Finite Group DatabasebetaFinite Group DatabasebetaOrder 4352.
484 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 59 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 | #14536830 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14530013 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #14527721 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #14526788 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C17×#1324 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11763390 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#10587561 | — | 2 | 1 | no | — |
| 8 | #11762945 | — | 2 | 2 | no | — |
| 9 | C2×#10587476 | — | 2 | 1 | no | — |
| 10 | #11762808 | — | 2 | 1 | no | — |
| 11 | #11762726 | — | 2 | 1 | no | — |
| 12 | #11762726 | C22 | 1 | 2 | yes | — |
| 13 | #11762722 | — | 2 | 2 | no | — |
| 14 | #11762719 | — | 2 | 2 | no | — |
| 15 | #11762578 | C22 | 1 | 1 | yes | characteristic |
| 16 | #11762544 | — | 2 | 2 | no | — |
| 17 | C17×#382 | C22 | 1 | 1 | yes | characteristic |
| 18 | C17×#324 | C22 | 1 | 2 | yes | — |
| 19 | C2×#63252 | — | 2 | 1 | no | — |
| 20 | C22×D68 | — | 2 | 1 | no | — |
| 21 | C22×D68 | — | 2 | 1 | no | — |
| 22 | C22×D68 | D4 | 1 | 2 | yes | — |
| 23 | C22×Dic34 | — | 2 | 1 | no | — |
| 24 | C2×C4⋊C4×C17 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×C4⋊C4×C17 | C23 | 1 | 1 | yes | characteristic |
| 26 | C2×C42×C17 | D4 | 1 | 1 | yes | characteristic |
| 27 | #10587616 | — | 2 | 2 | no | — |
| 28 | C2×#63227 | — | 2 | 2 | no | — |
| 29 | C2×#63227 | — | 2 | 2 | no | — |
| 30 | C2×#63227 | — | 2 | 2 | no | — |
| 31 | C2×#63227 | — | 2 | 2 | no | — |
| 32 | C2×#63223 | — | 2 | 1 | no | — |
| 33 | C2×D136 | — | 4 | 2 | no | — |
| 34 | #10587583 | — | 4 | 1 | no | — |
| 35 | #10587581 | — | 4 | 1 | no | — |
| 36 | #10587561 | — | 4 | 2 | no | — |
| 37 | #10587561 | — | 4 | 2 | no | — |
| 38 | C17×C22⋊C8 | — | 2 | 2 | no | — |
| 39 | C17×#95 | — | 2 | 2 | no | — |
| 40 | #10587502 | — | 2 | 1 | no | — |
| 41 | #10587502 | D4 | 1 | 2 | yes | — |
| 42 | #10587492 | — | 4 | 2 | no | — |
| 43 | #10587481 | — | 4 | 1 | no | — |
| 44 | #10587480 | — | 4 | 1 | no | — |
| 45 | #10587476 | — | 4 | 2 | no | — |
| 46 | C23×D17 | — | 2 | 2 | no | — |
| 47 | C22×C4×C17 | — | 2 | 1 | no | — |
| 48 | C22×C4×C17 | — | 2 | 1 | no | — |
| 49 | C22×C4×C17 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | C22×C4×C17 | C2×D4 | 1 | 2 | yes | — |
| 51 | C2×C22⋊D17 | — | 4 | 1 | no | — |
| 52 | C22×Dic17 | — | 2 | 2 | no | — |
| 53 | #63252 | — | 4 | 1 | no | — |
| 54 | #63252 | — | 4 | 1 | no | — |
| 55 | #63252 | — | 4 | 1 | no | — |
| 56 | #63252 | — | 4 | 1 | no | — |
| 57 | C2×D68 | — | 2 | 1 | no | — |
| 58 | C2×D68 | — | 2 | 2 | no | — |
| 59 | C2×D68 | — | 4 | 2 | no | — |
| 60 | C2×D68 | — | 4 | 2 | no | — |
| 61 | C2×D68 | — | 8 | 2 | no | — |
| 62 | C2×D68 | — | 8 | 1 | no | — |
| 63 | C2×D68 | — | 8 | 1 | no | — |
| 64 | C2×D68 | — | 8 | 2 | no | — |
| 65 | C2×D68 | — | 8 | 1 | no | — |
| 66 | C2×C4×D17 | — | 4 | 1 | no | — |
| 67 | C2×Dic34 | — | 2 | 1 | no | — |
| 68 | C2×Dic34 | — | 4 | 2 | no | — |
| 69 | C2×Dic34 | — | 8 | 1 | no | — |
| 70 | C2×Dic34 | — | 8 | 1 | no | — |
| 71 | C2×C8×C17 | — | 4 | 2 | no | — |
| 72 | C4⋊C4×C17 | — | 4 | 1 | no | — |
| 73 | C4⋊C4×C17 | — | 4 | 1 | no | — |
| 74 | C42×C17 | — | 2 | 1 | no | — |
| 75 | C42×C17 | D8 | 1 | 2 | yes | complemented |
| 76 | #63227 | — | 4 | 4 | no | — |
| 77 | #63227 | — | 4 | 2 | no | — |
| 78 | #63227 | — | 4 | 2 | no | — |
| 79 | #63227 | — | 4 | 2 | no | — |
| 80 | #63227 | — | 8 | 2 | no | — |
| 81 | #63227 | — | 8 | 2 | no | — |
| 82 | C4⋊Dic17 | — | 2 | 2 | no | — |
| 83 | #63223 | — | 4 | 1 | no | — |
| 84 | #63223 | — | 4 | 1 | no | — |
| 85 | C2×#3622 | — | 4 | 1 | no | — |
| 86 | D136 | — | 8 | 4 | no | — |
| 87 | #12156 | — | 17 | 1 | no | maximal, Sylow |
| 88 | C23×C17 | #120 | 1 | 1 | yes | characteristic |
| 89 | C22×D17 | — | 4 | 2 | no | — |
| 90 | C22×D17 | — | 4 | 2 | no | — |
| 91 | C22×D17 | — | 4 | 2 | no | — |
| 92 | C22×D17 | — | 4 | 2 | no | — |
| 93 | C22×D17 | — | 4 | 2 | no | — |
| 94 | C22×D17 | — | 4 | 2 | no | — |
| 95 | C22×D17 | — | 4 | 1 | no | — |
| 96 | C22×D17 | — | 4 | 2 | no | — |
| 97 | C2×C4×C17 | — | 2 | 1 | no | — |
| 98 | C2×C4×C17 | — | 2 | 1 | no | — |
| 99 | C2×C4×C17 | — | 2 | 2 | no | — |
| 100 | C2×C4×C17 | — | 2 | 2 | no | — |
| 101 | C2×C4×C17 | — | 4 | 1 | no | — |
| 102 | C2×C4×C17 | — | 4 | 1 | no | — |
| 103 | C2×C4×C17 | — | 4 | 2 | no | — |
| 104 | C2×C4×C17 | — | 4 | 2 | no | — |
| 105 | C2×C4×C17 | — | 4 | 1 | no | — |
| 106 | C2×C4×C17 | C2×D8 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C17 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 108 | C22⋊D17 | — | 8 | 1 | no | — |
| 109 | C22⋊D17 | — | 8 | 1 | no | — |
| 110 | C2×Dic17 | — | 4 | 2 | no | — |
| 111 | C2×Dic17 | — | 4 | 2 | no | — |
| 112 | C2×Dic17 | — | 4 | 2 | no | — |
| 113 | C2×Dic17 | — | 4 | 1 | no | — |
| 114 | D68 | — | 4 | 1 | no | — |
| 115 | D68 | — | 4 | 1 | no | — |
| 116 | D68 | — | 4 | 4 | no | — |
| 117 | D68 | — | 8 | 2 | no | — |
| 118 | D68 | — | 8 | 2 | no | — |
| 119 | D68 | — | 8 | 4 | no | — |
| 120 | D68 | — | 8 | 4 | no | — |
| 121 | C4×D17 | — | 8 | 1 | no | — |
| 122 | C4×D17 | — | 8 | 1 | no | — |
| 123 | Dic34 | — | 4 | 1 | no | — |
| 124 | Dic34 | — | 4 | 1 | no | — |
| 125 | Dic34 | — | 8 | 4 | no | — |
| 126 | C8×C17 | — | 8 | 2 | no | — |
| 127 | #3622 | — | 8 | 1 | no | — |
| 128 | #2819 | — | 17 | 1 | no | — |
| 129 | #2040 | — | 17 | 1 | no | — |
| 130 | #1598 | — | 17 | 2 | no | — |
| 131 | #1345 | — | 17 | 2 | no | — |
| 132 | #1324 | D17 | 1 | 1 | yes | characteristic, complemented |
| 133 | C22×C17 | — | 2 | 1 | no | — |
| 134 | C22×C17 | — | 2 | 1 | no | — |
| 135 | C22×C17 | #447 | 1 | 1 | yes | characteristic |
| 136 | C22×C17 | #454 | 1 | 1 | yes | characteristic |
| 137 | C22×C17 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 138 | C2×D17 | — | 8 | 2 | no | — |
| 139 | C2×D17 | — | 8 | 2 | no | — |
| 140 | C2×D17 | — | 8 | 2 | no | — |
| 141 | C2×D17 | — | 8 | 2 | no | — |
| 142 | C2×D17 | — | 8 | 1 | no | — |
| 143 | C2×D17 | — | 8 | 2 | no | — |
| 144 | C2×D17 | — | 8 | 2 | no | — |
| 145 | C2×D17 | — | 8 | 1 | no | — |
| 146 | C2×D17 | — | 8 | 2 | no | — |
| 147 | C2×D17 | — | 8 | 1 | no | — |
| 148 | C4×C17 | — | 2 | 1 | no | — |
| 149 | C4×C17 | — | 2 | 1 | no | — |
| 150 | C4×C17 | — | 4 | 2 | no | — |
| 151 | C4×C17 | — | 4 | 2 | no | — |
| 152 | C4×C17 | — | 8 | 1 | no | — |
| 153 | Dic17 | — | 8 | 2 | no | — |
| 154 | Dic17 | — | 8 | 1 | no | — |
| 155 | #537 | — | 17 | 1 | no | — |
| 156 | C2×#124 | — | 34 | 1 | no | — |
| 157 | #447 | — | 34 | 2 | no | — |
| 158 | C2×#104 | — | 34 | 1 | no | — |
| 159 | #417 | — | 34 | 1 | no | — |
| 160 | #394 | — | 17 | 2 | no | — |
| 161 | #394 | — | 34 | 1 | no | — |
| 162 | #393 | — | 34 | 2 | no | — |
| 163 | #392 | — | 34 | 2 | no | — |
| 164 | #382 | C2×D17 | 1 | 1 | yes | characteristic |
| 165 | #343 | — | 17 | 1 | no | — |
| 166 | #327 | — | 34 | 2 | no | — |
| 167 | #324 | C2×D17 | 1 | 2 | yes | — |
| 168 | C2×C17 | — | 2 | 1 | no | — |
| 169 | C2×C17 | — | 2 | 1 | no | — |
| 170 | C2×C17 | #1598 | 1 | 1 | yes | characteristic |
| 171 | C2×C17 | #2199 | 1 | 1 | yes | characteristic |
| 172 | C2×C17 | #2202 | 1 | 1 | yes | characteristic |
| 173 | D17 | — | 16 | 1 | no | — |
| 174 | D17 | — | 16 | 2 | no | — |
| 175 | C2×#41 | — | 34 | 1 | no | — |
| 176 | C22×Q8 | — | 34 | 1 | no | — |
| 177 | C22×D4 | — | 17 | 2 | no | — |
| 178 | C22×D4 | — | 34 | 1 | no | — |
| 179 | C22×D4 | — | 34 | 1 | no | — |
| 180 | C2×D8 | — | 68 | 2 | no | — |
| 181 | C2×M16 | — | 34 | 1 | no | — |
| 182 | #124 | — | 68 | 2 | no | — |
| 183 | #124 | — | 68 | 2 | no | — |
| 184 | C22⋊Q8 | — | 68 | 1 | no | — |
| 185 | #121 | — | 34 | 2 | no | — |
| 186 | #121 | — | 68 | 1 | no | — |
| 187 | C2×C4⋊C4 | D68 | 1 | 1 | yes | characteristic |
| 188 | C2×C4⋊C4 | C22×D17 | 1 | 1 | yes | characteristic |
| 189 | C2×C22⋊C4 | — | 34 | 2 | no | — |
| 190 | C2×C22⋊C4 | — | 34 | 2 | no | — |
| 191 | C2×C22⋊C4 | — | 34 | 2 | no | — |
| 192 | C2×C22⋊C4 | — | 34 | 2 | no | — |
| 193 | C2×C42 | D68 | 1 | 1 | yes | characteristic |
| 194 | #104 | — | 68 | 2 | no | — |
| 195 | #103 | — | 68 | 1 | no | — |
| 196 | #102 | — | 68 | 1 | no | — |
| 197 | #102 | — | 68 | 2 | no | — |
| 198 | C22⋊C8 | — | 2 | 2 | no | — |
| 199 | #95 | — | 2 | 2 | no | — |
| 200 | #95 | — | 17 | 2 | no | — |
| 201 | #95 | — | 34 | 1 | no | — |
| 202 | C17 | #12156 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 203 | C24 | — | 34 | 2 | no | — |
| 204 | #41 | — | 68 | 1 | no | — |
| 205 | #41 | — | 68 | 1 | no | — |
| 206 | #41 | — | 68 | 1 | no | — |
| 207 | #41 | — | 68 | 1 | no | — |
| 208 | C2×Q8 | — | 34 | 1 | no | — |
| 209 | C2×Q8 | — | 68 | 2 | no | — |
| 210 | C2×Q8 | — | 136 | 1 | no | — |
| 211 | C2×Q8 | — | 136 | 1 | no | — |
| 212 | C2×D4 | — | 34 | 1 | no | — |
| 213 | C2×D4 | — | 34 | 2 | no | — |
| 214 | C2×D4 | — | 68 | 2 | no | — |
| 215 | C2×D4 | — | 68 | 2 | no | — |
| 216 | C2×D4 | — | 68 | 1 | no | — |
| 217 | C2×D4 | — | 136 | 2 | no | — |
| 218 | C2×D4 | — | 136 | 1 | no | — |
| 219 | C2×D4 | — | 136 | 1 | no | — |
| 220 | C2×D4 | — | 136 | 2 | no | — |
| 221 | C2×D4 | — | 136 | 1 | no | — |
| 222 | C22×C4 | — | 2 | 1 | no | — |
| 223 | C22×C4 | — | 2 | 1 | no | — |
| 224 | C22×C4 | — | 34 | 2 | no | — |
| 225 | C22×C4 | — | 68 | 1 | no | — |
| 226 | C22×C4 | C2×D68 | 1 | 1 | yes | characteristic |
| 227 | C22×C4 | D4×D17 | 1 | 2 | yes | — |
| 228 | D8 | — | 136 | 4 | no | — |
| 229 | M16 | — | 68 | 1 | no | — |
| 230 | M16 | — | 68 | 1 | no | — |
| 231 | C2×C8 | — | 4 | 2 | no | — |
| 232 | C2×C8 | — | 68 | 1 | no | — |
| 233 | C4⋊C4 | — | 4 | 1 | no | — |
| 234 | C4⋊C4 | — | 4 | 1 | no | — |
| 235 | C4⋊C4 | — | 34 | 2 | no | — |
| 236 | C22⋊C4 | — | 68 | 4 | no | — |
| 237 | C22⋊C4 | — | 68 | 2 | no | — |
| 238 | C22⋊C4 | — | 68 | 2 | no | — |
| 239 | C22⋊C4 | — | 68 | 2 | no | — |
| 240 | C22⋊C4 | — | 136 | 2 | no | — |
| 241 | C22⋊C4 | — | 136 | 2 | no | — |
| 242 | C42 | — | 2 | 1 | no | — |
| 243 | C42 | D136 | 1 | 2 | yes | complemented |
| 244 | C23 | — | 68 | 2 | no | — |
| 245 | C23 | — | 68 | 2 | no | — |
| 246 | C23 | — | 68 | 2 | no | — |
| 247 | C23 | — | 68 | 2 | no | — |
| 248 | C23 | — | 68 | 2 | no | — |
| 249 | C23 | — | 68 | 2 | no | — |
| 250 | C23 | — | 68 | 1 | no | — |
| 251 | C23 | — | 68 | 2 | no | — |
| 252 | C23 | #10587568 | 1 | 1 | yes | characteristic |
| 253 | Q8 | — | 68 | 1 | no | — |
| 254 | Q8 | — | 68 | 1 | no | — |
| 255 | Q8 | — | 136 | 4 | no | — |
| 256 | D4 | — | 68 | 1 | no | — |
| 257 | D4 | — | 68 | 1 | no | — |
| 258 | D4 | — | 68 | 4 | no | — |
| 259 | D4 | — | 136 | 2 | no | — |
| 260 | D4 | — | 136 | 2 | no | — |
| 261 | D4 | — | 136 | 4 | no | — |
| 262 | D4 | — | 136 | 4 | no | — |
| 263 | D4 | — | 136 | 1 | no | — |
| 264 | D4 | — | 136 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 1 | no | — |
| 267 | C2×C4 | — | 2 | 2 | no | — |
| 268 | C2×C4 | — | 2 | 2 | no | — |
| 269 | C2×C4 | — | 4 | 1 | no | — |
| 270 | C2×C4 | — | 4 | 1 | no | — |
| 271 | C2×C4 | — | 4 | 2 | no | — |
| 272 | C2×C4 | — | 4 | 2 | no | — |
| 273 | C2×C4 | — | 4 | 1 | no | — |
| 274 | C2×C4 | — | 68 | 2 | no | — |
| 275 | C2×C4 | — | 68 | 2 | no | — |
| 276 | C2×C4 | — | 68 | 2 | no | — |
| 277 | C2×C4 | — | 68 | 1 | no | — |
| 278 | C2×C4 | — | 136 | 1 | no | — |
| 279 | C2×C4 | — | 136 | 1 | no | — |
| 280 | C2×C4 | C2×D136 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #10587594 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 8 | 2 | no | — |
| 283 | C8 | — | 136 | 1 | no | — |
| 284 | C22 | — | 2 | 1 | no | — |
| 285 | C22 | — | 2 | 1 | no | — |
| 286 | C22 | — | 136 | 2 | no | — |
| 287 | C22 | — | 136 | 2 | no | — |
| 288 | C22 | — | 136 | 2 | no | — |
| 289 | C22 | — | 136 | 2 | no | — |
| 290 | C22 | — | 136 | 2 | no | — |
| 291 | C22 | — | 136 | 1 | no | — |
| 292 | C22 | — | 136 | 1 | no | — |
| 293 | C22 | — | 136 | 2 | no | — |
| 294 | C22 | — | 136 | 2 | no | — |
| 295 | C22 | — | 136 | 1 | no | — |
| 296 | C22 | #11762945 | 1 | 1 | yes | characteristic |
| 297 | C22 | #11762984 | 1 | 1 | yes | characteristic |
| 298 | C22 | #11762997 | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 2 | 1 | no | — |
| 300 | C4 | — | 2 | 1 | no | — |
| 301 | C4 | — | 4 | 2 | no | — |
| 302 | C4 | — | 4 | 2 | no | — |
| 303 | C4 | — | 8 | 1 | no | — |
| 304 | C4 | — | 136 | 2 | no | — |
| 305 | C4 | — | 136 | 1 | no | — |
| 306 | C2 | — | 2 | 1 | no | — |
| 307 | C2 | — | 2 | 1 | no | — |
| 308 | C2 | — | 272 | 2 | no | — |
| 309 | C2 | — | 272 | 1 | no | — |
| 310 | C2 | #14527721 | 1 | 1 | yes | characteristic |
| 311 | C2 | #14530787 | 1 | 1 | yes | characteristic |
| 312 | C2 | #14530802 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14536830 | C2 | semidirect |
| 2. #14530013 | C2 | semidirect |
| 3. #14527721 | C2 | semidirect |
| 4. #14526788 | C2 | semidirect |
| 5. C17×#1324 | C2 | semidirect |
| 75. C42×C17 | D8 | semidirect |
| 132. #1324 | D17 | semidirect |
| 202. C17 | #12156 | semidirect |
| 243. C42 | D136 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14536830 | 2 |
| #14530013 | 2 |
| #14527721 | 2 |
| #14526788 | 2 |
| C17×#1324 | 2 |
| #12156 | 17 |