Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
424 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 424 conjugacy classes of subgroups — too many to draw. Only its 55 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 | #26031117 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #26022200 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #26021356 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #26018244 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C89×#1321 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19152398 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | #19152396 | — | 2 | 1 | no | — |
| 8 | #19152162 | — | 2 | 1 | no | — |
| 9 | #19152036 | — | 2 | 2 | no | — |
| 10 | #19151923 | — | 2 | 2 | no | — |
| 11 | #19151851 | C22 | 1 | 2 | yes | — |
| 12 | #19151834 | — | 2 | 1 | no | — |
| 13 | #19151734 | — | 2 | 2 | no | — |
| 14 | #19151564 | — | 2 | 1 | no | — |
| 15 | #19151409 | — | 2 | 2 | no | — |
| 16 | #19151356 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | #408×C89 | C22 | 1 | 1 | yes | characteristic |
| 18 | #349×C89 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×#13083574 | — | 2 | 1 | no | — |
| 20 | C22×D356 | — | 2 | 1 | no | — |
| 21 | #15668045 | — | 4 | 1 | no | — |
| 22 | #15668043 | — | 4 | 1 | no | — |
| 23 | #15668042 | — | 4 | 1 | no | — |
| 24 | #15668040 | — | 4 | 1 | no | — |
| 25 | #15668038 | — | 4 | 1 | no | — |
| 26 | #15668037 | — | 4 | 1 | no | — |
| 27 | #15668034 | — | 2 | 2 | no | — |
| 28 | #15668032 | — | 4 | 2 | no | — |
| 29 | C2×D712 | — | 2 | 1 | no | — |
| 30 | C2×D712 | D4 | 1 | 2 | yes | — |
| 31 | C2×#13083558 | — | 2 | 1 | no | — |
| 32 | #15668008 | — | 4 | 1 | no | — |
| 33 | #15668001 | — | 4 | 1 | no | — |
| 34 | #15667986 | — | 2 | 2 | no | — |
| 35 | #15667986 | — | 2 | 2 | no | — |
| 36 | #15667980 | — | 4 | 1 | no | — |
| 37 | #15667980 | — | 4 | 1 | no | — |
| 38 | #15667971 | — | 2 | 1 | no | — |
| 39 | #15667957 | — | 4 | 2 | no | — |
| 40 | D1424 | — | 4 | 2 | no | — |
| 41 | #15667946 | — | 2 | 1 | no | — |
| 42 | #15667944 | D4 | 1 | 2 | yes | — |
| 43 | #15667934 | — | 2 | 2 | no | — |
| 44 | #15667934 | — | 2 | 2 | no | — |
| 45 | #15667924 | — | 4 | 2 | no | — |
| 46 | C2×#41×C89 | D4 | 1 | 1 | yes | characteristic |
| 47 | C2×M16×C89 | C23 | 1 | 1 | yes | characteristic |
| 48 | C22×C8×C89 | D4 | 1 | 1 | yes | characteristic |
| 49 | M32×C89 | — | 2 | 2 | no | — |
| 50 | C22⋊C8×C89 | — | 2 | 2 | no | — |
| 51 | C23×D89 | — | 4 | 1 | no | — |
| 52 | #13083574 | — | 4 | 1 | no | — |
| 53 | #13083574 | — | 4 | 1 | no | — |
| 54 | #13083574 | — | 4 | 1 | no | — |
| 55 | #13083574 | — | 4 | 1 | no | — |
| 56 | C2×Dic178 | — | 2 | 1 | no | — |
| 57 | C2×D356 | — | 2 | 1 | no | — |
| 58 | C2×D356 | — | 2 | 2 | no | — |
| 59 | C2×D356 | — | 4 | 1 | no | — |
| 60 | C2×D356 | — | 8 | 1 | no | — |
| 61 | C2×D356 | — | 8 | 1 | no | — |
| 62 | C2×C22⋊D89 | — | 4 | 1 | no | — |
| 63 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 64 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 65 | C2×C4×D89 | — | 4 | 1 | no | — |
| 66 | Dic356 | — | 2 | 2 | no | — |
| 67 | #13083564 | — | 4 | 2 | no | — |
| 68 | #13083563 | — | 4 | 2 | no | — |
| 69 | #13083562 | — | 4 | 2 | no | — |
| 70 | #13083561 | — | 4 | 2 | no | — |
| 71 | D712 | — | 2 | 2 | no | — |
| 72 | D712 | — | 4 | 2 | no | — |
| 73 | D712 | — | 8 | 1 | no | — |
| 74 | #13083559 | — | 4 | 2 | no | — |
| 75 | #13083558 | — | 4 | 1 | no | — |
| 76 | #13083558 | — | 4 | 1 | no | — |
| 77 | C2×#10623494 | — | 2 | 2 | no | — |
| 78 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 79 | C22⋊Dic89 | — | 4 | 1 | no | — |
| 80 | #13083551 | — | 4 | 2 | no | — |
| 81 | #13083551 | — | 4 | 2 | no | — |
| 82 | C4×Dic89 | — | 4 | 1 | no | — |
| 83 | #41×C89 | — | 4 | 1 | no | — |
| 84 | #41×C89 | — | 4 | 1 | no | — |
| 85 | C2×Q8×C89 | D8 | 1 | 1 | yes | characteristic |
| 86 | C2×D4×C89 | — | 2 | 1 | no | — |
| 87 | C2×D4×C89 | D8 | 1 | 1 | yes | characteristic |
| 88 | C22×C4×C89 | — | 2 | 1 | no | — |
| 89 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | M16×C89 | — | 4 | 1 | no | — |
| 91 | C2×C8×C89 | — | 2 | 1 | no | — |
| 92 | C2×C8×C89 | — | 4 | 1 | no | — |
| 93 | C2×C8×C89 | C2×D4 | 1 | 2 | yes | — |
| 94 | C16×C89 | — | 4 | 2 | no | — |
| 95 | C23×C89 | — | 2 | 1 | no | — |
| 96 | C23×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 97 | C22×D89 | — | 4 | 1 | no | — |
| 98 | C22×D89 | — | 4 | 2 | no | — |
| 99 | C22×D89 | — | 8 | 2 | no | — |
| 100 | C22×D89 | — | 16 | 2 | no | — |
| 101 | Q8×C89 | — | 4 | 1 | no | — |
| 102 | D4×C89 | — | 4 | 1 | no | — |
| 103 | D4×C89 | — | 8 | 1 | no | — |
| 104 | C2×C4×C89 | — | 2 | 1 | no | — |
| 105 | C2×C4×C89 | — | 2 | 1 | no | — |
| 106 | C2×C4×C89 | — | 2 | 1 | no | — |
| 107 | C2×C4×C89 | — | 4 | 1 | no | — |
| 108 | C2×C4×C89 | — | 4 | 1 | no | — |
| 109 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C89 | C2×D8 | 1 | 1 | yes | characteristic |
| 111 | C22⋊D89 | — | 8 | 1 | no | — |
| 112 | C22⋊D89 | — | 8 | 1 | no | — |
| 113 | C22⋊D89 | — | 16 | 2 | no | — |
| 114 | C22⋊D89 | — | 16 | 2 | no | — |
| 115 | C2×Dic89 | — | 4 | 1 | no | — |
| 116 | C2×Dic89 | — | 4 | 2 | no | — |
| 117 | D356 | — | 4 | 1 | no | — |
| 118 | D356 | — | 4 | 2 | no | — |
| 119 | D356 | — | 4 | 1 | no | — |
| 120 | D356 | — | 4 | 2 | no | — |
| 121 | D356 | — | 8 | 2 | no | — |
| 122 | C4×D89 | — | 8 | 1 | no | — |
| 123 | C4×D89 | — | 8 | 1 | no | — |
| 124 | Dic178 | — | 4 | 1 | no | — |
| 125 | Dic178 | — | 4 | 1 | no | — |
| 126 | C8×C89 | — | 2 | 2 | no | — |
| 127 | C8×C89 | — | 4 | 1 | no | — |
| 128 | #10623494 | — | 4 | 2 | no | — |
| 129 | C22×C89 | — | 2 | 1 | no | — |
| 130 | C22×C89 | — | 4 | 1 | no | — |
| 131 | C22×C89 | — | 4 | 2 | no | — |
| 132 | C22×C89 | — | 4 | 1 | no | — |
| 133 | C22×C89 | #447 | 1 | 1 | yes | characteristic |
| 134 | C2×D89 | — | 8 | 2 | no | — |
| 135 | C2×D89 | — | 8 | 2 | no | — |
| 136 | C2×D89 | — | 8 | 2 | no | — |
| 137 | C2×D89 | — | 8 | 1 | no | — |
| 138 | C2×D89 | — | 8 | 2 | no | — |
| 139 | C2×D89 | — | 16 | 4 | no | — |
| 140 | C4×C89 | — | 2 | 1 | no | — |
| 141 | C4×C89 | — | 4 | 1 | no | — |
| 142 | C4×C89 | #453 | 1 | 1 | yes | characteristic |
| 143 | C4×C89 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 144 | Dic89 | — | 8 | 2 | no | — |
| 145 | Dic89 | — | 8 | 1 | no | — |
| 146 | #12117 | — | 89 | 1 | no | maximal, Sylow |
| 147 | C2×C89 | — | 2 | 1 | no | — |
| 148 | C2×C89 | — | 4 | 1 | no | — |
| 149 | C2×C89 | — | 8 | 1 | no | — |
| 150 | C2×C89 | #1595 | 1 | 1 | yes | characteristic |
| 151 | D89 | — | 16 | 1 | no | — |
| 152 | D89 | — | 16 | 2 | no | — |
| 153 | #3199 | — | 89 | 1 | no | — |
| 154 | #2190 | — | 89 | 2 | no | — |
| 155 | #2017 | — | 89 | 1 | no | — |
| 156 | #1341 | — | 89 | 2 | no | — |
| 157 | #1321 | D89 | 1 | 1 | yes | characteristic, complemented |
| 158 | C89 | #12117 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 159 | #575 | — | 89 | 1 | no | — |
| 160 | #575 | — | 178 | 1 | no | — |
| 161 | #535 | — | 178 | 1 | no | — |
| 162 | #509 | — | 178 | 2 | no | — |
| 163 | #482 | — | 178 | 2 | no | — |
| 164 | #469 | — | 89 | 2 | no | — |
| 165 | #466 | — | 178 | 1 | no | — |
| 166 | #447 | — | 178 | 2 | no | — |
| 167 | #408 | — | 178 | 1 | no | — |
| 168 | #408 | C2×D89 | 1 | 1 | yes | characteristic |
| 169 | #360 | — | 178 | 2 | no | — |
| 170 | #349 | C2×D89 | 1 | 2 | yes | complemented |
| 171 | #338 | — | 89 | 1 | no | — |
| 172 | C2×#41 | — | 178 | 1 | no | — |
| 173 | C2×#41 | D356 | 1 | 1 | yes | characteristic |
| 174 | C22×D4 | — | 178 | 1 | no | — |
| 175 | #137 | — | 356 | 1 | no | — |
| 176 | #137 | — | 356 | 1 | no | — |
| 177 | #137 | — | 356 | 1 | no | — |
| 178 | AGL(1,ℤ/8) | — | 356 | 1 | no | — |
| 179 | AGL(1,ℤ/8) | — | 356 | 1 | no | — |
| 180 | AGL(1,ℤ/8) | — | 356 | 1 | no | — |
| 181 | #135 | — | 356 | 2 | no | — |
| 182 | #135 | — | 178 | 2 | no | — |
| 183 | C2×D8 | — | 89 | 2 | no | — |
| 184 | C2×D8 | — | 178 | 1 | no | — |
| 185 | C2×M16 | — | 178 | 1 | no | — |
| 186 | C2×M16 | C22×D89 | 1 | 1 | yes | characteristic |
| 187 | C22×C8 | D356 | 1 | 1 | yes | characteristic |
| 188 | #127 | — | 356 | 1 | no | — |
| 189 | #124 | — | 356 | 1 | no | — |
| 190 | #121 | — | 178 | 2 | no | — |
| 191 | #121 | — | 178 | 2 | no | — |
| 192 | #120 | — | 356 | 1 | no | — |
| 193 | #120 | — | 356 | 1 | no | — |
| 194 | #117 | — | 178 | 1 | no | — |
| 195 | SD32 | — | 356 | 2 | no | — |
| 196 | D16 | — | 356 | 2 | no | — |
| 197 | M32 | — | 2 | 2 | no | — |
| 198 | #107 | — | 178 | 1 | no | — |
| 199 | #106 | — | 89 | 2 | no | — |
| 200 | #102 | — | 178 | 2 | no | — |
| 201 | #102 | — | 178 | 2 | no | — |
| 202 | C22⋊C8 | — | 2 | 2 | no | — |
| 203 | C22⋊C8 | — | 356 | 2 | no | — |
| 204 | C24 | — | 356 | 1 | no | — |
| 205 | #41 | — | 4 | 1 | no | — |
| 206 | #41 | — | 4 | 1 | no | — |
| 207 | #41 | — | 356 | 1 | no | — |
| 208 | #41 | — | 356 | 1 | no | — |
| 209 | #41 | — | 356 | 1 | no | — |
| 210 | #41 | — | 356 | 1 | no | — |
| 211 | C2×Q8 | — | 178 | 1 | no | — |
| 212 | C2×Q8 | D712 | 1 | 1 | yes | characteristic |
| 213 | C2×D4 | — | 2 | 1 | no | — |
| 214 | C2×D4 | — | 356 | 1 | no | — |
| 215 | C2×D4 | — | 356 | 2 | no | — |
| 216 | C2×D4 | — | 356 | 2 | no | — |
| 217 | C2×D4 | — | 356 | 1 | no | — |
| 218 | C2×D4 | — | 178 | 1 | no | — |
| 219 | C2×D4 | — | 178 | 2 | no | — |
| 220 | C2×D4 | — | 712 | 1 | no | — |
| 221 | C2×D4 | — | 712 | 1 | no | — |
| 222 | C2×D4 | D712 | 1 | 1 | yes | characteristic |
| 223 | C22×C4 | — | 2 | 1 | no | — |
| 224 | C22×C4 | — | 356 | 1 | no | — |
| 225 | C22×C4 | C2×D356 | 1 | 1 | yes | characteristic |
| 226 | Q16 | — | 356 | 2 | no | — |
| 227 | Q16 | — | 178 | 2 | no | — |
| 228 | SD16 | — | 356 | 2 | no | — |
| 229 | SD16 | — | 356 | 2 | no | — |
| 230 | SD16 | — | 356 | 2 | no | — |
| 231 | D8 | — | 356 | 2 | no | — |
| 232 | D8 | — | 356 | 2 | no | — |
| 233 | D8 | — | 178 | 2 | no | — |
| 234 | D8 | — | 712 | 1 | no | — |
| 235 | M16 | — | 4 | 1 | no | — |
| 236 | M16 | — | 356 | 1 | no | — |
| 237 | M16 | — | 356 | 1 | no | — |
| 238 | C2×C8 | — | 2 | 1 | no | — |
| 239 | C2×C8 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 178 | 2 | no | — |
| 241 | C2×C8 | D4×D89 | 1 | 2 | yes | — |
| 242 | C4⋊C4 | — | 178 | 2 | no | — |
| 243 | C22⋊C4 | — | 356 | 2 | no | — |
| 244 | C22⋊C4 | — | 356 | 1 | no | — |
| 245 | C22⋊C4 | — | 356 | 2 | no | — |
| 246 | C42 | — | 356 | 1 | no | — |
| 247 | C16 | — | 4 | 2 | no | — |
| 248 | C23 | — | 2 | 1 | no | — |
| 249 | C23 | — | 356 | 1 | no | — |
| 250 | C23 | — | 356 | 2 | no | — |
| 251 | C23 | — | 1424 | 2 | no | — |
| 252 | C23 | — | 712 | 2 | no | — |
| 253 | C23 | #15668040 | 1 | 1 | yes | characteristic |
| 254 | Q8 | — | 4 | 1 | no | — |
| 255 | Q8 | — | 356 | 1 | no | — |
| 256 | Q8 | — | 356 | 1 | no | — |
| 257 | D4 | — | 4 | 1 | no | — |
| 258 | D4 | — | 8 | 1 | no | — |
| 259 | D4 | — | 356 | 1 | no | — |
| 260 | D4 | — | 356 | 2 | no | — |
| 261 | D4 | — | 356 | 2 | no | — |
| 262 | D4 | — | 356 | 1 | no | — |
| 263 | D4 | — | 1424 | 2 | no | — |
| 264 | D4 | — | 1424 | 2 | no | — |
| 265 | D4 | — | 712 | 2 | no | — |
| 266 | D4 | — | 712 | 1 | no | — |
| 267 | D4 | — | 712 | 1 | no | — |
| 268 | C2×C4 | — | 2 | 1 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 4 | 1 | no | — |
| 272 | C2×C4 | — | 4 | 1 | no | — |
| 273 | C2×C4 | — | 356 | 1 | no | — |
| 274 | C2×C4 | — | 356 | 2 | no | — |
| 275 | C2×C4 | — | 712 | 1 | no | — |
| 276 | C2×C4 | — | 712 | 1 | no | — |
| 277 | C2×C4 | #15667982 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | C2×D712 | 1 | 1 | yes | characteristic |
| 279 | C8 | — | 2 | 2 | no | — |
| 280 | C8 | — | 4 | 1 | no | — |
| 281 | C8 | — | 356 | 2 | no | — |
| 282 | C22 | — | 2 | 1 | no | — |
| 283 | C22 | — | 4 | 1 | no | — |
| 284 | C22 | — | 4 | 2 | no | — |
| 285 | C22 | — | 4 | 1 | no | — |
| 286 | C22 | — | 1424 | 4 | no | — |
| 287 | C22 | — | 712 | 2 | no | — |
| 288 | C22 | — | 712 | 2 | no | — |
| 289 | C22 | — | 712 | 2 | no | — |
| 290 | C22 | — | 712 | 1 | no | — |
| 291 | C22 | — | 712 | 2 | no | — |
| 292 | C22 | #19151734 | 1 | 1 | yes | characteristic |
| 293 | C4 | — | 2 | 1 | no | — |
| 294 | C4 | — | 4 | 1 | no | — |
| 295 | C4 | — | 712 | 2 | no | — |
| 296 | C4 | — | 712 | 1 | no | — |
| 297 | C4 | #19151768 | 1 | 1 | yes | characteristic |
| 298 | C4 | #19151786 | 1 | 1 | yes | characteristic |
| 299 | C2 | — | 2 | 1 | no | — |
| 300 | C2 | — | 4 | 1 | no | — |
| 301 | C2 | — | 8 | 1 | no | — |
| 302 | C2 | — | 1424 | 2 | no | — |
| 303 | C2 | — | 1424 | 1 | no | — |
| 304 | C2 | #26019172 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26031117 | C2 | semidirect |
| 2. #26022200 | C2 | semidirect |
| 3. #26021356 | C2 | semidirect |
| 4. #26018244 | C2 | semidirect |
| 5. C89×#1321 | C2 | semidirect |
| 6. #19152398 | C22 | semidirect |
| 16. #19151356 | C22 | semidirect |
| 18. #349×C89 | C22 | semidirect |
| 157. #1321 | D89 | semidirect |
| 158. C89 | #12117 | semidirect |
| 170. #349 | C2×D89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26031117 | 2 |
| #26022200 | 2 |
| #26021356 | 2 |
| #26018244 | 2 |
| C89×#1321 | 2 |
| #12117 | 89 |