Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
460 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 92 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 | #10617309 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604549 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10603920 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10599951 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10598570 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×#4133 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×#4131 | C22 | 1 | 1 | yes | characteristic |
| 8 | #66002 | C22 | 1 | 1 | yes | characteristic |
| 9 | #65787 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #65777 | C22 | 1 | 2 | yes | — |
| 11 | #65776 | C22 | 1 | 1 | yes | characteristic |
| 12 | #65695 | — | 2 | 2 | no | — |
| 13 | #65692 | — | 2 | 2 | no | — |
| 14 | #65691 | — | 2 | 2 | no | — |
| 15 | #65682 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | C4×D5⋊C8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #65029 | C22 | 1 | 2 | yes | — |
| 18 | #64929 | C4 | 1 | 2 | yes | complemented |
| 19 | #64867 | — | 2 | 2 | no | — |
| 20 | #64856 | C4 | 1 | 2 | yes | complemented |
| 21 | C5×#354 | C4 | 1 | 1 | yes | characteristic |
| 22 | #64761 | C4 | 1 | 1 | yes | characteristic |
| 23 | #64694 | C4 | 1 | 2 | yes | — |
| 24 | #12633 | — | 5 | 1 | no | maximal, Sylow |
| 25 | C2×#713 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×#711 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#704 | — | 2 | 2 | no | — |
| 28 | C2×#704 | D4 | 1 | 2 | yes | — |
| 29 | C22×#170 | D4 | 1 | 2 | yes | — |
| 30 | #4133 | — | 4 | 1 | no | — |
| 31 | #4133 | — | 4 | 1 | no | — |
| 32 | #4131 | — | 4 | 1 | no | — |
| 33 | #4131 | — | 4 | 1 | no | — |
| 34 | C2×#700 | — | 2 | 1 | no | — |
| 35 | C2×#700 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×#700 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C2×D5⋊C8 | — | 2 | 1 | no | — |
| 38 | C2×D5⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C2×D5⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C5×#124 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | #4090 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | #4086 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | #4030 | — | 2 | 2 | no | — |
| 44 | #4027 | — | 2 | 2 | no | — |
| 45 | #4023 | — | 2 | 2 | no | — |
| 46 | #4021 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C42×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | #4013 | — | 2 | 2 | no | — |
| 49 | C22⋊#170 | — | 2 | 2 | no | — |
| 50 | #4005 | C2×C4 | 1 | 4 | yes | — |
| 51 | #4004 | — | 2 | 2 | no | — |
| 52 | #4003 | — | 2 | 2 | no | — |
| 53 | #4002 | — | 4 | 2 | no | — |
| 54 | C4×#170 | — | 2 | 1 | no | — |
| 55 | C4×#170 | — | 2 | 1 | no | — |
| 56 | C4×#170 | — | 4 | 2 | no | — |
| 57 | C5×#99 | C2×C4 | 1 | 2 | yes | — |
| 58 | #3966 | C2×C4 | 1 | 2 | yes | — |
| 59 | #3938 | — | 2 | 2 | no | — |
| 60 | #2931 | — | 5 | 1 | no | — |
| 61 | #2132 | — | 5 | 1 | no | — |
| 62 | #2126 | — | 5 | 1 | no | — |
| 63 | #1757 | — | 5 | 2 | no | — |
| 64 | #1395 | — | 5 | 2 | no | — |
| 65 | C2×C5×Q8 | C42 | 1 | 1 | yes | characteristic |
| 66 | C2×C5×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 67 | C2×C22⋊D5 | — | 2 | 2 | no | — |
| 68 | C2×C22⋊D5 | #41 | 1 | 2 | yes | — |
| 69 | C22×Dic5 | C2×D4 | 1 | 2 | yes | — |
| 70 | #713 | — | 4 | 1 | no | — |
| 71 | #713 | — | 4 | 1 | no | — |
| 72 | #711 | — | 4 | 1 | no | — |
| 73 | #711 | — | 4 | 1 | no | — |
| 74 | C2×D20 | — | 2 | 1 | no | — |
| 75 | C2×D20 | C42 | 1 | 1 | yes | characteristic |
| 76 | C2×C4×D5 | — | 2 | 1 | no | — |
| 77 | C2×C4×D5 | — | 2 | 1 | no | — |
| 78 | C2×C4×D5 | C22×C4 | 1 | 1 | yes | characteristic |
| 79 | C2×Dic10 | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | #704 | — | 2 | 4 | no | — |
| 81 | #704 | — | 4 | 2 | no | — |
| 82 | #704 | — | 4 | 2 | no | — |
| 83 | C2×#170 | — | 2 | 2 | no | — |
| 84 | C2×#170 | — | 2 | 2 | no | — |
| 85 | C2×#170 | — | 2 | 1 | no | — |
| 86 | C2×#170 | — | 2 | 1 | no | — |
| 87 | C2×#170 | — | 4 | 2 | no | — |
| 88 | C2×#170 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C2×#170 | C22⋊C4 | 1 | 2 | yes | — |
| 90 | C2×#170 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 91 | #700 | — | 4 | 1 | no | — |
| 92 | #700 | — | 4 | 2 | no | — |
| 93 | #700 | — | 4 | 1 | no | — |
| 94 | D5⋊C8 | — | 2 | 1 | no | — |
| 95 | D5⋊C8 | — | 2 | 1 | no | — |
| 96 | D5⋊C8 | — | 4 | 1 | no | — |
| 97 | D5⋊C8 | — | 4 | 1 | no | — |
| 98 | D5⋊C8 | — | 4 | 1 | no | — |
| 99 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 100 | C5×C22⋊C4 | — | 2 | 2 | no | — |
| 101 | C42×C5 | C42 | 1 | 1 | yes | characteristic |
| 102 | C22⋊Dic5 | — | 2 | 2 | no | — |
| 103 | #685 | — | 2 | 2 | no | — |
| 104 | #683 | — | 2 | 2 | no | — |
| 105 | #683 | — | 4 | 2 | no | — |
| 106 | C4×Dic5 | — | 2 | 1 | no | — |
| 107 | C4×Dic5 | — | 4 | 2 | no | — |
| 108 | C4×Dic5 | C42 | 1 | 1 | yes | characteristic |
| 109 | C2×#131 | — | 5 | 1 | no | — |
| 110 | C2×#131 | — | 5 | 1 | no | — |
| 111 | #532 | — | 5 | 1 | no | — |
| 112 | #436 | — | 10 | 2 | no | — |
| 113 | C8×D4 | — | 10 | 2 | no | — |
| 114 | #423 | — | 5 | 1 | no | — |
| 115 | #413 | — | 5 | 2 | no | — |
| 116 | #410 | — | 5 | 2 | no | — |
| 117 | #408 | — | 5 | 1 | no | — |
| 118 | #408 | — | 5 | 1 | no | — |
| 119 | #405 | — | 10 | 2 | no | — |
| 120 | C2×C4×C8 | — | 5 | 1 | no | — |
| 121 | #356 | — | 5 | 2 | no | — |
| 122 | #355 | — | 5 | 2 | no | — |
| 123 | #354 | — | 5 | 1 | no | — |
| 124 | #354 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 125 | #353 | — | 5 | 2 | no | — |
| 126 | #344 | — | 10 | 2 | no | — |
| 127 | C23×C5 | C4×D4 | 1 | 2 | yes | — |
| 128 | C22×D5 | — | 2 | 1 | no | — |
| 129 | C22×D5 | #117 | 1 | 1 | yes | characteristic |
| 130 | C5×Q8 | — | 4 | 1 | no | — |
| 131 | C5×D4 | — | 4 | 1 | no | — |
| 132 | C2×C4×C5 | — | 2 | 1 | no | — |
| 133 | C2×C4×C5 | — | 2 | 1 | no | — |
| 134 | C2×C4×C5 | — | 4 | 2 | no | — |
| 135 | C2×C4×C5 | C2×C42 | 1 | 1 | yes | characteristic |
| 136 | C22⋊D5 | — | 4 | 2 | no | — |
| 137 | C22⋊D5 | — | 4 | 4 | no | — |
| 138 | C2×Dic5 | — | 2 | 1 | no | — |
| 139 | C2×Dic5 | — | 2 | 2 | no | — |
| 140 | C2×Dic5 | — | 2 | 2 | no | — |
| 141 | C2×Dic5 | — | 4 | 2 | no | — |
| 142 | C2×Dic5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 143 | C2×Dic5 | C4×D4 | 1 | 2 | yes | — |
| 144 | D20 | — | 4 | 1 | no | — |
| 145 | D20 | — | 4 | 2 | no | — |
| 146 | C4×D5 | — | 2 | 1 | no | — |
| 147 | C4×D5 | — | 2 | 1 | no | — |
| 148 | C4×D5 | — | 4 | 2 | no | — |
| 149 | C4×D5 | — | 4 | 1 | no | — |
| 150 | C4×D5 | — | 4 | 1 | no | — |
| 151 | Dic10 | — | 4 | 1 | no | — |
| 152 | #170 | — | 2 | 1 | no | — |
| 153 | #170 | — | 2 | 2 | no | — |
| 154 | #170 | — | 4 | 4 | no | — |
| 155 | #170 | — | 4 | 1 | no | — |
| 156 | #170 | — | 4 | 1 | no | — |
| 157 | #170 | #99 | 1 | 1 | yes | characteristic |
| 158 | #170 | #99 | 1 | 1 | yes | characteristic |
| 159 | C2×#41 | — | 5 | 1 | no | — |
| 160 | C2×#41 | — | 5 | 1 | no | — |
| 161 | #131 | — | 20 | 1 | no | — |
| 162 | #131 | — | 20 | 1 | no | — |
| 163 | #131 | — | 20 | 1 | no | — |
| 164 | #131 | — | 20 | 1 | no | — |
| 165 | C2×M16 | — | 5 | 1 | no | — |
| 166 | C2×M16 | — | 5 | 2 | no | — |
| 167 | C2×M16 | — | 5 | 1 | no | — |
| 168 | C2×M16 | — | 10 | 1 | no | — |
| 169 | C2×M16 | — | 10 | 2 | no | — |
| 170 | C22×C8 | — | 5 | 1 | no | — |
| 171 | C22×C8 | — | 5 | 2 | no | — |
| 172 | C22×C8 | — | 5 | 1 | no | — |
| 173 | C22×C8 | — | 10 | 1 | no | — |
| 174 | C4⋊Q8 | — | 5 | 1 | no | — |
| 175 | #127 | — | 5 | 1 | no | — |
| 176 | #124 | — | 5 | 1 | no | — |
| 177 | #124 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 178 | #121 | — | 10 | 2 | no | — |
| 179 | C4×D4 | — | 10 | 2 | no | — |
| 180 | #117 | — | 10 | 2 | no | — |
| 181 | C2×C42 | — | 5 | 1 | no | — |
| 182 | C4⋊C8 | — | 10 | 2 | no | — |
| 183 | #101 | — | 5 | 4 | no | — |
| 184 | #100 | — | 10 | 2 | no | — |
| 185 | #99 | — | 5 | 2 | no | — |
| 186 | #99 | — | 10 | 2 | no | — |
| 187 | #99 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 188 | C22⋊C8 | — | 10 | 2 | no | — |
| 189 | C22⋊C8 | — | 10 | 2 | no | — |
| 190 | #97 | — | 20 | 2 | no | — |
| 191 | C4×C8 | — | 10 | 1 | no | — |
| 192 | C4×C8 | — | 10 | 1 | no | — |
| 193 | C4×C8 | — | 20 | 2 | no | — |
| 194 | C22×C5 | — | 2 | 2 | no | — |
| 195 | C22×C5 | — | 4 | 2 | no | — |
| 196 | C22×C5 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 197 | C2×D5 | — | 2 | 1 | no | — |
| 198 | C2×D5 | — | 4 | 1 | no | — |
| 199 | C2×D5 | — | 4 | 2 | no | — |
| 200 | C2×D5 | — | 4 | 1 | no | — |
| 201 | C4×C5 | — | 2 | 1 | no | — |
| 202 | C4×C5 | — | 4 | 1 | no | — |
| 203 | C4×C5 | — | 4 | 1 | no | — |
| 204 | C4×C5 | — | 8 | 2 | no | — |
| 205 | Dic5 | — | 2 | 2 | no | — |
| 206 | Dic5 | — | 2 | 2 | no | — |
| 207 | Dic5 | — | 8 | 2 | no | — |
| 208 | Dic5 | C2×#99 | 1 | 1 | yes | characteristic |
| 209 | Dic5 | #410 | 1 | 1 | yes | characteristic |
| 210 | #41 | — | 20 | 1 | no | — |
| 211 | #41 | — | 20 | 1 | no | — |
| 212 | #41 | — | 20 | 1 | no | — |
| 213 | #41 | — | 20 | 1 | no | — |
| 214 | C2×Q8 | — | 5 | 1 | no | — |
| 215 | C2×Q8 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 216 | C2×D4 | — | 5 | 2 | no | — |
| 217 | C2×D4 | — | 5 | 1 | no | — |
| 218 | C2×D4 | — | 10 | 1 | no | — |
| 219 | C2×D4 | — | 10 | 2 | no | — |
| 220 | C2×D4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 221 | C22×C4 | — | 5 | 2 | no | — |
| 222 | C22×C4 | — | 5 | 1 | no | — |
| 223 | C22×C4 | — | 10 | 1 | no | — |
| 224 | C22×C4 | — | 10 | 1 | no | — |
| 225 | M16 | — | 10 | 4 | no | — |
| 226 | M16 | — | 20 | 1 | no | — |
| 227 | M16 | — | 20 | 2 | no | — |
| 228 | M16 | — | 20 | 2 | no | — |
| 229 | M16 | — | 20 | 2 | no | — |
| 230 | M16 | — | 20 | 1 | no | — |
| 231 | C2×C8 | — | 5 | 1 | no | — |
| 232 | C2×C8 | — | 5 | 1 | no | — |
| 233 | C2×C8 | — | 5 | 2 | no | — |
| 234 | C2×C8 | — | 10 | 2 | no | — |
| 235 | C2×C8 | — | 10 | 2 | no | — |
| 236 | C2×C8 | — | 10 | 1 | no | — |
| 237 | C2×C8 | — | 10 | 1 | no | — |
| 238 | C2×C8 | — | 10 | 1 | no | — |
| 239 | C2×C8 | — | 10 | 1 | no | — |
| 240 | C2×C8 | — | 20 | 1 | no | — |
| 241 | C2×C8 | — | 20 | 1 | no | — |
| 242 | C2×C8 | — | 20 | 2 | no | — |
| 243 | C2×C8 | — | 20 | 1 | no | — |
| 244 | C4⋊C4 | — | 10 | 2 | no | — |
| 245 | C4⋊C4 | — | 20 | 2 | no | — |
| 246 | C22⋊C4 | — | 2 | 2 | no | — |
| 247 | C22⋊C4 | — | 2 | 2 | no | — |
| 248 | C22⋊C4 | — | 10 | 2 | no | — |
| 249 | C22⋊C4 | — | 10 | 2 | no | — |
| 250 | C42 | — | 5 | 1 | no | — |
| 251 | C42 | — | 10 | 1 | no | — |
| 252 | C42 | — | 20 | 2 | no | — |
| 253 | C42 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 254 | C2×C5 | — | 2 | 1 | no | — |
| 255 | C2×C5 | — | 4 | 2 | no | — |
| 256 | C2×C5 | C4×#99 | 1 | 1 | yes | characteristic |
| 257 | D5 | — | 4 | 1 | no | — |
| 258 | D5 | — | 8 | 1 | no | — |
| 259 | C23 | — | 5 | 1 | no | — |
| 260 | C23 | — | 10 | 1 | no | — |
| 261 | C23 | D4×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 262 | Q8 | — | 4 | 1 | no | — |
| 263 | Q8 | — | 20 | 1 | no | — |
| 264 | D4 | — | 4 | 1 | no | — |
| 265 | D4 | — | 20 | 1 | no | — |
| 266 | D4 | — | 20 | 2 | no | — |
| 267 | D4 | — | 20 | 2 | no | — |
| 268 | D4 | — | 20 | 4 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 4 | 2 | no | — |
| 272 | C2×C4 | — | 5 | 1 | no | — |
| 273 | C2×C4 | — | 5 | 2 | no | — |
| 274 | C2×C4 | — | 10 | 1 | no | — |
| 275 | C2×C4 | — | 10 | 1 | no | — |
| 276 | C2×C4 | — | 10 | 2 | no | — |
| 277 | C2×C4 | — | 10 | 2 | no | — |
| 278 | C2×C4 | — | 10 | 1 | no | — |
| 279 | C2×C4 | — | 20 | 1 | no | — |
| 280 | C2×C4 | — | 20 | 1 | no | — |
| 281 | C2×C4 | — | 20 | 2 | no | — |
| 282 | C2×C4 | — | 20 | 2 | no | — |
| 283 | C2×C4 | C2×C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 284 | C8 | — | 5 | 1 | no | — |
| 285 | C8 | — | 5 | 1 | no | — |
| 286 | C8 | — | 10 | 1 | no | — |
| 287 | C8 | — | 10 | 2 | no | — |
| 288 | C8 | — | 20 | 1 | no | — |
| 289 | C8 | — | 20 | 1 | no | — |
| 290 | C8 | — | 20 | 4 | no | — |
| 291 | C5 | #12633 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 292 | C22 | — | 2 | 2 | no | — |
| 293 | C22 | — | 4 | 2 | no | — |
| 294 | C22 | — | 10 | 1 | no | — |
| 295 | C22 | — | 20 | 1 | no | — |
| 296 | C22 | — | 20 | 2 | no | — |
| 297 | C22 | — | 20 | 1 | no | — |
| 298 | C22 | C22⋊C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 2 | 1 | no | — |
| 300 | C4 | — | 4 | 1 | no | — |
| 301 | C4 | — | 4 | 1 | no | — |
| 302 | C4 | — | 5 | 1 | no | — |
| 303 | C4 | — | 5 | 1 | no | — |
| 304 | C4 | — | 8 | 2 | no | — |
| 305 | C4 | — | 10 | 2 | no | — |
| 306 | C4 | — | 10 | 2 | no | — |
| 307 | C4 | — | 40 | 2 | no | — |
| 308 | C2 | — | 2 | 1 | no | — |
| 309 | C2 | — | 4 | 2 | no | — |
| 310 | C2 | — | 20 | 1 | no | — |
| 311 | C2 | — | 40 | 1 | no | — |
| 312 | C2 | AGL(1,𝔽5)×#99 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #10603920 | C2 | semidirect |
| 5. #10598570 | C2 | semidirect |
| 6. C2×#4133 | C4 | semidirect |
| 9. #65787 | C4 | semidirect |
| 15. #65682 | C4 | semidirect |
| 16. C4×D5⋊C8 | C4 | semidirect |
| 18. #64929 | C4 | semidirect |
| 20. #64856 | C4 | semidirect |
| 291. C5 | #12633 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617309 | 2 |
| #10604549 | 2 |
| #10603920 | 2 |
| #10599951 | 2 |
| #10598570 | 2 |
| #12633 | 5 |