Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
872 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 224 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 | #10617791 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10617547 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10617547 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10604352 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10604350 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #10604266 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10604265 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C5×#520 | C4 | 1 | 1 | yes | characteristic |
| 9 | #66161 | C4 | 1 | 1 | yes | characteristic |
| 10 | #41×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 11 | #65904 | — | 2 | 1 | no | — |
| 12 | #65903 | — | 2 | 1 | no | — |
| 13 | #65877 | — | 2 | 3 | no | — |
| 14 | #65873 | — | 2 | 3 | no | — |
| 15 | #65853 | C4 | 1 | 3 | yes | — |
| 16 | #65851 | C22 | 1 | 3 | yes | — |
| 17 | #65834 | C4 | 1 | 3 | yes | — |
| 18 | C2×#4036 | C22 | 1 | 3 | yes | — |
| 19 | Q8×#170 | — | 2 | 1 | no | — |
| 20 | Q8×#170 | — | 2 | 1 | no | — |
| 21 | #65774 | — | 2 | 3 | no | — |
| 22 | #65774 | — | 2 | 3 | no | — |
| 23 | #65759 | C22 | 1 | 3 | yes | — |
| 24 | #65759 | C22 | 1 | 3 | yes | — |
| 25 | C4×#704 | C22 | 1 | 3 | yes | — |
| 26 | C4×#704 | C22 | 1 | 3 | yes | — |
| 27 | #64870 | C22 | 1 | 1 | yes | characteristic |
| 28 | #64870 | C22 | 1 | 3 | yes | — |
| 29 | #64870 | C22 | 1 | 3 | yes | — |
| 30 | #64870 | C22 | 1 | 3 | yes | — |
| 31 | #64870 | C22 | 1 | 6 | yes | — |
| 32 | #17692 | — | 5 | 1 | no | maximal, Sylow |
| 33 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×#704 | — | 2 | 3 | no | — |
| 35 | C2×#704 | — | 2 | 3 | no | — |
| 36 | C4×C5×Q8 | — | 2 | 1 | no | — |
| 37 | C4×C5×D4 | — | 2 | 3 | no | — |
| 38 | C5×#117 | C2×C4 | 1 | 3 | yes | — |
| 39 | C2×C5×C4⋊C4 | C2×C4 | 1 | 3 | yes | — |
| 40 | Q8×Dic5 | — | 2 | 1 | no | — |
| 41 | Q8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 42 | Q8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 43 | D4×Dic5 | — | 2 | 3 | no | — |
| 44 | D4×Dic5 | D4 | 1 | 3 | yes | — |
| 45 | D4×Dic5 | D4 | 1 | 3 | yes | — |
| 46 | #4072 | C2×C4 | 1 | 3 | yes | — |
| 47 | #4072 | C2×C4 | 1 | 3 | yes | — |
| 48 | C2×C4⋊Dic5 | — | 2 | 3 | no | — |
| 49 | C2×C4⋊Dic5 | C2×C4 | 1 | 3 | yes | — |
| 50 | #4070 | — | 2 | 6 | no | — |
| 51 | C2×#683 | — | 2 | 3 | no | — |
| 52 | C2×C4×Dic5 | C23 | 1 | 3 | yes | — |
| 53 | #4036 | — | 2 | 3 | no | — |
| 54 | #4036 | — | 2 | 3 | no | — |
| 55 | #4036 | — | 2 | 3 | no | — |
| 56 | #4036 | C2×C4 | 1 | 6 | yes | — |
| 57 | #4036 | C23 | 1 | 3 | yes | — |
| 58 | #4036 | C23 | 1 | 1 | yes | characteristic |
| 59 | #4034 | — | 2 | 3 | no | — |
| 60 | #4034 | C2×C4 | 1 | 3 | yes | — |
| 61 | #4034 | C2×C4 | 1 | 3 | yes | — |
| 62 | #4024 | — | 2 | 3 | no | — |
| 63 | #4024 | — | 2 | 3 | no | — |
| 64 | #4024 | — | 2 | 6 | no | — |
| 65 | #4016 | — | 2 | 3 | no | — |
| 66 | #4015 | — | 2 | 3 | no | — |
| 67 | C4×Dic10 | — | 2 | 3 | no | — |
| 68 | C4×Dic10 | — | 2 | 3 | no | — |
| 69 | C22⋊#170 | — | 2 | 3 | no | — |
| 70 | C22⋊#170 | — | 2 | 3 | no | — |
| 71 | #4002 | C23 | 1 | 1 | yes | characteristic |
| 72 | #4002 | C23 | 1 | 1 | yes | characteristic |
| 73 | #4002 | C23 | 1 | 3 | yes | — |
| 74 | #4002 | C23 | 1 | 3 | yes | — |
| 75 | C4⋊#170 | — | 2 | 3 | no | — |
| 76 | C4⋊#170 | — | 2 | 3 | no | — |
| 77 | C4×#170 | — | 2 | 3 | no | — |
| 78 | C4×#170 | — | 2 | 3 | no | — |
| 79 | #3504 | — | 5 | 1 | no | — |
| 80 | #2971 | — | 5 | 1 | no | — |
| 81 | #2971 | — | 5 | 1 | no | — |
| 82 | #1537 | — | 5 | 3 | no | — |
| 83 | #1528 | — | 5 | 3 | no | — |
| 84 | #1528 | — | 5 | 3 | no | — |
| 85 | #1526 | — | 5 | 3 | no | — |
| 86 | C5×#41 | — | 2 | 4 | no | — |
| 87 | C2×C5×Q8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C5×D4 | C22⋊C4 | 1 | 3 | yes | — |
| 89 | C22×C4×C5 | — | 2 | 3 | no | — |
| 90 | C22×C4×C5 | C22×C4 | 1 | 3 | yes | — |
| 91 | C22×Dic5 | — | 2 | 3 | no | — |
| 92 | C22×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 93 | C22×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 94 | C2×Dic10 | — | 2 | 3 | no | — |
| 95 | C2×Dic10 | — | 2 | 3 | no | — |
| 96 | #704 | — | 4 | 6 | no | — |
| 97 | #704 | — | 4 | 6 | no | — |
| 98 | C2×#170 | — | 2 | 1 | no | — |
| 99 | C2×#170 | — | 2 | 1 | no | — |
| 100 | C2×#170 | — | 2 | 3 | no | — |
| 101 | C2×#170 | — | 2 | 3 | no | — |
| 102 | C5×C4⋊C4 | — | 2 | 3 | no | — |
| 103 | C5×C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 104 | C5×C4⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 105 | C5×C22⋊C4 | — | 2 | 3 | no | — |
| 106 | C42×C5 | — | 2 | 3 | no | — |
| 107 | C22⋊Dic5 | — | 2 | 3 | no | — |
| 108 | C22⋊Dic5 | C22⋊C4 | 1 | 6 | yes | — |
| 109 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 110 | C4⋊Dic5 | — | 2 | 6 | no | — |
| 111 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 112 | C4⋊Dic5 | — | 2 | 3 | no | — |
| 113 | C4⋊Dic5 | C22⋊C4 | 1 | 3 | yes | — |
| 114 | C4⋊Dic5 | C22⋊C4 | 1 | 3 | yes | — |
| 115 | C4⋊Dic5 | C22×C4 | 1 | 1 | yes | characteristic |
| 116 | C4⋊Dic5 | C22×C4 | 1 | 3 | yes | — |
| 117 | #683 | — | 2 | 3 | no | — |
| 118 | #683 | — | 2 | 3 | no | — |
| 119 | #683 | — | 2 | 6 | no | — |
| 120 | #683 | — | 2 | 1 | no | — |
| 121 | #683 | — | 2 | 3 | no | — |
| 122 | C4×Dic5 | — | 2 | 3 | no | — |
| 123 | C4×Dic5 | C22×C4 | 1 | 3 | yes | — |
| 124 | C4×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 125 | C4×Dic5 | C2×D4 | 1 | 3 | yes | — |
| 126 | C4×Dic5 | C24 | 1 | 1 | yes | characteristic |
| 127 | #557 | — | 10 | 1 | no | — |
| 128 | #557 | — | 10 | 1 | no | — |
| 129 | #554 | — | 10 | 3 | no | — |
| 130 | #554 | — | 10 | 3 | no | — |
| 131 | #544 | — | 5 | 3 | no | — |
| 132 | #544 | — | 5 | 3 | no | — |
| 133 | #533 | — | 5 | 3 | no | — |
| 134 | C2×#125 | — | 5 | 3 | no | — |
| 135 | #520 | — | 5 | 1 | no | — |
| 136 | #520 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 137 | C4×#41 | — | 5 | 1 | no | — |
| 138 | C8×Q8 | — | 10 | 1 | no | — |
| 139 | C8×Q8 | — | 10 | 1 | no | — |
| 140 | #436 | — | 10 | 3 | no | — |
| 141 | #436 | — | 10 | 3 | no | — |
| 142 | #432 | — | 5 | 3 | no | — |
| 143 | #432 | — | 5 | 3 | no | — |
| 144 | C4×M16 | — | 5 | 3 | no | — |
| 145 | C4×M16 | — | 5 | 3 | no | — |
| 146 | #330 | — | 5 | 3 | no | — |
| 147 | #330 | — | 5 | 6 | no | — |
| 148 | #330 | — | 5 | 1 | no | — |
| 149 | #330 | — | 5 | 3 | no | — |
| 150 | #330 | — | 5 | 3 | no | — |
| 151 | C23×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 152 | C5×Q8 | #101 | 1 | 4 | yes | — |
| 153 | C5×D4 | — | 2 | 6 | no | — |
| 154 | C2×C4×C5 | — | 2 | 6 | no | — |
| 155 | C2×C4×C5 | — | 2 | 3 | no | — |
| 156 | C2×C4×C5 | — | 2 | 1 | no | — |
| 157 | C2×C4×C5 | — | 8 | 3 | no | — |
| 158 | C2×C4×C5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 159 | C2×C4×C5 | C23×C4 | 1 | 1 | yes | characteristic |
| 160 | C2×Dic5 | — | 2 | 1 | no | — |
| 161 | C2×Dic5 | — | 2 | 3 | no | — |
| 162 | C2×Dic5 | — | 4 | 3 | no | — |
| 163 | C2×Dic5 | — | 4 | 3 | no | — |
| 164 | C2×Dic5 | — | 8 | 3 | no | — |
| 165 | C2×Dic5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 166 | C2×Dic5 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 167 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 168 | C2×Dic5 | C22×D4 | 1 | 1 | yes | characteristic |
| 169 | Dic10 | — | 8 | 3 | no | — |
| 170 | Dic10 | — | 8 | 3 | no | — |
| 171 | #170 | — | 2 | 2 | no | — |
| 172 | #170 | — | 2 | 2 | no | — |
| 173 | #170 | — | 4 | 3 | no | — |
| 174 | #170 | — | 4 | 3 | no | — |
| 175 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 176 | C2×M16 | — | 10 | 3 | no | — |
| 177 | C2×M16 | — | 10 | 3 | no | — |
| 178 | C4⋊Q8 | — | 5 | 3 | no | — |
| 179 | C4⋊Q8 | — | 5 | 3 | no | — |
| 180 | C4⋊Q8 | — | 10 | 3 | no | — |
| 181 | C4⋊Q8 | — | 10 | 3 | no | — |
| 182 | #125 | — | 5 | 3 | no | — |
| 183 | #125 | — | 5 | 6 | no | — |
| 184 | #125 | — | 5 | 1 | no | — |
| 185 | #125 | — | 10 | 3 | no | — |
| 186 | #125 | — | 10 | 3 | no | — |
| 187 | #125 | — | 10 | 3 | no | — |
| 188 | #125 | — | 10 | 3 | no | — |
| 189 | C22⋊Q8 | — | 10 | 6 | no | — |
| 190 | C22⋊Q8 | — | 10 | 3 | no | — |
| 191 | C22⋊Q8 | — | 10 | 3 | no | — |
| 192 | C22⋊Q8 | — | 10 | 6 | no | — |
| 193 | C4×Q8 | — | 2 | 1 | no | — |
| 194 | C4×Q8 | — | 5 | 1 | no | — |
| 195 | C4×Q8 | — | 5 | 1 | no | — |
| 196 | C4×Q8 | — | 10 | 3 | no | — |
| 197 | C4×Q8 | — | 10 | 3 | no | — |
| 198 | C4×Q8 | — | 10 | 1 | no | — |
| 199 | C4×D4 | — | 2 | 3 | no | — |
| 200 | C4×D4 | — | 5 | 3 | no | — |
| 201 | C4×D4 | — | 5 | 3 | no | — |
| 202 | C4×D4 | — | 10 | 3 | no | — |
| 203 | #117 | — | 5 | 3 | no | — |
| 204 | #117 | — | 5 | 3 | no | — |
| 205 | #117 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 206 | C2×C4⋊C4 | — | 5 | 3 | no | — |
| 207 | C2×C4⋊C4 | — | 10 | 3 | no | — |
| 208 | C2×C4⋊C4 | — | 10 | 3 | no | — |
| 209 | C2×C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 210 | C2×C42 | — | 5 | 3 | no | — |
| 211 | C4⋊C8 | — | 10 | 3 | no | — |
| 212 | C4⋊C8 | — | 10 | 3 | no | — |
| 213 | C22⋊C8 | — | 10 | 3 | no | — |
| 214 | C22⋊C8 | — | 10 | 3 | no | — |
| 215 | #97 | — | 5 | 1 | no | — |
| 216 | #97 | — | 5 | 1 | no | — |
| 217 | #97 | — | 5 | 3 | no | — |
| 218 | #97 | — | 5 | 3 | no | — |
| 219 | C4×C8 | — | 10 | 3 | no | — |
| 220 | C4×C8 | — | 10 | 3 | no | — |
| 221 | C22×C5 | — | 2 | 3 | no | — |
| 222 | C22×C5 | — | 2 | 3 | no | — |
| 223 | C22×C5 | — | 2 | 3 | no | — |
| 224 | C22×C5 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 225 | C4×C5 | — | 2 | 1 | no | — |
| 226 | C4×C5 | — | 4 | 1 | no | — |
| 227 | C4×C5 | — | 4 | 3 | no | — |
| 228 | C4×C5 | C2×#101 | 1 | 6 | yes | — |
| 229 | Dic5 | — | 2 | 1 | no | — |
| 230 | Dic5 | — | 2 | 1 | no | — |
| 231 | Dic5 | — | 2 | 3 | no | — |
| 232 | Dic5 | — | 2 | 3 | no | — |
| 233 | Dic5 | — | 4 | 1 | no | — |
| 234 | Dic5 | — | 4 | 3 | no | — |
| 235 | #41 | — | 2 | 4 | no | — |
| 236 | C2×Q8 | — | 10 | 3 | no | — |
| 237 | C2×Q8 | — | 10 | 3 | no | — |
| 238 | C2×Q8 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 239 | C2×D4 | C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 240 | C22×C4 | — | 2 | 3 | no | — |
| 241 | C22×C4 | — | 5 | 3 | no | — |
| 242 | C22×C4 | — | 5 | 3 | no | — |
| 243 | C22×C4 | — | 10 | 3 | no | — |
| 244 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 245 | M16 | — | 20 | 6 | no | — |
| 246 | M16 | — | 20 | 6 | no | — |
| 247 | C2×C8 | — | 10 | 1 | no | — |
| 248 | C2×C8 | — | 10 | 1 | no | — |
| 249 | C2×C8 | — | 10 | 3 | no | — |
| 250 | C2×C8 | — | 10 | 3 | no | — |
| 251 | C4⋊C4 | — | 2 | 3 | no | — |
| 252 | C4⋊C4 | — | 5 | 3 | no | — |
| 253 | C4⋊C4 | — | 5 | 3 | no | — |
| 254 | C4⋊C4 | — | 5 | 3 | no | — |
| 255 | C4⋊C4 | — | 5 | 1 | no | — |
| 256 | C4⋊C4 | — | 10 | 3 | no | — |
| 257 | C4⋊C4 | — | 10 | 3 | no | — |
| 258 | C4⋊C4 | — | 10 | 6 | no | — |
| 259 | C4⋊C4 | — | 10 | 6 | no | — |
| 260 | C4⋊C4 | — | 10 | 1 | no | — |
| 261 | C4⋊C4 | — | 10 | 1 | no | — |
| 262 | C4⋊C4 | — | 10 | 3 | no | — |
| 263 | C4⋊C4 | — | 10 | 3 | no | — |
| 264 | C4⋊C4 | — | 10 | 3 | no | — |
| 265 | C4⋊C4 | C22×AGL(1,𝔽5) | 1 | 3 | yes | — |
| 266 | C4⋊C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 267 | C22⋊C4 | — | 2 | 3 | no | — |
| 268 | C22⋊C4 | — | 5 | 6 | no | — |
| 269 | C22⋊C4 | — | 10 | 3 | no | — |
| 270 | C42 | — | 2 | 3 | no | — |
| 271 | C42 | — | 5 | 3 | no | — |
| 272 | C42 | — | 5 | 3 | no | — |
| 273 | C42 | — | 5 | 1 | no | — |
| 274 | C42 | — | 5 | 3 | no | — |
| 275 | C42 | — | 10 | 3 | no | — |
| 276 | C2×C5 | — | 4 | 3 | no | — |
| 277 | C2×C5 | C22×#101 | 1 | 1 | yes | characteristic |
| 278 | C2×C5 | #2901 | 1 | 1 | yes | characteristic |
| 279 | C2×C5 | #2901 | 1 | 1 | yes | characteristic |
| 280 | C23 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 281 | Q8 | — | 40 | 3 | no | — |
| 282 | Q8 | — | 40 | 3 | no | — |
| 283 | Q8 | #4005 | 1 | 4 | yes | — |
| 284 | D4 | — | 2 | 6 | no | — |
| 285 | C2×C4 | — | 2 | 6 | no | — |
| 286 | C2×C4 | — | 2 | 1 | no | — |
| 287 | C2×C4 | — | 2 | 3 | no | — |
| 288 | C2×C4 | — | 5 | 1 | no | — |
| 289 | C2×C4 | — | 5 | 1 | no | — |
| 290 | C2×C4 | — | 5 | 3 | no | — |
| 291 | C2×C4 | — | 5 | 3 | no | — |
| 292 | C2×C4 | — | 8 | 3 | no | — |
| 293 | C2×C4 | — | 10 | 1 | no | — |
| 294 | C2×C4 | — | 10 | 3 | no | — |
| 295 | C2×C4 | — | 20 | 3 | no | — |
| 296 | C2×C4 | — | 20 | 3 | no | — |
| 297 | C2×C4 | — | 40 | 3 | no | — |
| 298 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 3 | yes | — |
| 299 | C2×C4 | C23×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 10 | 2 | no | — |
| 301 | C8 | — | 10 | 2 | no | — |
| 302 | C8 | — | 20 | 3 | no | — |
| 303 | C8 | — | 20 | 3 | no | — |
| 304 | C5 | #17692 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 305 | C22 | — | 2 | 3 | no | — |
| 306 | C22 | — | 2 | 3 | no | — |
| 307 | C22 | — | 2 | 3 | no | — |
| 308 | C22 | C22×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 2 | 1 | no | — |
| 310 | C4 | — | 4 | 3 | no | — |
| 311 | C4 | — | 4 | 1 | no | — |
| 312 | C4 | — | 10 | 1 | no | — |
| 313 | C4 | — | 10 | 3 | no | — |
| 314 | C4 | — | 10 | 1 | no | — |
| 315 | C4 | — | 10 | 3 | no | — |
| 316 | C4 | — | 20 | 1 | no | — |
| 317 | C4 | — | 20 | 3 | no | — |
| 318 | C4 | C2×#4005 | 1 | 6 | yes | — |
| 319 | C2 | — | 4 | 3 | no | — |
| 320 | C2 | C22×#4005 | 1 | 1 | yes | characteristic |
| 321 | C2 | #10617560 | 1 | 1 | yes | characteristic |
| 322 | C2 | #10617562 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #10604352 | C2 | semidirect |
| 5. #10604350 | C2 | semidirect |
| 6. #10604266 | C2 | semidirect |
| 7. #10604265 | C2 | semidirect |
| 304. C5 | #17692 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617791 | 2 |
| #10617547 | 2 |
| #10604352 | 2 |
| #10604350 | 2 |
| #10604266 | 2 |
| #10604265 | 2 |
| #17692 | 5 |