Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
397 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 74 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 | C2×#2098 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#1391 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#1315 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7438 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 5 | C2×#430 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#424 | — | 2 | 1 | no | — |
| 7 | C2×#421 | — | 2 | 1 | no | — |
| 8 | C2×#410 | — | 2 | 1 | no | — |
| 9 | C2×#408 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C22⋊C16 | — | 2 | 1 | no | — |
| 11 | #2098 | — | 2 | 1 | no | — |
| 12 | #2098 | — | 2 | 1 | no | — |
| 13 | #2098 | C4 | 1 | 2 | yes | complemented |
| 14 | #2098 | C22 | 1 | 2 | yes | — |
| 15 | C2×H(3,ℤ/4) | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | #1391 | — | 2 | 1 | no | — |
| 17 | #1391 | C22 | 1 | 2 | yes | — |
| 18 | #1315 | — | 2 | 1 | no | — |
| 19 | #1315 | C22 | 1 | 2 | yes | — |
| 20 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 21 | C22×M16 | — | 2 | 1 | no | — |
| 22 | C22×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×#117 | — | 2 | 1 | no | — |
| 25 | C2×#117 | — | 4 | 1 | no | — |
| 26 | C22×C16 | — | 4 | 1 | no | — |
| 27 | #430 | — | 2 | 1 | no | — |
| 28 | #430 | — | 4 | 1 | no | — |
| 29 | #430 | C2×C4 | 1 | 2 | yes | complemented |
| 30 | C2×#108 | — | 4 | 1 | no | — |
| 31 | #424 | — | 2 | 2 | no | — |
| 32 | #424 | — | 2 | 2 | no | — |
| 33 | #424 | — | 4 | 2 | no | — |
| 34 | C2×C4⋊C8 | — | 4 | 2 | no | — |
| 35 | #421 | — | 2 | 2 | no | — |
| 36 | #421 | — | 2 | 2 | no | — |
| 37 | #421 | — | 4 | 2 | no | — |
| 38 | C2×#104 | — | 4 | 2 | no | — |
| 39 | #410 | — | 2 | 2 | no | — |
| 40 | #410 | — | 2 | 2 | no | — |
| 41 | #410 | — | 4 | 1 | no | — |
| 42 | #410 | — | 4 | 1 | no | — |
| 43 | C2×#99 | — | 2 | 2 | no | — |
| 44 | C2×#99 | — | 4 | 1 | no | — |
| 45 | #408 | — | 4 | 1 | no | — |
| 46 | #408 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | #408 | D4 | 1 | 1 | yes | characteristic |
| 48 | #408 | D4 | 1 | 1 | yes | characteristic |
| 49 | #408 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×C22⋊C8 | — | 4 | 1 | no | — |
| 51 | C22⋊C16 | — | 2 | 2 | no | — |
| 52 | C22⋊C16 | — | 4 | 1 | no | — |
| 53 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 54 | H(3,ℤ/4) | C2×C4 | 1 | 2 | yes | complemented |
| 55 | C2×#41 | — | 2 | 1 | no | — |
| 56 | C2×#41 | — | 2 | 1 | no | — |
| 57 | C2×#41 | — | 4 | 1 | no | — |
| 58 | C2×#41 | — | 4 | 1 | no | — |
| 59 | C2×#41 | — | 4 | 1 | no | — |
| 60 | C2×#41 | — | 4 | 1 | no | — |
| 61 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C2×#41 | SD16 | 1 | 1 | yes | characteristic |
| 63 | C2×#41 | Q16 | 1 | 1 | yes | characteristic |
| 64 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C22×Q8 | Q16 | 1 | 1 | yes | characteristic |
| 66 | C22×D4 | — | 2 | 1 | no | — |
| 67 | C22×D4 | SD16 | 1 | 1 | yes | characteristic |
| 68 | C23×C4 | — | 2 | 1 | no | — |
| 69 | C23×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×M16 | — | 2 | 1 | no | — |
| 71 | C2×M16 | — | 2 | 2 | no | — |
| 72 | C2×M16 | — | 4 | 1 | no | — |
| 73 | C2×M16 | — | 4 | 1 | no | — |
| 74 | C2×M16 | — | 4 | 1 | no | — |
| 75 | C2×M16 | — | 4 | 1 | no | — |
| 76 | C2×M16 | — | 4 | 1 | no | — |
| 77 | C2×M16 | — | 4 | 1 | no | — |
| 78 | C2×M16 | — | 4 | 1 | no | — |
| 79 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 80 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 81 | C22×C8 | — | 2 | 1 | no | — |
| 82 | C22×C8 | — | 2 | 1 | no | — |
| 83 | C22×C8 | — | 2 | 1 | no | — |
| 84 | C22×C8 | — | 4 | 1 | no | — |
| 85 | C22×C8 | — | 4 | 1 | no | — |
| 86 | C22×C8 | — | 4 | 1 | no | — |
| 87 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | #117 | — | 2 | 2 | no | — |
| 90 | #117 | — | 2 | 2 | no | — |
| 91 | #117 | — | 4 | 1 | no | — |
| 92 | #117 | — | 4 | 1 | no | — |
| 93 | #117 | — | 4 | 2 | no | — |
| 94 | #117 | — | 4 | 2 | no | — |
| 95 | #117 | — | 8 | 1 | no | — |
| 96 | #117 | — | 8 | 1 | no | — |
| 97 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 98 | C2×C4⋊C4 | — | 8 | 1 | no | — |
| 99 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 100 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 101 | C2×C42 | — | 2 | 2 | no | — |
| 102 | C2×C42 | — | 8 | 1 | no | — |
| 103 | C2×C16 | — | 4 | 2 | no | — |
| 104 | C2×C16 | — | 8 | 1 | no | — |
| 105 | C2×C16 | — | 8 | 1 | no | — |
| 106 | #108 | — | 4 | 2 | no | — |
| 107 | #108 | — | 4 | 2 | no | — |
| 108 | C4⋊C8 | — | 4 | 4 | no | — |
| 109 | C4⋊C8 | — | 4 | 4 | no | — |
| 110 | #104 | — | 4 | 4 | no | — |
| 111 | #104 | — | 4 | 4 | no | — |
| 112 | #99 | — | 2 | 4 | no | — |
| 113 | #99 | — | 4 | 2 | no | — |
| 114 | #99 | — | 4 | 2 | no | — |
| 115 | #99 | — | 4 | 2 | no | — |
| 116 | C22⋊C8 | — | 4 | 1 | no | — |
| 117 | C22⋊C8 | — | 4 | 1 | no | — |
| 118 | C22⋊C8 | — | 4 | 1 | no | — |
| 119 | C22⋊C8 | — | 4 | 1 | no | — |
| 120 | C24 | — | 2 | 1 | no | — |
| 121 | C24 | #104 | 1 | 1 | yes | characteristic |
| 122 | #41 | — | 4 | 1 | no | — |
| 123 | #41 | — | 4 | 1 | no | — |
| 124 | #41 | — | 4 | 1 | no | — |
| 125 | #41 | — | 4 | 1 | no | — |
| 126 | #41 | — | 4 | 1 | no | — |
| 127 | #41 | — | 4 | 1 | no | — |
| 128 | #41 | — | 4 | 1 | no | — |
| 129 | #41 | — | 4 | 1 | no | — |
| 130 | #41 | — | 8 | 1 | no | — |
| 131 | #41 | — | 8 | 1 | no | — |
| 132 | #41 | — | 8 | 1 | no | — |
| 133 | #41 | — | 8 | 1 | no | — |
| 134 | C2×Q8 | — | 2 | 1 | no | — |
| 135 | C2×Q8 | — | 4 | 1 | no | — |
| 136 | C2×Q8 | — | 4 | 1 | no | — |
| 137 | C2×Q8 | #103 | 1 | 1 | yes | characteristic |
| 138 | C2×Q8 | C2×Q16 | 1 | 1 | yes | characteristic |
| 139 | C2×D4 | — | 2 | 1 | no | — |
| 140 | C2×D4 | — | 2 | 1 | no | — |
| 141 | C2×D4 | — | 2 | 1 | no | — |
| 142 | C2×D4 | — | 2 | 1 | no | — |
| 143 | C2×D4 | — | 2 | 1 | no | — |
| 144 | C2×D4 | — | 4 | 1 | no | — |
| 145 | C2×D4 | — | 4 | 1 | no | — |
| 146 | C2×D4 | — | 8 | 1 | no | — |
| 147 | C2×D4 | — | 8 | 1 | no | — |
| 148 | C2×D4 | #103 | 1 | 1 | yes | characteristic |
| 149 | C2×D4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 150 | C22×C4 | — | 2 | 1 | no | — |
| 151 | C22×C4 | — | 2 | 1 | no | — |
| 152 | C22×C4 | — | 2 | 1 | no | — |
| 153 | C22×C4 | — | 2 | 1 | no | — |
| 154 | C22×C4 | — | 2 | 1 | no | — |
| 155 | C22×C4 | — | 2 | 1 | no | — |
| 156 | C22×C4 | — | 2 | 1 | no | — |
| 157 | C22×C4 | — | 2 | 1 | no | — |
| 158 | C22×C4 | — | 2 | 1 | no | — |
| 159 | C22×C4 | — | 4 | 1 | no | — |
| 160 | C22×C4 | — | 4 | 1 | no | — |
| 161 | C22×C4 | — | 4 | 1 | no | — |
| 162 | C22×C4 | — | 4 | 1 | no | — |
| 163 | C22×C4 | — | 4 | 2 | no | — |
| 164 | C22×C4 | — | 8 | 2 | no | — |
| 165 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 166 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 167 | C22×C4 | #103 | 1 | 1 | yes | characteristic |
| 168 | C22×C4 | #103 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | #104 | 1 | 1 | yes | characteristic |
| 170 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 171 | M16 | — | 4 | 2 | no | — |
| 172 | M16 | — | 4 | 1 | no | — |
| 173 | M16 | — | 4 | 1 | no | — |
| 174 | M16 | — | 4 | 2 | no | — |
| 175 | M16 | — | 4 | 2 | no | — |
| 176 | M16 | — | 8 | 1 | no | — |
| 177 | M16 | — | 8 | 1 | no | — |
| 178 | C2×C8 | — | 2 | 1 | no | — |
| 179 | C2×C8 | — | 2 | 1 | no | — |
| 180 | C2×C8 | — | 2 | 1 | no | — |
| 181 | C2×C8 | — | 2 | 1 | no | — |
| 182 | C2×C8 | — | 4 | 1 | no | — |
| 183 | C2×C8 | — | 4 | 1 | no | — |
| 184 | C2×C8 | — | 4 | 1 | no | — |
| 185 | C2×C8 | — | 4 | 1 | no | — |
| 186 | C2×C8 | — | 4 | 1 | no | — |
| 187 | C2×C8 | — | 4 | 1 | no | — |
| 188 | C2×C8 | — | 4 | 2 | no | — |
| 189 | C2×C8 | — | 8 | 1 | no | — |
| 190 | C2×C8 | — | 8 | 1 | no | — |
| 191 | C2×C8 | — | 8 | 1 | no | — |
| 192 | C4⋊C4 | — | 4 | 2 | no | — |
| 193 | C4⋊C4 | — | 4 | 2 | no | — |
| 194 | C4⋊C4 | — | 8 | 2 | no | — |
| 195 | C4⋊C4 | — | 8 | 2 | no | — |
| 196 | C22⋊C4 | — | 4 | 2 | no | — |
| 197 | C22⋊C4 | — | 4 | 2 | no | — |
| 198 | C22⋊C4 | — | 4 | 4 | no | — |
| 199 | C22⋊C4 | — | 8 | 2 | no | — |
| 200 | C42 | — | 2 | 4 | no | — |
| 201 | C42 | — | 4 | 2 | no | — |
| 202 | C42 | — | 8 | 2 | no | — |
| 203 | C42 | — | 8 | 2 | no | — |
| 204 | C16 | — | 8 | 2 | no | — |
| 205 | C23 | — | 2 | 1 | no | — |
| 206 | C23 | — | 2 | 1 | no | — |
| 207 | C23 | — | 2 | 1 | no | — |
| 208 | C23 | — | 2 | 1 | no | — |
| 209 | C23 | — | 4 | 2 | no | — |
| 210 | C23 | — | 4 | 1 | no | — |
| 211 | C23 | — | 4 | 1 | no | — |
| 212 | C23 | — | 8 | 2 | no | — |
| 213 | C23 | #328 | 1 | 1 | yes | characteristic |
| 214 | C23 | #328 | 1 | 1 | yes | characteristic |
| 215 | C23 | C2×#104 | 1 | 1 | yes | characteristic |
| 216 | Q8 | — | 4 | 1 | no | — |
| 217 | Q8 | — | 4 | 1 | no | — |
| 218 | Q8 | — | 8 | 1 | no | — |
| 219 | D4 | — | 4 | 1 | no | — |
| 220 | D4 | — | 4 | 1 | no | — |
| 221 | D4 | — | 8 | 1 | no | — |
| 222 | D4 | — | 8 | 1 | no | — |
| 223 | D4 | — | 8 | 1 | no | — |
| 224 | D4 | — | 8 | 1 | no | — |
| 225 | D4 | — | 8 | 1 | no | — |
| 226 | C2×C4 | — | 2 | 1 | no | — |
| 227 | C2×C4 | — | 2 | 1 | no | — |
| 228 | C2×C4 | — | 2 | 1 | no | — |
| 229 | C2×C4 | — | 2 | 1 | no | — |
| 230 | C2×C4 | — | 2 | 1 | no | — |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 2 | 1 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 2 | 1 | no | — |
| 235 | C2×C4 | — | 2 | 1 | no | — |
| 236 | C2×C4 | — | 2 | 1 | no | — |
| 237 | C2×C4 | — | 2 | 1 | no | — |
| 238 | C2×C4 | — | 4 | 1 | no | — |
| 239 | C2×C4 | — | 4 | 1 | no | — |
| 240 | C2×C4 | — | 4 | 1 | no | — |
| 241 | C2×C4 | — | 4 | 1 | no | — |
| 242 | C2×C4 | — | 4 | 1 | no | — |
| 243 | C2×C4 | — | 4 | 1 | no | — |
| 244 | C2×C4 | — | 4 | 1 | no | — |
| 245 | C2×C4 | — | 4 | 1 | no | — |
| 246 | C2×C4 | — | 4 | 1 | no | — |
| 247 | C2×C4 | — | 4 | 1 | no | — |
| 248 | C2×C4 | — | 4 | 4 | no | — |
| 249 | C2×C4 | — | 8 | 4 | no | — |
| 250 | C2×C4 | — | 8 | 4 | no | — |
| 251 | C2×C4 | — | 16 | 4 | no | — |
| 252 | C2×C4 | #328 | 1 | 1 | yes | characteristic |
| 253 | C2×C4 | #328 | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | #352 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | #352 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | #353 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | #353 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | C2×#103 | 1 | 1 | yes | characteristic |
| 260 | C8 | — | 4 | 1 | no | — |
| 261 | C8 | — | 4 | 1 | no | — |
| 262 | C8 | — | 4 | 1 | no | — |
| 263 | C8 | — | 4 | 1 | no | — |
| 264 | C8 | — | 8 | 2 | no | — |
| 265 | C22 | — | 2 | 2 | no | — |
| 266 | C22 | — | 2 | 1 | no | — |
| 267 | C22 | — | 2 | 1 | no | — |
| 268 | C22 | — | 4 | 1 | no | — |
| 269 | C22 | — | 4 | 1 | no | — |
| 270 | C22 | — | 4 | 2 | no | — |
| 271 | C22 | — | 4 | 1 | no | — |
| 272 | C22 | — | 4 | 1 | no | — |
| 273 | C22 | — | 8 | 1 | no | — |
| 274 | C22 | — | 8 | 1 | no | — |
| 275 | C22 | — | 8 | 1 | no | — |
| 276 | C22 | — | 8 | 2 | no | — |
| 277 | C22 | — | 16 | 1 | no | — |
| 278 | C22 | #1352 | 1 | 1 | yes | characteristic |
| 279 | C22 | #1352 | 1 | 1 | yes | characteristic |
| 280 | C22 | C2×#328 | 1 | 1 | yes | characteristic |
| 281 | C4 | — | 2 | 1 | no | — |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 4 | 1 | no | — |
| 284 | C4 | — | 4 | 1 | no | — |
| 285 | C4 | — | 8 | 4 | no | — |
| 286 | C4 | — | 16 | 4 | no | — |
| 287 | C4 | #1352 | 1 | 1 | yes | characteristic |
| 288 | C4 | #1352 | 1 | 1 | yes | characteristic |
| 289 | C4 | C2×#352 | 1 | 1 | yes | characteristic |
| 290 | C4 | C2×#353 | 1 | 1 | yes | characteristic |
| 291 | C2 | — | 2 | 1 | no | — |
| 292 | C2 | — | 2 | 1 | no | — |
| 293 | C2 | — | 4 | 1 | no | — |
| 294 | C2 | — | 4 | 1 | no | — |
| 295 | C2 | — | 8 | 1 | no | — |
| 296 | C2 | — | 8 | 1 | no | — |
| 297 | C2 | #7438 | 1 | 2 | yes | direct product factor |
| 298 | C2 | C2×#1352 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| socle | C22 | #1352 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7438 | C2 | direct (N × H) |
| 5. C2×#430 | C4 | semidirect |
| 13. #2098 | C4 | semidirect |
| 15. C2×H(3,ℤ/4) | C4 | semidirect |
| 29. #430 | C2×C4 | semidirect |
| 54. H(3,ℤ/4) | C2×C4 | semidirect |
| 297. C2 | #7438 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2098 | 2 |
| C2×#1391 | 2 |
| C2×#1315 | 2 |
| #7438 | 2 |