Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
620 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 84 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 | D181⋊#104 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #26042270 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | D181⋊#100 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #26042231 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | H(5,𝔽2)×D181 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | D181⋊M16 | C22 | 1 | 2 | yes | complemented |
| 7 | C2×C4⋊#10624382 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C2×C22⋊#10624382 | — | 2 | 2 | no | — |
| 9 | C2×C4×#10624382 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #19153881 | — | 2 | 2 | no | — |
| 11 | #19153880 | — | 2 | 2 | no | — |
| 12 | #19153879 | — | 2 | 2 | no | — |
| 13 | #19153878 | — | 2 | 2 | no | — |
| 14 | #19153876 | — | 2 | 2 | no | — |
| 15 | #19153862 | — | 2 | 2 | no | — |
| 16 | #19153862 | C4 | 1 | 2 | yes | complemented |
| 17 | #19153861 | — | 2 | 2 | no | — |
| 18 | #19153861 | — | 2 | 2 | no | — |
| 19 | #19153861 | C4 | 1 | 1 | yes | characteristic, complemented |
| 20 | #41×D181 | — | 2 | 2 | no | — |
| 21 | #41×D181 | C22 | 1 | 2 | yes | — |
| 22 | C2×D4×D181 | — | 2 | 2 | no | — |
| 23 | C2×D4×D181 | — | 2 | 2 | no | — |
| 24 | C2×D4×D181 | C22 | 1 | 1 | yes | characteristic |
| 25 | H(5,𝔽2)×C181 | C4 | 1 | 1 | yes | characteristic, complemented |
| 26 | C22×#10624382 | — | 2 | 1 | no | — |
| 27 | C22×#10624382 | — | 2 | 1 | no | — |
| 28 | #15668459 | — | 2 | 2 | no | — |
| 29 | #15668458 | — | 2 | 2 | no | — |
| 30 | D181⋊C8 | — | 2 | 2 | no | — |
| 31 | C4⋊#10624382 | — | 2 | 2 | no | — |
| 32 | C22⋊#10624382 | — | 4 | 4 | no | — |
| 33 | C4×#10624382 | — | 2 | 2 | no | — |
| 34 | C23×D181 | — | 2 | 2 | no | — |
| 35 | C23×D181 | D4 | 1 | 2 | yes | — |
| 36 | #15668450 | — | 2 | 2 | no | — |
| 37 | #15668450 | D4 | 1 | 2 | yes | — |
| 38 | #15668449 | — | 2 | 4 | no | — |
| 39 | #15668449 | — | 2 | 2 | no | — |
| 40 | #15668449 | — | 2 | 2 | no | — |
| 41 | #15668449 | C2×C4 | 1 | 2 | yes | — |
| 42 | #15668448 | — | 2 | 4 | no | — |
| 43 | #15668448 | — | 2 | 2 | no | — |
| 44 | #15668448 | — | 2 | 2 | no | — |
| 45 | #15668448 | D4 | 1 | 2 | yes | — |
| 46 | Q8×D181 | D4 | 1 | 2 | yes | — |
| 47 | C2×D724 | — | 2 | 2 | no | — |
| 48 | C2×D724 | — | 2 | 2 | no | — |
| 49 | C2×D724 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C2×C22⋊D181 | — | 2 | 2 | no | — |
| 51 | C2×C22⋊D181 | — | 2 | 4 | no | — |
| 52 | C2×C22⋊D181 | — | 2 | 2 | no | — |
| 53 | C2×C22⋊D181 | D4 | 1 | 2 | yes | — |
| 54 | D4×D181 | — | 2 | 4 | no | — |
| 55 | D4×D181 | — | 2 | 4 | no | — |
| 56 | D4×D181 | — | 2 | 4 | no | — |
| 57 | D4×D181 | — | 2 | 4 | no | — |
| 58 | D4×D181 | — | 2 | 2 | no | — |
| 59 | D4×D181 | — | 2 | 2 | no | — |
| 60 | D4×D181 | — | 2 | 2 | no | — |
| 61 | D4×D181 | — | 2 | 2 | no | — |
| 62 | D4×D181 | — | 2 | 2 | no | — |
| 63 | D4×D181 | D4 | 1 | 2 | yes | — |
| 64 | C2×C4×D181 | — | 2 | 2 | no | — |
| 65 | C2×C4×D181 | — | 2 | 2 | no | — |
| 66 | C2×C4×D181 | C23 | 1 | 1 | yes | characteristic |
| 67 | #41×C181 | — | 2 | 2 | no | — |
| 68 | #41×C181 | C2×C4 | 1 | 2 | yes | — |
| 69 | C2×D4×C181 | — | 2 | 2 | no | — |
| 70 | C2×D4×C181 | — | 2 | 2 | no | — |
| 71 | C2×D4×C181 | C2×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×#10624382 | — | 2 | 2 | no | — |
| 73 | C2×#10624382 | — | 4 | 1 | no | — |
| 74 | C2×#10624382 | — | 4 | 1 | no | — |
| 75 | C2×#10624382 | — | 4 | 2 | no | — |
| 76 | C2×#10624382 | — | 4 | 1 | no | — |
| 77 | #13089685 | — | 4 | 2 | no | — |
| 78 | C22×D181 | — | 2 | 4 | no | — |
| 79 | C22×D181 | — | 2 | 2 | no | — |
| 80 | C22×D181 | — | 2 | 2 | no | — |
| 81 | C22×D181 | — | 2 | 2 | no | — |
| 82 | C22×D181 | — | 2 | 2 | no | — |
| 83 | C22×D181 | — | 4 | 2 | no | — |
| 84 | C22×D181 | — | 4 | 2 | no | — |
| 85 | C22×D181 | — | 8 | 2 | no | — |
| 86 | C22×D181 | — | 8 | 2 | no | — |
| 87 | C22×D181 | C22⋊C4 | 1 | 2 | yes | — |
| 88 | C22×D181 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | Dic362 | — | 2 | 2 | no | — |
| 90 | Dic362 | C22⋊C4 | 1 | 2 | yes | complemented |
| 91 | D724 | — | 2 | 4 | no | — |
| 92 | D724 | — | 2 | 2 | no | — |
| 93 | D724 | — | 2 | 2 | no | — |
| 94 | D724 | C22⋊C4 | 1 | 2 | yes | complemented |
| 95 | C22⋊D181 | — | 2 | 4 | no | — |
| 96 | C22⋊D181 | — | 2 | 4 | no | — |
| 97 | C22⋊D181 | — | 2 | 2 | no | — |
| 98 | C22⋊D181 | — | 2 | 2 | no | — |
| 99 | C22⋊D181 | — | 2 | 4 | no | — |
| 100 | C22⋊D181 | — | 2 | 2 | no | — |
| 101 | C2×Dic181 | — | 2 | 2 | no | — |
| 102 | C2×Dic181 | — | 2 | 2 | no | — |
| 103 | C2×Dic181 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C4×D181 | — | 2 | 4 | no | — |
| 105 | C4×D181 | — | 2 | 2 | no | — |
| 106 | C4×D181 | — | 2 | 2 | no | — |
| 107 | C4×D181 | — | 2 | 2 | no | — |
| 108 | C4×D181 | C2×D4 | 1 | 2 | yes | — |
| 109 | C4×D181 | C2×D4 | 1 | 2 | yes | — |
| 110 | C23×C181 | — | 2 | 2 | no | — |
| 111 | C23×C181 | C22⋊C4 | 1 | 2 | yes | — |
| 112 | Q8×C181 | C22⋊C4 | 1 | 2 | yes | complemented |
| 113 | D4×C181 | — | 2 | 4 | no | — |
| 114 | D4×C181 | — | 2 | 2 | no | — |
| 115 | D4×C181 | — | 2 | 2 | no | — |
| 116 | D4×C181 | C22⋊C4 | 1 | 2 | yes | complemented |
| 117 | C2×C4×C181 | — | 2 | 2 | no | — |
| 118 | C2×C4×C181 | — | 2 | 2 | no | — |
| 119 | C2×C4×C181 | C22×C4 | 1 | 1 | yes | characteristic |
| 120 | C22×C181 | — | 2 | 2 | no | — |
| 121 | C22×C181 | — | 2 | 2 | no | — |
| 122 | C22×C181 | — | 4 | 2 | no | — |
| 123 | C22×C181 | — | 8 | 2 | no | — |
| 124 | C22×C181 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 125 | C2×D181 | — | 2 | 1 | no | — |
| 126 | C2×D181 | — | 2 | 1 | no | — |
| 127 | C2×D181 | — | 2 | 2 | no | — |
| 128 | C2×D181 | — | 2 | 2 | no | — |
| 129 | C2×D181 | — | 4 | 2 | no | — |
| 130 | C2×D181 | — | 4 | 2 | no | — |
| 131 | C2×D181 | — | 4 | 2 | no | — |
| 132 | C2×D181 | — | 4 | 2 | no | — |
| 133 | C2×D181 | — | 4 | 2 | no | — |
| 134 | C2×D181 | — | 8 | 2 | no | — |
| 135 | C2×D181 | — | 8 | 4 | no | — |
| 136 | C2×D181 | — | 8 | 2 | no | — |
| 137 | C2×D181 | #120 | 1 | 1 | yes | characteristic |
| 138 | C2×D181 | #120 | 1 | 1 | yes | characteristic |
| 139 | #10624382 | — | 4 | 2 | no | — |
| 140 | #10624382 | — | 8 | 1 | no | — |
| 141 | C4×C181 | — | 2 | 2 | no | — |
| 142 | C4×C181 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 143 | Dic181 | — | 2 | 2 | no | — |
| 144 | Dic181 | #120 | 1 | 2 | yes | — |
| 145 | C2×C181 | — | 2 | 1 | no | — |
| 146 | C2×C181 | — | 4 | 2 | no | — |
| 147 | C2×C181 | — | 4 | 2 | no | — |
| 148 | C2×C181 | #379 | 1 | 1 | yes | characteristic |
| 149 | D181 | — | 2 | 1 | no | — |
| 150 | D181 | — | 4 | 2 | no | — |
| 151 | D181 | — | 4 | 2 | no | — |
| 152 | D181 | #453 | 1 | 1 | yes | characteristic |
| 153 | D181 | #455 | 1 | 1 | yes | characteristic |
| 154 | C181 | #1792 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | #1792 | — | 181 | 1 | no | maximal, Sylow |
| 156 | C2×H(5,𝔽2) | — | 181 | 1 | no | — |
| 157 | C2×#104 | — | 181 | 2 | no | — |
| 158 | #417 | — | 181 | 2 | no | — |
| 159 | C2×#100 | — | 181 | 1 | no | — |
| 160 | #390 | — | 181 | 1 | no | — |
| 161 | H(5,𝔽2) | — | 362 | 2 | no | — |
| 162 | H(5,𝔽2) | — | 362 | 2 | no | — |
| 163 | H(5,𝔽2) | — | 362 | 2 | no | — |
| 164 | H(5,𝔽2) | — | 181 | 2 | no | — |
| 165 | H(5,𝔽2) | — | 181 | 1 | no | — |
| 166 | H(5,𝔽2) | #10624382 | 1 | 1 | yes | characteristic, complemented |
| 167 | C2×#41 | — | 362 | 2 | no | — |
| 168 | C2×#41 | — | 181 | 2 | no | — |
| 169 | C22×D4 | — | 362 | 2 | no | — |
| 170 | C22×D4 | — | 362 | 2 | no | — |
| 171 | C22×D4 | — | 181 | 1 | no | — |
| 172 | C2×M16 | — | 181 | 2 | no | — |
| 173 | C2×C4⋊C4 | — | 181 | 1 | no | — |
| 174 | C2×C22⋊C4 | — | 362 | 2 | no | — |
| 175 | C2×C42 | — | 181 | 1 | no | — |
| 176 | #104 | — | 362 | 2 | no | — |
| 177 | #104 | — | 362 | 2 | no | — |
| 178 | #103 | — | 362 | 2 | no | — |
| 179 | #102 | — | 362 | 2 | no | — |
| 180 | #100 | — | 362 | 2 | no | — |
| 181 | C24 | — | 362 | 2 | no | — |
| 182 | C24 | — | 181 | 2 | no | — |
| 183 | #41 | — | 2 | 2 | no | — |
| 184 | #41 | — | 362 | 4 | no | — |
| 185 | #41 | — | 362 | 2 | no | — |
| 186 | #41 | — | 362 | 2 | no | — |
| 187 | #41 | — | 362 | 4 | no | — |
| 188 | #41 | — | 362 | 2 | no | — |
| 189 | #41 | — | 362 | 2 | no | — |
| 190 | #41 | — | 362 | 2 | no | — |
| 191 | #41 | — | 181 | 2 | no | — |
| 192 | #41 | — | 181 | 2 | no | — |
| 193 | #41 | — | 181 | 2 | no | — |
| 194 | #41 | C2×#10624382 | 1 | 2 | yes | — |
| 195 | C2×Q8 | — | 181 | 2 | no | — |
| 196 | C2×D4 | — | 2 | 2 | no | — |
| 197 | C2×D4 | — | 2 | 2 | no | — |
| 198 | C2×D4 | — | 362 | 2 | no | — |
| 199 | C2×D4 | — | 362 | 2 | no | — |
| 200 | C2×D4 | — | 362 | 4 | no | — |
| 201 | C2×D4 | — | 362 | 4 | no | — |
| 202 | C2×D4 | — | 362 | 4 | no | — |
| 203 | C2×D4 | — | 362 | 2 | no | — |
| 204 | C2×D4 | — | 362 | 4 | no | — |
| 205 | C2×D4 | — | 362 | 2 | no | — |
| 206 | C2×D4 | — | 362 | 2 | no | — |
| 207 | C2×D4 | — | 362 | 4 | no | — |
| 208 | C2×D4 | — | 362 | 2 | no | — |
| 209 | C2×D4 | — | 362 | 2 | no | — |
| 210 | C2×D4 | — | 362 | 2 | no | — |
| 211 | C2×D4 | — | 362 | 2 | no | — |
| 212 | C2×D4 | — | 181 | 2 | no | — |
| 213 | C2×D4 | — | 181 | 2 | no | — |
| 214 | C2×D4 | — | 181 | 1 | no | — |
| 215 | C2×D4 | C2×#10624382 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | — | 362 | 2 | no | — |
| 217 | C22×C4 | — | 362 | 2 | no | — |
| 218 | C22×C4 | — | 362 | 1 | no | — |
| 219 | C22×C4 | — | 362 | 1 | no | — |
| 220 | C22×C4 | — | 181 | 1 | no | — |
| 221 | M16 | — | 362 | 2 | no | — |
| 222 | M16 | — | 362 | 2 | no | — |
| 223 | C2×C8 | — | 362 | 2 | no | — |
| 224 | C4⋊C4 | — | 362 | 2 | no | — |
| 225 | C22⋊C4 | — | 724 | 4 | no | — |
| 226 | C42 | — | 362 | 2 | no | — |
| 227 | C23 | — | 2 | 2 | no | — |
| 228 | C23 | — | 362 | 2 | no | — |
| 229 | C23 | — | 362 | 4 | no | — |
| 230 | C23 | — | 362 | 2 | no | — |
| 231 | C23 | — | 362 | 2 | no | — |
| 232 | C23 | — | 362 | 2 | no | — |
| 233 | C23 | — | 1448 | 2 | no | — |
| 234 | C23 | — | 1448 | 2 | no | — |
| 235 | C23 | — | 181 | 1 | no | — |
| 236 | C23 | — | 181 | 2 | no | — |
| 237 | C23 | — | 724 | 2 | no | — |
| 238 | C23 | — | 724 | 2 | no | — |
| 239 | C23 | C22⋊#10624382 | 1 | 2 | yes | — |
| 240 | Q8 | — | 362 | 2 | no | — |
| 241 | Q8 | — | 181 | 2 | no | — |
| 242 | Q8 | C22⋊#10624382 | 1 | 2 | yes | complemented |
| 243 | D4 | — | 2 | 4 | no | — |
| 244 | D4 | — | 2 | 2 | no | — |
| 245 | D4 | — | 2 | 2 | no | — |
| 246 | D4 | — | 362 | 4 | no | — |
| 247 | D4 | — | 362 | 2 | no | — |
| 248 | D4 | — | 362 | 4 | no | — |
| 249 | D4 | — | 362 | 4 | no | — |
| 250 | D4 | — | 362 | 2 | no | — |
| 251 | D4 | — | 362 | 2 | no | — |
| 252 | D4 | — | 362 | 4 | no | — |
| 253 | D4 | — | 362 | 2 | no | — |
| 254 | D4 | — | 362 | 2 | no | — |
| 255 | D4 | — | 181 | 2 | no | — |
| 256 | D4 | C22⋊#10624382 | 1 | 2 | yes | complemented |
| 257 | C2×C4 | — | 2 | 2 | no | — |
| 258 | C2×C4 | — | 2 | 2 | no | — |
| 259 | C2×C4 | — | 362 | 4 | no | — |
| 260 | C2×C4 | — | 362 | 2 | no | — |
| 261 | C2×C4 | — | 362 | 2 | no | — |
| 262 | C2×C4 | — | 362 | 2 | no | — |
| 263 | C2×C4 | — | 362 | 2 | no | — |
| 264 | C2×C4 | — | 362 | 2 | no | — |
| 265 | C2×C4 | — | 362 | 2 | no | — |
| 266 | C2×C4 | — | 181 | 1 | no | — |
| 267 | C2×C4 | — | 181 | 2 | no | — |
| 268 | C2×C4 | — | 181 | 2 | no | — |
| 269 | C2×C4 | — | 724 | 2 | no | — |
| 270 | C2×C4 | — | 724 | 1 | no | — |
| 271 | C2×C4 | — | 724 | 1 | no | — |
| 272 | C2×C4 | — | 724 | 1 | no | — |
| 273 | C2×C4 | C22×#10624382 | 1 | 1 | yes | characteristic |
| 274 | C8 | — | 724 | 2 | no | — |
| 275 | C22 | — | 2 | 2 | no | — |
| 276 | C22 | — | 2 | 2 | no | — |
| 277 | C22 | — | 4 | 2 | no | — |
| 278 | C22 | — | 8 | 2 | no | — |
| 279 | C22 | — | 362 | 1 | no | — |
| 280 | C22 | — | 362 | 1 | no | — |
| 281 | C22 | — | 362 | 2 | no | — |
| 282 | C22 | — | 362 | 2 | no | — |
| 283 | C22 | — | 1448 | 2 | no | — |
| 284 | C22 | — | 1448 | 2 | no | — |
| 285 | C22 | — | 1448 | 4 | no | — |
| 286 | C22 | — | 181 | 1 | no | — |
| 287 | C22 | — | 181 | 1 | no | — |
| 288 | C22 | — | 724 | 2 | no | — |
| 289 | C22 | — | 724 | 2 | no | — |
| 290 | C22 | — | 724 | 2 | no | — |
| 291 | C22 | — | 724 | 2 | no | — |
| 292 | C22 | — | 724 | 2 | no | — |
| 293 | C22 | C2×C22⋊#10624382 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 362 | 2 | no | — |
| 296 | C4 | — | 1448 | 1 | no | — |
| 297 | C4 | — | 181 | 2 | no | — |
| 298 | C4 | — | 724 | 2 | no | — |
| 299 | C4 | C2×C22⋊#10624382 | 1 | 2 | yes | — |
| 300 | C2 | — | 2 | 1 | no | — |
| 301 | C2 | — | 4 | 2 | no | — |
| 302 | C2 | — | 4 | 2 | no | — |
| 303 | C2 | — | 362 | 1 | no | — |
| 304 | C2 | — | 181 | 1 | no | — |
| 305 | C2 | — | 181 | 1 | no | — |
| 306 | C2 | — | 724 | 2 | no | — |
| 307 | C2 | — | 724 | 2 | no | — |
| 308 | C2 | #26042220 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D181⋊#104 | C2 | semidirect |
| 2. #26042270 | C2 | semidirect |
| 3. D181⋊#100 | C2 | semidirect |
| 4. #26042231 | C2 | semidirect |
| 6. D181⋊M16 | C22 | semidirect |
| 7. C2×C4⋊#10624382 | C22 | semidirect |
| 9. C2×C4×#10624382 | C22 | semidirect |
| 16. #19153862 | C4 | semidirect |
| 19. #19153861 | C4 | semidirect |
| 25. H(5,𝔽2)×C181 | C4 | semidirect |
| 90. Dic362 | C22⋊C4 | semidirect |
| 94. D724 | C22⋊C4 | semidirect |
| 112. Q8×C181 | C22⋊C4 | semidirect |
| 116. D4×C181 | C22⋊C4 | semidirect |
| 154. C181 | #1792 | semidirect |
| 166. H(5,𝔽2) | #10624382 | semidirect |
| 242. Q8 | C22⋊#10624382 | semidirect |
| 256. D4 | C22⋊#10624382 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D181⋊#104 | 2 |
| #26042270 | 2 |
| D181⋊#100 | 2 |
| #26042231 | 2 |
| H(5,𝔽2)×D181 | 2 |
| #1792 | 181 |