Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
890 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 890 conjugacy classes of subgroups — too many to draw. Only its 140 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 | #10619067 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10605010 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10605007 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10604745 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#64898 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #66283 | C4 | 1 | 1 | yes | characteristic |
| 7 | C5×#568 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×#4094 | C22 | 1 | 1 | yes | characteristic |
| 9 | #66084 | — | 2 | 4 | no | — |
| 10 | D5×#131 | — | 2 | 4 | no | — |
| 11 | #66079 | — | 2 | 2 | no | — |
| 12 | #66079 | C4 | 1 | 2 | yes | — |
| 13 | C2×#4053 | — | 2 | 2 | no | — |
| 14 | C2×#4053 | C22 | 1 | 2 | yes | — |
| 15 | C2×#3998 | C22 | 1 | 2 | yes | complemented |
| 16 | C2×#3998 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C22×#674 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #64912 | — | 2 | 2 | no | — |
| 19 | C22⋊#674 | — | 2 | 2 | no | — |
| 20 | #64898 | — | 2 | 4 | no | — |
| 21 | #64898 | — | 2 | 4 | no | — |
| 22 | #12685 | — | 5 | 1 | no | maximal, Sylow |
| 23 | C2×C5×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | #4150 | — | 2 | 2 | no | — |
| 25 | #4149 | — | 2 | 2 | no | — |
| 26 | D5×#41 | — | 2 | 4 | no | — |
| 27 | #4147 | C8 | 1 | 2 | yes | — |
| 28 | #4144 | — | 2 | 2 | no | — |
| 29 | #4144 | C8 | 1 | 2 | yes | — |
| 30 | C2×#709 | — | 2 | 2 | no | — |
| 31 | C2×#709 | C2×C4 | 1 | 2 | yes | — |
| 32 | C5×#131 | — | 2 | 4 | no | — |
| 33 | C2×C5×M16 | — | 2 | 2 | no | — |
| 34 | C2×C5×M16 | C2×C4 | 1 | 2 | yes | — |
| 35 | #4094 | — | 2 | 2 | no | — |
| 36 | #4094 | — | 2 | 2 | no | — |
| 37 | C2×#681 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C2×#681 | D4 | 1 | 2 | yes | — |
| 39 | C22×#168 | D4 | 1 | 2 | yes | — |
| 40 | C22×#168 | C23 | 1 | 1 | yes | characteristic |
| 41 | #4053 | — | 2 | 4 | no | — |
| 42 | #4053 | — | 2 | 4 | no | — |
| 43 | #4053 | — | 2 | 4 | no | — |
| 44 | #4053 | — | 2 | 2 | no | — |
| 45 | #4053 | — | 2 | 2 | no | — |
| 46 | #4053 | D4 | 1 | 4 | yes | — |
| 47 | #4053 | D4 | 1 | 4 | yes | — |
| 48 | D5×M16 | — | 2 | 8 | no | — |
| 49 | D5×M16 | — | 2 | 4 | no | — |
| 50 | #4047 | — | 2 | 8 | no | — |
| 51 | #4047 | — | 2 | 4 | no | — |
| 52 | C2×#676 | — | 2 | 4 | no | — |
| 53 | C2×#676 | — | 2 | 2 | no | — |
| 54 | C2×C8×D5 | — | 2 | 4 | no | — |
| 55 | C2×C8×D5 | — | 2 | 2 | no | — |
| 56 | #3998 | — | 2 | 2 | no | — |
| 57 | #3998 | — | 2 | 2 | no | — |
| 58 | #3998 | — | 2 | 2 | no | — |
| 59 | C2×#674 | — | 2 | 2 | no | — |
| 60 | C2×#674 | — | 2 | 2 | no | — |
| 61 | C2×#674 | — | 2 | 1 | no | — |
| 62 | C2×#674 | — | 2 | 1 | no | — |
| 63 | #3573 | — | 5 | 1 | no | — |
| 64 | #2117 | — | 5 | 2 | no | — |
| 65 | C2×#350 | — | 5 | 2 | no | — |
| 66 | #2115 | — | 5 | 1 | no | — |
| 67 | #2113 | — | 5 | 1 | no | — |
| 68 | C5×#41 | — | 2 | 2 | no | — |
| 69 | C5×#41 | — | 2 | 2 | no | — |
| 70 | C2×C5×Q8 | C2×C8 | 1 | 1 | yes | characteristic |
| 71 | C2×C5×D4 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C2×C5×D4 | M16 | 1 | 2 | yes | — |
| 73 | C22×C4×C5 | C22⋊C4 | 1 | 2 | yes | — |
| 74 | C22×C4×C5 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C2×C22⋊D5 | — | 2 | 4 | no | — |
| 76 | C2×C22⋊D5 | — | 2 | 2 | no | — |
| 77 | #713 | — | 2 | 4 | no | — |
| 78 | Q8×D5 | — | 2 | 4 | no | — |
| 79 | #711 | — | 2 | 8 | no | — |
| 80 | #711 | — | 2 | 4 | no | — |
| 81 | D4×D5 | — | 2 | 8 | no | — |
| 82 | D4×D5 | — | 2 | 4 | no | — |
| 83 | #709 | — | 2 | 4 | no | — |
| 84 | #709 | — | 2 | 4 | no | — |
| 85 | #709 | — | 2 | 4 | no | — |
| 86 | #709 | — | 2 | 2 | no | — |
| 87 | #709 | — | 2 | 2 | no | — |
| 88 | #709 | C22⋊C4 | 1 | 4 | yes | — |
| 89 | #709 | C2×C8 | 1 | 4 | yes | — |
| 90 | C2×D20 | — | 2 | 2 | no | — |
| 91 | C2×D20 | M16 | 1 | 2 | yes | — |
| 92 | C2×C4×D5 | — | 2 | 4 | no | — |
| 93 | C2×C4×D5 | — | 2 | 2 | no | — |
| 94 | C2×Dic10 | — | 2 | 2 | no | — |
| 95 | C2×Dic10 | M16 | 1 | 2 | yes | — |
| 96 | C5×M16 | — | 2 | 4 | no | — |
| 97 | C5×M16 | — | 2 | 2 | no | — |
| 98 | C5×M16 | C22⋊C4 | 1 | 4 | yes | — |
| 99 | C2×C5×C8 | — | 2 | 4 | no | — |
| 100 | C2×C5×C8 | — | 2 | 2 | no | — |
| 101 | #681 | — | 2 | 2 | no | — |
| 102 | #681 | — | 2 | 4 | no | — |
| 103 | C2×#168 | — | 2 | 2 | no | — |
| 104 | C2×#168 | — | 2 | 2 | no | — |
| 105 | C2×#168 | C22⋊C4 | 1 | 2 | yes | — |
| 106 | C2×#168 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C2×#168 | C2×D4 | 1 | 2 | yes | — |
| 108 | C2×#168 | C2×D4 | 1 | 2 | yes | — |
| 109 | C2×#168 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | #676 | — | 2 | 8 | no | — |
| 111 | #676 | — | 2 | 4 | no | — |
| 112 | #676 | — | 2 | 4 | no | — |
| 113 | C8×D5 | — | 2 | 8 | no | — |
| 114 | C8×D5 | — | 2 | 4 | no | — |
| 115 | C8×D5 | — | 2 | 4 | no | — |
| 116 | #674 | — | 4 | 1 | no | — |
| 117 | #674 | — | 4 | 1 | no | — |
| 118 | #674 | — | 4 | 2 | no | — |
| 119 | #585 | — | 5 | 1 | no | — |
| 120 | #568 | — | 5 | 2 | no | — |
| 121 | #568 | — | 10 | 2 | no | — |
| 122 | #568 | — | 10 | 4 | no | — |
| 123 | #568 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 124 | C2×#131 | — | 5 | 1 | no | — |
| 125 | C2×#131 | — | 5 | 2 | no | — |
| 126 | C2×#131 | — | 10 | 2 | no | — |
| 127 | C2×#131 | — | 10 | 4 | no | — |
| 128 | C2×M32 | — | 5 | 1 | no | — |
| 129 | C2×M32 | — | 5 | 2 | no | — |
| 130 | C22×C16 | — | 5 | 1 | no | — |
| 131 | #350 | — | 10 | 4 | no | — |
| 132 | #350 | — | 10 | 4 | no | — |
| 133 | #349 | — | 10 | 2 | no | — |
| 134 | C22⋊C16 | — | 10 | 2 | no | — |
| 135 | C23×C5 | C22⋊C8 | 1 | 2 | yes | — |
| 136 | C23×C5 | C2×M16 | 1 | 1 | yes | characteristic |
| 137 | C22×D5 | — | 2 | 4 | no | — |
| 138 | C22×D5 | — | 2 | 2 | no | — |
| 139 | C5×Q8 | — | 2 | 2 | no | — |
| 140 | C5×D4 | — | 2 | 4 | no | — |
| 141 | C5×D4 | — | 2 | 2 | no | — |
| 142 | C2×C4×C5 | — | 2 | 2 | no | — |
| 143 | C2×C4×C5 | — | 2 | 2 | no | — |
| 144 | C2×C4×C5 | C22⋊C8 | 1 | 2 | yes | — |
| 145 | C2×C4×C5 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 146 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C5 | C22×C8 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C5 | C2×M16 | 1 | 2 | yes | — |
| 149 | C22⋊D5 | — | 2 | 8 | no | — |
| 150 | C22⋊D5 | — | 2 | 4 | no | — |
| 151 | C22⋊D5 | — | 2 | 4 | no | — |
| 152 | C2×Dic5 | — | 2 | 4 | no | — |
| 153 | C2×Dic5 | — | 2 | 2 | no | — |
| 154 | D20 | — | 2 | 4 | no | — |
| 155 | D20 | — | 2 | 2 | no | — |
| 156 | D20 | C22⋊C8 | 1 | 4 | yes | — |
| 157 | C4×D5 | — | 2 | 8 | no | — |
| 158 | C4×D5 | — | 2 | 4 | no | — |
| 159 | C4×D5 | — | 2 | 4 | no | — |
| 160 | Dic10 | — | 2 | 4 | no | — |
| 161 | Dic10 | — | 2 | 2 | no | — |
| 162 | Dic10 | C22⋊C8 | 1 | 4 | yes | — |
| 163 | C5×C8 | — | 2 | 4 | no | — |
| 164 | #168 | — | 2 | 2 | no | — |
| 165 | #168 | #120 | 1 | 1 | yes | characteristic |
| 166 | #168 | #120 | 1 | 2 | yes | — |
| 167 | #168 | #120 | 1 | 1 | yes | characteristic |
| 168 | 2−1+4 | — | 5 | 2 | no | — |
| 169 | 2−1+4 | — | 10 | 2 | no | — |
| 170 | H(5,𝔽2) | — | 5 | 2 | no | — |
| 171 | H(5,𝔽2) | — | 10 | 2 | no | — |
| 172 | H(5,𝔽2) | — | 10 | 2 | no | — |
| 173 | C2×#41 | — | 5 | 2 | no | — |
| 174 | C2×#41 | — | 10 | 4 | no | — |
| 175 | C2×#41 | — | 10 | 2 | no | — |
| 176 | C2×#41 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 177 | #131 | — | 2 | 4 | no | — |
| 178 | #131 | — | 5 | 4 | no | — |
| 179 | #131 | — | 5 | 4 | no | — |
| 180 | #131 | — | 10 | 4 | no | — |
| 181 | #131 | — | 10 | 8 | no | — |
| 182 | #131 | — | 10 | 2 | no | — |
| 183 | #131 | — | 10 | 2 | no | — |
| 184 | #131 | — | 10 | 4 | no | — |
| 185 | #131 | — | 10 | 2 | no | — |
| 186 | #131 | — | 10 | 2 | no | — |
| 187 | #131 | — | 10 | 4 | no | — |
| 188 | #131 | — | 10 | 4 | no | — |
| 189 | C2×M16 | — | 2 | 2 | no | — |
| 190 | C2×M16 | — | 5 | 2 | no | — |
| 191 | C2×M16 | — | 5 | 1 | no | — |
| 192 | C2×M16 | — | 10 | 2 | no | — |
| 193 | C2×M16 | — | 10 | 4 | no | — |
| 194 | C2×M16 | — | 10 | 8 | no | — |
| 195 | C2×M16 | — | 10 | 4 | no | — |
| 196 | C2×M16 | C2×AGL(1,𝔽5) | 1 | 2 | yes | — |
| 197 | C22×C8 | — | 5 | 1 | no | — |
| 198 | C22×C8 | — | 5 | 2 | no | — |
| 199 | C22×C8 | — | 10 | 2 | no | — |
| 200 | C22×C8 | — | 10 | 4 | no | — |
| 201 | M32 | — | 10 | 2 | no | — |
| 202 | M32 | — | 10 | 2 | no | — |
| 203 | M32 | — | 10 | 2 | no | — |
| 204 | C2×C16 | — | 10 | 1 | no | — |
| 205 | C2×C16 | — | 10 | 1 | no | — |
| 206 | C2×C16 | — | 10 | 2 | no | — |
| 207 | C2×C16 | — | 10 | 2 | no | — |
| 208 | C22×C5 | — | 2 | 2 | no | — |
| 209 | C22×C5 | — | 4 | 1 | no | — |
| 210 | C22×C5 | — | 4 | 2 | no | — |
| 211 | C22×C5 | C2×C22⋊C8 | 1 | 1 | yes | characteristic |
| 212 | C22×C5 | #407 | 1 | 2 | yes | — |
| 213 | C2×D5 | — | 2 | 4 | no | — |
| 214 | C2×D5 | — | 8 | 2 | no | — |
| 215 | C2×D5 | — | 8 | 4 | no | — |
| 216 | C4×C5 | — | 2 | 2 | no | — |
| 217 | C4×C5 | #379 | 1 | 1 | yes | characteristic |
| 218 | C4×C5 | C2×C22⋊C8 | 1 | 2 | yes | — |
| 219 | C4×C5 | #407 | 1 | 1 | yes | characteristic |
| 220 | Dic5 | — | 2 | 4 | no | — |
| 221 | #41 | — | 2 | 2 | no | — |
| 222 | #41 | — | 2 | 2 | no | — |
| 223 | #41 | — | 5 | 4 | no | — |
| 224 | #41 | — | 5 | 4 | no | — |
| 225 | #41 | — | 10 | 4 | no | — |
| 226 | #41 | — | 10 | 4 | no | — |
| 227 | #41 | — | 10 | 4 | no | — |
| 228 | #41 | — | 10 | 8 | no | — |
| 229 | #41 | — | 10 | 4 | no | — |
| 230 | #41 | — | 10 | 2 | no | — |
| 231 | #41 | — | 10 | 2 | no | — |
| 232 | #41 | — | 10 | 4 | no | — |
| 233 | C2×Q8 | — | 5 | 2 | no | — |
| 234 | C2×Q8 | — | 10 | 2 | no | — |
| 235 | C2×Q8 | — | 10 | 4 | no | — |
| 236 | C2×Q8 | D5⋊C8 | 1 | 1 | yes | characteristic |
| 237 | C2×D4 | — | 5 | 2 | no | — |
| 238 | C2×D4 | — | 10 | 2 | no | — |
| 239 | C2×D4 | — | 10 | 4 | no | — |
| 240 | C2×D4 | — | 10 | 8 | no | — |
| 241 | C2×D4 | — | 10 | 2 | no | — |
| 242 | C2×D4 | — | 10 | 4 | no | — |
| 243 | C2×D4 | D5⋊C8 | 1 | 1 | yes | characteristic |
| 244 | C2×D4 | #700 | 1 | 2 | yes | — |
| 245 | C22×C4 | — | 10 | 4 | no | — |
| 246 | C22×C4 | — | 10 | 2 | no | — |
| 247 | C22×C4 | C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 248 | C22×C4 | C22×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 249 | M16 | — | 2 | 4 | no | — |
| 250 | M16 | — | 2 | 2 | no | — |
| 251 | M16 | — | 10 | 4 | no | — |
| 252 | M16 | — | 10 | 4 | no | — |
| 253 | M16 | — | 10 | 8 | no | — |
| 254 | M16 | — | 10 | 4 | no | — |
| 255 | M16 | — | 10 | 2 | no | — |
| 256 | M16 | C22⋊AGL(1,𝔽5) | 1 | 4 | yes | — |
| 257 | C2×C8 | — | 2 | 4 | no | — |
| 258 | C2×C8 | — | 2 | 2 | no | — |
| 259 | C2×C8 | — | 5 | 1 | no | — |
| 260 | C2×C8 | — | 5 | 2 | no | — |
| 261 | C2×C8 | — | 5 | 2 | no | — |
| 262 | C2×C8 | — | 5 | 1 | no | — |
| 263 | C2×C8 | — | 5 | 2 | no | — |
| 264 | C2×C8 | — | 10 | 4 | no | — |
| 265 | C2×C8 | — | 10 | 4 | no | — |
| 266 | C2×C8 | — | 10 | 8 | no | — |
| 267 | C2×C8 | — | 10 | 2 | no | — |
| 268 | C2×C8 | — | 10 | 2 | no | — |
| 269 | C16 | — | 20 | 1 | no | — |
| 270 | C16 | — | 20 | 1 | no | — |
| 271 | C16 | — | 20 | 2 | no | — |
| 272 | C2×C5 | — | 2 | 2 | no | — |
| 273 | C2×C5 | — | 2 | 1 | no | — |
| 274 | C2×C5 | — | 4 | 2 | no | — |
| 275 | C2×C5 | #1779 | 1 | 1 | yes | characteristic |
| 276 | D5 | — | 4 | 4 | no | — |
| 277 | C23 | — | 10 | 4 | no | — |
| 278 | C23 | — | 10 | 2 | no | — |
| 279 | C23 | #4003 | 1 | 2 | yes | — |
| 280 | C23 | C2×#700 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 2 | 2 | no | — |
| 282 | Q8 | — | 5 | 4 | no | — |
| 283 | Q8 | — | 10 | 4 | no | — |
| 284 | Q8 | — | 10 | 2 | no | — |
| 285 | D4 | — | 2 | 4 | no | — |
| 286 | D4 | — | 2 | 2 | no | — |
| 287 | D4 | — | 5 | 4 | no | — |
| 288 | D4 | — | 10 | 4 | no | — |
| 289 | D4 | — | 10 | 8 | no | — |
| 290 | D4 | — | 10 | 4 | no | — |
| 291 | D4 | — | 10 | 4 | no | — |
| 292 | D4 | — | 10 | 2 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 10 | 8 | no | — |
| 296 | C2×C4 | — | 10 | 4 | no | — |
| 297 | C2×C4 | — | 10 | 2 | no | — |
| 298 | C2×C4 | — | 10 | 4 | no | — |
| 299 | C2×C4 | — | 10 | 4 | no | — |
| 300 | C2×C4 | #4003 | 1 | 2 | yes | — |
| 301 | C2×C4 | C2×D5⋊C8 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | D5⋊M16 | 1 | 2 | yes | — |
| 303 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 2 | yes | — |
| 304 | C2×C4 | C2×C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 305 | C8 | — | 2 | 4 | no | — |
| 306 | C8 | — | 5 | 1 | no | — |
| 307 | C8 | — | 5 | 1 | no | — |
| 308 | C8 | — | 5 | 2 | no | — |
| 309 | C8 | — | 10 | 2 | no | — |
| 310 | C5 | #12685 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 4 | 1 | no | — |
| 313 | C22 | — | 4 | 2 | no | — |
| 314 | C22 | — | 10 | 4 | no | — |
| 315 | C22 | — | 40 | 2 | no | — |
| 316 | C22 | — | 40 | 4 | no | — |
| 317 | C22 | C2×#4003 | 1 | 1 | yes | characteristic |
| 318 | C22 | #65756 | 1 | 2 | yes | — |
| 319 | C4 | — | 2 | 2 | no | — |
| 320 | C4 | — | 10 | 4 | no | — |
| 321 | C4 | D5⋊(C22⋊C8) | 1 | 2 | yes | — |
| 322 | C4 | #65755 | 1 | 1 | yes | characteristic |
| 323 | C4 | #65800 | 1 | 1 | yes | characteristic |
| 324 | C2 | — | 2 | 1 | no | — |
| 325 | C2 | — | 2 | 2 | no | — |
| 326 | C2 | — | 4 | 2 | no | — |
| 327 | C2 | — | 20 | 4 | no | — |
| 328 | C2 | #10604898 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10605010 | C2 | semidirect |
| 3. #10605007 | C2 | semidirect |
| 4. #10604745 | C2 | semidirect |
| 5. C2×#64898 | C2 | semidirect |
| 15. C2×#3998 | C22 | semidirect |
| 16. C2×#3998 | C22 | semidirect |
| 17. C22×#674 | C22 | semidirect |
| 310. C5 | #12685 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10619067 | 2 |
| #10605010 | 2 |
| #10605007 | 2 |
| #10604745 | 2 |
| C2×#64898 | 2 |
| #12685 | 5 |