no | — |
| 156 | C2×C4×C5 | — | 4 | 6 | no | — |
| 157 | C2×C4×C5 | — | 4 | 12 | no | — |
| 158 | C2×C4×C5 | — | 4 | 3 | no | — |
| 159 | C2×C4×C5 | — | 4 | 3 | no | — |
| 160 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 161 | C2×C4×C5 | C22×D4 | 1 | 3 | yes | — |
| 162 | C22⋊D5 | — | 4 | 6 | no | — |
| 163 | C22⋊D5 | — | 4 | 2 | no | — |
| 164 | C2×Dic5 | — | 4 | 3 | no | — |
| 165 | D20 | — | 2 | 6 | no | — |
| 166 | D20 | — | 4 | 1 | no | — |
| 167 | C4×D5 | — | 4 | 6 | no | — |
| 168 | C4×D5 | — | 4 | 2 | no | — |
| 169 | Dic10 | — | 2 | 6 | no | — |
| 170 | Dic10 | — | 4 | 1 | no | — |
| 171 | #168 | — | 4 | 6 | no | — |
| 172 | 2−1+4 | — | 5 | 2 | no | — |
| 173 | 2−1+4 | — | 20 | 1 | no | — |
| 174 | H(5,𝔽2) | — | 10 | 3 | no | — |
| 175 | H(5,𝔽2) | — | 20 | 1 | no | — |
| 176 | C2×#41 | — | 2 | 3 | no | — |
| 177 | C2×#41 | — | 2 | 1 | no | — |
| 178 | C2×#41 | — | 10 | 3 | no | — |
| 179 | C2×#41 | — | 20 | 2 | no | — |
| 180 | C2×#41 | C22⋊D5 | 1 | 3 | yes | — |
| 181 | C2×#41 | C22×D5 | 1 | 1 | yes | characteristic |
| 182 | #137 | — | 10 | 6 | no | — |
| 183 | #137 | — | 20 | 3 | no | — |
| 184 | AGL(1,ℤ/8) | — | 10 | 6 | no | — |
| 185 | AGL(1,ℤ/8) | — | 20 | 3 | no | — |
| 186 | #135 | — | 20 | 6 | no | — |
| 187 | C2×M16 | — | 10 | 3 | no | — |
| 188 | C4⋊Q8 | — | 2 | 3 | no | — |
| 189 | C4⋊Q8 | C22×D5 | 1 | 3 | yes | — |
| 190 | #127 | — | 2 | 3 | no | — |
| 191 | #127 | C22⋊D5 | 1 | 3 | yes | — |
| 192 | #124 | — | 2 | 3 | no | — |
| 193 | #124 | — | 2 | 3 | no | — |
| 194 | #124 | C22⋊D5 | 1 | 3 | yes | — |
| 195 | #124 | C22×D5 | 1 | 3 | yes | — |
| 196 | #121 | — | 2 | 6 | no | — |
| 197 | #121 | — | 2 | 6 | no | — |
| 198 | #121 | — | 2 | 6 | no | — |
| 199 | C4×Q8 | — | 2 | 1 | no | — |
| 200 | C4×Q8 | — | 2 | 3 | no | — |
| 201 | C4×D4 | — | 2 | 3 | no | — |
| 202 | C4×D4 | — | 2 | 6 | no | — |
| 203 | C4×D4 | — | 2 | 6 | no | — |
| 204 | C4×D4 | — | 2 | 3 | no | — |
| 205 | #117 | — | 2 | 3 | no | — |
| 206 | #117 | — | 2 | 6 | no | — |
| 207 | C2×C42 | — | 2 | 3 | no | — |
| 208 | C2×C42 | C22⋊D5 | 1 | 3 | yes | — |
| 209 | #104 | — | 10 | 6 | no | — |
| 210 | #104 | — | 10 | 6 | no | — |
| 211 | #101 | — | 5 | 6 | no | — |
| 212 | #100 | — | 10 | 3 | no | — |
| 213 | C22×C5 | — | 2 | 3 | no | — |
| 214 | C22×C5 | — | 4 | 3 | no | — |
| 215 | C22×C5 | — | 4 | 6 | no | — |
| 216 | C22×C5 | — | 4 | 3 | no | — |
| 217 | C22×C5 | C2×#120 | 1 | 1 | yes | characteristic |
| 218 | C2×D5 | — | 4 | 2 | no | — |
| 219 | C2×D5 | — | 16 | 3 | no | — |
| 220 | C4×C5 | — | 2 | 6 | no | — |
| 221 | C4×C5 | — | 2 | 3 | no | — |
| 222 | C4×C5 | — | 4 | 3 | no | — |
| 223 | C4×C5 | — | 4 | 3 | no | — |
| 224 | C4×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 225 | C4×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 226 | Dic5 | — | 4 | 2 | no | — |
| 227 | #41 | — | 4 | 1 | no | — |
| 228 | #41 | — | 4 | 1 | no | — |
| 229 | #41 | — | 4 | 3 | no | — |
| 230 | #41 | — | 4 | 3 | no | — |
| 231 | #41 | — | 4 | 12 | no | — |
| 232 | #41 | — | 8 | 1 | no | — |
| 233 | #41 | — | 8 | 1 | no | — |
| 234 | #41 | — | 10 | 6 | no | — |
| 235 | #41 | — | 10 | 6 | no | — |
| 236 | #41 | — | 20 | 3 | no | — |
| 237 | #41 | — | 20 | 6 | no | — |
| 238 | #41 | — | 20 | 2 | no | — |
| 239 | #41 | — | 20 | 1 | no | — |
| 240 | C2×Q8 | — | 2 | 1 | no | — |
| 241 | C2×Q8 | — | 2 | 1 | no | — |
| 242 | C2×Q8 | — | 10 | 3 | no | — |
| 243 | C2×Q8 | — | 20 | 2 | no | — |
| 244 | C2×Q8 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 245 | C2×Q8 | C23×D5 | 1 | 1 | yes | characteristic |
| 246 | C2×D4 | — | 2 | 3 | no | — |
| 247 | C2×D4 | — | 2 | 6 | no | — |
| 248 | C2×D4 | — | 2 | 3 | no | — |
| 249 | C2×D4 | — | 2 | 3 | no | — |
| 250 | C2×D4 | — | 10 | 3 | no | — |
| 251 | C2×D4 | — | 20 | 6 | no | — |
| 252 | C2×D4 | — | 20 | 3 | no | — |
| 253 | C2×D4 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 254 | C2×D4 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 255 | C22×C4 | — | 2 | 6 | no | — |
| 256 | C22×C4 | — | 2 | 3 | no | — |
| 257 | C22×C4 | — | 2 | 3 | no | — |
| 258 | C22×C4 | — | 20 | 3 | no | — |
| 259 | C22×C4 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 260 | Q16 | — | 20 | 6 | no | — |
| 261 | SD16 | — | 20 | 6 | no | — |
| 262 | SD16 | — | 20 | 6 | no | — |
| 263 | D8 | — | 20 | 6 | no | — |
| 264 | M16 | — | 10 | 6 | no | — |
| 265 | M16 | — | 20 | 3 | no | — |
| 266 | C2×C8 | — | 20 | 3 | no | — |
| 267 | C4⋊C4 | — | 2 | 3 | no | — |
| 268 | C4⋊C4 | — | 2 | 3 | no | — |
| 269 | C4⋊C4 | — | 2 | 6 | no | — |
| 270 | C22⋊C4 | — | 2 | 6 | no | — |
| 271 | C22⋊C4 | — | 2 | 6 | no | — |
| 272 | C22⋊C4 | — | 2 | 6 | no | — |
| 273 | C42 | — | 2 | 3 | no | — |
| 274 | C42 | — | 2 | 3 | no | — |
| 275 | C42 | — | 2 | 3 | no | — |
| 276 | C42 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 277 | C42 | C2×C22⋊D5 | 1 | 3 | yes | — |
| 278 | C2×C5 | — | 2 | 1 | no | — |
| 279 | C2×C5 | — | 4 | 3 | no | — |
| 280 | C2×C5 | — | 8 | 3 | no | — |
| 281 | C2×C5 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 282 | D5 | — | 8 | 2 | no | — |
| 283 | C23 | — | 2 | 3 | no | — |
| 284 | C23 | — | 20 | 3 | no | — |
| 285 | C23 | #4099 | 1 | 3 | yes | — |
| 286 | Q8 | — | 2 | 4 | no | — |
| 287 | Q8 | — | 4 | 1 | no | — |
| 288 | Q8 | — | 4 | 3 | no | — |
| 289 | Q8 | — | 8 | 1 | no | — |
| 290 | Q8 | — | 10 | 6 | no | — |
| 291 | Q8 | — | 20 | 1 | no | — |
| 292 | D4 | — | 4 | 3 | no | — |
| 293 | D4 | — | 4 | 3 | no | — |
| 294 | D4 | — | 4 | 12 | no | — |
| 295 | D4 | — | 4 | 6 | no | — |
| 296 | D4 | — | 8 | 3 | no | — |
| 297 | D4 | — | 8 | 3 | no | — |
| 298 | D4 | — | 10 | 6 | no | — |
| 299 | D4 | — | 20 | 6 | no | — |
| 300 | D4 | — | 20 | 2 | no | — |
| 301 | D4 | — | 20 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 3 | no | — |
| 303 | C2×C4 | — | 2 | 3 | no | — |
| 304 | C2×C4 | — | 2 | 3 | no | — |
| 305 | C2×C4 | — | 2 | 3 | no | — |
| 306 | C2×C4 | — | 2 | 3 | no | — |
| 307 | C2×C4 | — | 4 | 6 | no | — |
| 308 | C2×C4 | — | 4 | 12 | no | — |
| 309 | C2×C4 | — | 4 | 3 | no | — |
| 310 | C2×C4 | — | 4 | 3 | no | — |
| 311 | C2×C4 | — | 20 | 6 | no | — |
| 312 | C2×C4 | — | 20 | 3 | no | — |
| 313 | C2×C4 | — | 20 | 2 | no | — |
| 314 | C2×C4 | #4099 | 1 | 1 | yes | characteristic |
| 315 | C2×C4 | C22×C22⋊D5 | 1 | 3 | yes | — |
| 316 | C8 | — | 20 | 6 | no | — |
| 317 | C5 | #33567 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 318 | C22 | — | 2 | 3 | no | — |
| 319 | C22 | — | 4 | 3 | no | — |
| 320 | C22 | — | 4 | 6 | no | — |
| 321 | C22 | — | 4 | 3 | no | — |
| 322 | C22 | — | 20 | 2 | no | — |
| 323 | C22 | — | 80 | 3 | no | — |
| 324 | C22 | C2×#4099 | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 2 | 6 | no | — |
| 326 | C4 | — | 2 | 3 | no | — |
| 327 | C4 | — | 4 | 3 | no | — |
| 328 | C4 | — | 4 | 3 | no | — |
| 329 | C4 | — | 20 | 2 | no | — |
| 330 | C4 | #65533 | 1 | 1 | yes | characteristic |
| 331 | C4 | #65533 | 1 | 1 | yes | characteristic |
| 332 | C2 | — | 2 | 1 | no | — |
| 333 | C2 | — | 4 | 3 | no | — |
| 334 | C2 | — | 8 | 3 | no | — |
| 335 | C2 | — | 40 | 2 | no | — |
| 336 | C2 | C2×#65533 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#3483 | C2 | semidirect |
| 2. #10615053 | C2 | semidirect |
| 3. #10602735 | C2 | semidirect |
| 4. #10602723 | C2 | semidirect |
| 5. #10602613 | C2 | semidirect |
| 7. C5×Q82 | C22 | semidirect |
| 9. C5×#555 | C22 | semidirect |
| 21. #64768 | C22 | semidirect |
| 22. #64762 | C22 | semidirect |
| 65. #3483 | D5 | semidirect |
| 123. Q82 | C2×D5 | semidirect |
| 125. #555 | C2×D5 | semidirect |
| 317. C5 | #33567 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#3483 | 2 |
| #10615053 | 2 |
| #10602735 | 2 |
| #10602723 | 2 |
| #10602613 | 2 |
| #33567 | 5 |