Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
650 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 105 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 | #26031172 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #26030698 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #26021076 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #26018813 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C89×#1281 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#15668034 | — | 2 | 2 | no | — |
| 7 | #19152148 | C22 | 1 | 2 | yes | — |
| 8 | C2×#15668009 | C22 | 1 | 1 | yes | characteristic |
| 9 | #19151992 | — | 2 | 2 | no | — |
| 10 | #19151987 | — | 2 | 2 | no | — |
| 11 | #19151975 | — | 2 | 2 | no | — |
| 12 | #19151945 | — | 2 | 2 | no | — |
| 13 | #19151794 | — | 2 | 4 | no | — |
| 14 | #19151734 | — | 2 | 4 | no | — |
| 15 | C2×#15667934 | — | 2 | 2 | no | — |
| 16 | #19151464 | C22 | 1 | 1 | yes | characteristic |
| 17 | #19151333 | — | 2 | 4 | no | — |
| 18 | #431×C89 | C22 | 1 | 2 | yes | — |
| 19 | C2×#97×C89 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#13083574 | — | 2 | 2 | no | — |
| 21 | C22×D356 | — | 2 | 1 | no | — |
| 22 | C22×D356 | — | 2 | 2 | no | — |
| 23 | #15668034 | — | 4 | 4 | no | — |
| 24 | #15668034 | — | 4 | 4 | no | — |
| 25 | C2×Dic356 | — | 2 | 2 | no | — |
| 26 | C2×#13083563 | — | 2 | 4 | no | — |
| 27 | C2×D712 | — | 2 | 2 | no | — |
| 28 | C2×D712 | — | 2 | 8 | no | — |
| 29 | #15668012 | D4 | 1 | 2 | yes | — |
| 30 | #15668009 | — | 2 | 2 | no | — |
| 31 | #15668009 | — | 2 | 1 | no | — |
| 32 | #15668009 | D4 | 1 | 2 | yes | — |
| 33 | #15668009 | D4 | 1 | 2 | yes | — |
| 34 | #15668000 | — | 2 | 2 | no | — |
| 35 | #15667986 | — | 2 | 2 | no | — |
| 36 | #15667986 | — | 2 | 2 | no | — |
| 37 | C4×D356 | — | 2 | 2 | no | — |
| 38 | C4×D356 | — | 2 | 2 | no | — |
| 39 | C2×C4⋊Dic89 | — | 2 | 1 | no | — |
| 40 | #15667934 | — | 2 | 8 | no | — |
| 41 | #15667934 | — | 4 | 4 | no | — |
| 42 | #15667919 | — | 2 | 2 | no | — |
| 43 | C22×C8×C89 | D4 | 1 | 2 | yes | — |
| 44 | C2×C42×C89 | C23 | 1 | 1 | yes | characteristic |
| 45 | C4⋊C8×C89 | — | 2 | 2 | no | — |
| 46 | C22⋊C8×C89 | D4 | 1 | 4 | yes | — |
| 47 | #97×C89 | — | 2 | 2 | no | — |
| 48 | C4×C8×C89 | — | 2 | 2 | no | — |
| 49 | C23×D89 | — | 4 | 1 | no | — |
| 50 | #13083574 | — | 4 | 4 | no | — |
| 51 | #13083574 | — | 4 | 4 | no | — |
| 52 | C2×Dic178 | — | 2 | 2 | no | — |
| 53 | C2×D356 | — | 2 | 2 | no | — |
| 54 | C2×D356 | — | 2 | 2 | no | — |
| 55 | C2×D356 | — | 2 | 4 | no | — |
| 56 | C2×D356 | — | 2 | 2 | no | — |
| 57 | C2×D356 | — | 2 | 2 | no | — |
| 58 | C2×D356 | — | 4 | 2 | no | — |
| 59 | C2×D356 | — | 4 | 2 | no | — |
| 60 | C2×D356 | — | 4 | 1 | no | — |
| 61 | C2×D356 | — | 4 | 2 | no | — |
| 62 | C2×D356 | — | 8 | 4 | no | — |
| 63 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 64 | C22×Dic89 | — | 4 | 1 | no | — |
| 65 | C2×C4×D89 | — | 4 | 2 | no | — |
| 66 | Dic356 | — | 4 | 4 | no | — |
| 67 | #13083563 | — | 4 | 8 | no | — |
| 68 | D712 | — | 4 | 4 | no | — |
| 69 | D712 | — | 8 | 8 | no | — |
| 70 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 71 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 72 | C4⋊Dic89 | — | 4 | 2 | no | — |
| 73 | #13083551 | — | 4 | 2 | no | — |
| 74 | #13083551 | — | 4 | 2 | no | — |
| 75 | C22×C4×C89 | C2×D4 | 1 | 2 | yes | — |
| 76 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×C8×C89 | — | 2 | 2 | no | — |
| 78 | C2×C8×C89 | — | 2 | 2 | no | — |
| 79 | C2×C8×C89 | — | 2 | 4 | no | — |
| 80 | C2×C8×C89 | D8 | 1 | 4 | yes | complemented |
| 81 | C42×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C42×C89 | C2×D4 | 1 | 2 | yes | — |
| 83 | C42×C89 | #41 | 1 | 1 | yes | characteristic |
| 84 | C23×C89 | #127 | 1 | 1 | yes | characteristic |
| 85 | C22×D89 | — | 4 | 2 | no | — |
| 86 | C22×D89 | — | 4 | 2 | no | — |
| 87 | C22×D89 | — | 8 | 4 | no | — |
| 88 | C22×D89 | — | 8 | 1 | no | — |
| 89 | C22×D89 | — | 8 | 1 | no | — |
| 90 | C2×C4×C89 | — | 4 | 2 | no | — |
| 91 | C2×C4×C89 | #120 | 1 | 2 | yes | — |
| 92 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C89 | #121 | 1 | 2 | yes | — |
| 94 | C2×C4×C89 | #121 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C89 | C2×D8 | 1 | 2 | yes | — |
| 96 | C2×C4×C89 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 97 | C22⋊D89 | — | 8 | 2 | no | — |
| 98 | C22⋊D89 | — | 8 | 2 | no | — |
| 99 | C2×Dic89 | — | 4 | 2 | no | — |
| 100 | C2×Dic89 | — | 8 | 2 | no | — |
| 101 | C2×Dic89 | — | 8 | 1 | no | — |
| 102 | D356 | — | 4 | 4 | no | — |
| 103 | D356 | — | 4 | 8 | no | — |
| 104 | D356 | — | 4 | 4 | no | — |
| 105 | D356 | — | 4 | 4 | no | — |
| 106 | D356 | — | 8 | 2 | no | — |
| 107 | D356 | — | 8 | 4 | no | — |
| 108 | D356 | — | 8 | 2 | no | — |
| 109 | C4×D89 | — | 8 | 4 | no | — |
| 110 | Dic178 | — | 4 | 4 | no | — |
| 111 | C8×C89 | — | 2 | 4 | no | — |
| 112 | C8×C89 | — | 4 | 4 | no | — |
| 113 | C22×C89 | — | 2 | 2 | no | — |
| 114 | C22×C89 | #392 | 1 | 1 | yes | characteristic |
| 115 | C22×C89 | #493 | 1 | 1 | yes | characteristic |
| 116 | C22×C89 | #496 | 1 | 1 | yes | characteristic |
| 117 | C2×D89 | — | 8 | 4 | no | — |
| 118 | C2×D89 | — | 8 | 2 | no | — |
| 119 | C2×D89 | — | 8 | 2 | no | — |
| 120 | C2×D89 | — | 8 | 2 | no | — |
| 121 | C2×D89 | — | 8 | 2 | no | — |
| 122 | C2×D89 | — | 16 | 2 | no | — |
| 123 | C2×D89 | — | 16 | 2 | no | — |
| 124 | C4×C89 | — | 2 | 2 | no | — |
| 125 | C4×C89 | — | 2 | 2 | no | — |
| 126 | C4×C89 | #447 | 1 | 2 | yes | — |
| 127 | C4×C89 | #459 | 1 | 2 | yes | — |
| 128 | Dic89 | — | 8 | 2 | no | — |
| 129 | Dic89 | — | 16 | 1 | no | — |
| 130 | #10218 | — | 89 | 1 | no | maximal, Sylow |
| 131 | C2×C89 | — | 2 | 1 | no | — |
| 132 | C2×C89 | — | 2 | 1 | no | — |
| 133 | C2×C89 | #2008 | 1 | 2 | yes | — |
| 134 | C2×C89 | #2011 | 1 | 1 | yes | characteristic |
| 135 | D89 | — | 16 | 2 | no | — |
| 136 | D89 | — | 16 | 2 | no | — |
| 137 | #3205 | — | 89 | 2 | no | — |
| 138 | #3150 | — | 89 | 1 | no | — |
| 139 | #1961 | — | 89 | 1 | no | — |
| 140 | #1517 | — | 89 | 2 | no | — |
| 141 | #1281 | D89 | 1 | 1 | yes | characteristic, complemented |
| 142 | C89 | #10218 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | C2×#135 | — | 178 | 2 | no | — |
| 144 | #532 | — | 89 | 2 | no | — |
| 145 | C2×#127 | — | 89 | 1 | no | — |
| 146 | #497 | — | 178 | 2 | no | — |
| 147 | #496 | — | 178 | 2 | no | — |
| 148 | #493 | — | 178 | 2 | no | — |
| 149 | #486 | — | 178 | 2 | no | — |
| 150 | #459 | — | 178 | 4 | no | — |
| 151 | #447 | — | 178 | 4 | no | — |
| 152 | #431 | C2×D89 | 1 | 2 | yes | — |
| 153 | C2×#102 | — | 178 | 2 | no | — |
| 154 | C2×#97 | C2×D89 | 1 | 1 | yes | characteristic |
| 155 | #382 | — | 89 | 1 | no | — |
| 156 | #327 | — | 178 | 4 | no | — |
| 157 | C2×#41 | — | 178 | 2 | no | — |
| 158 | C22×D4 | — | 178 | 1 | no | — |
| 159 | C22×D4 | — | 178 | 2 | no | — |
| 160 | #135 | — | 356 | 4 | no | — |
| 161 | #135 | — | 356 | 4 | no | — |
| 162 | C2×Q16 | — | 178 | 2 | no | — |
| 163 | C2×SD16 | — | 178 | 4 | no | — |
| 164 | C2×D8 | — | 178 | 2 | no | — |
| 165 | C2×D8 | — | 178 | 8 | no | — |
| 166 | C22×C8 | D356 | 1 | 2 | yes | — |
| 167 | C4⋊Q8 | — | 89 | 2 | no | — |
| 168 | #127 | — | 89 | 2 | no | — |
| 169 | #127 | — | 89 | 2 | no | — |
| 170 | #127 | — | 178 | 2 | no | — |
| 171 | #127 | — | 178 | 1 | no | — |
| 172 | #124 | — | 178 | 2 | no | — |
| 173 | #121 | — | 178 | 2 | no | — |
| 174 | #121 | — | 178 | 2 | no | — |
| 175 | C4×D4 | — | 178 | 2 | no | — |
| 176 | C4×D4 | — | 178 | 2 | no | — |
| 177 | C2×C4⋊C4 | — | 178 | 1 | no | — |
| 178 | C2×C42 | C22×D89 | 1 | 1 | yes | characteristic |
| 179 | C4⋊C8 | — | 2 | 2 | no | — |
| 180 | #102 | — | 356 | 4 | no | — |
| 181 | #102 | — | 178 | 8 | no | — |
| 182 | C22⋊C8 | D356 | 1 | 4 | yes | — |
| 183 | #97 | — | 2 | 2 | no | — |
| 184 | C4×C8 | — | 2 | 2 | no | — |
| 185 | #95 | — | 178 | 2 | no | — |
| 186 | C24 | — | 356 | 1 | no | — |
| 187 | #41 | — | 356 | 4 | no | — |
| 188 | #41 | — | 356 | 4 | no | — |
| 189 | C2×Q8 | — | 178 | 2 | no | — |
| 190 | C2×D4 | — | 356 | 2 | no | — |
| 191 | C2×D4 | — | 356 | 1 | no | — |
| 192 | C2×D4 | — | 356 | 2 | no | — |
| 193 | C2×D4 | — | 356 | 2 | no | — |
| 194 | C2×D4 | — | 356 | 2 | no | — |
| 195 | C2×D4 | — | 178 | 4 | no | — |
| 196 | C2×D4 | — | 178 | 2 | no | — |
| 197 | C2×D4 | — | 178 | 2 | no | — |
| 198 | C2×D4 | — | 178 | 2 | no | — |
| 199 | C2×D4 | — | 178 | 2 | no | — |
| 200 | C2×D4 | — | 712 | 4 | no | — |
| 201 | C22×C4 | — | 356 | 2 | no | — |
| 202 | C22×C4 | — | 356 | 1 | no | — |
| 203 | C22×C4 | C2×D356 | 1 | 2 | yes | — |
| 204 | C22×C4 | C2×D356 | 1 | 1 | yes | characteristic |
| 205 | Q16 | — | 356 | 4 | no | — |
| 206 | SD16 | — | 356 | 8 | no | — |
| 207 | D8 | — | 356 | 4 | no | — |
| 208 | D8 | — | 712 | 8 | no | — |
| 209 | C2×C8 | — | 2 | 2 | no | — |
| 210 | C2×C8 | — | 2 | 2 | no | — |
| 211 | C2×C8 | — | 2 | 4 | no | — |
| 212 | C2×C8 | D712 | 1 | 4 | yes | complemented |
| 213 | C4⋊C4 | — | 356 | 2 | no | — |
| 214 | C4⋊C4 | — | 178 | 2 | no | — |
| 215 | C4⋊C4 | — | 178 | 2 | no | — |
| 216 | C22⋊C4 | — | 356 | 2 | no | — |
| 217 | C22⋊C4 | — | 356 | 2 | no | — |
| 218 | C42 | D4×D89 | 1 | 2 | yes | — |
| 219 | C42 | D4×D89 | 1 | 1 | yes | characteristic |
| 220 | C42 | #13083575 | 1 | 1 | yes | characteristic |
| 221 | C23 | — | 356 | 2 | no | — |
| 222 | C23 | — | 356 | 2 | no | — |
| 223 | C23 | — | 712 | 4 | no | — |
| 224 | C23 | — | 712 | 1 | no | — |
| 225 | C23 | — | 712 | 1 | no | — |
| 226 | C23 | #15668009 | 1 | 1 | yes | characteristic |
| 227 | Q8 | — | 356 | 4 | no | — |
| 228 | D4 | — | 356 | 4 | no | — |
| 229 | D4 | — | 356 | 8 | no | — |
| 230 | D4 | — | 356 | 4 | no | — |
| 231 | D4 | — | 356 | 4 | no | — |
| 232 | D4 | — | 712 | 2 | no | — |
| 233 | D4 | — | 712 | 4 | no | — |
| 234 | D4 | — | 712 | 2 | no | — |
| 235 | D4 | — | 712 | 2 | no | — |
| 236 | D4 | — | 712 | 2 | no | — |
| 237 | C2×C4 | — | 4 | 2 | no | — |
| 238 | C2×C4 | — | 356 | 2 | no | — |
| 239 | C2×C4 | — | 712 | 2 | no | — |
| 240 | C2×C4 | — | 712 | 1 | no | — |
| 241 | C2×C4 | — | 712 | 4 | no | — |
| 242 | C2×C4 | #15667982 | 1 | 2 | yes | — |
| 243 | C2×C4 | #15667982 | 1 | 1 | yes | characteristic |
| 244 | C2×C4 | #15667987 | 1 | 2 | yes | — |
| 245 | C2×C4 | #15667987 | 1 | 1 | yes | characteristic |
| 246 | C2×C4 | C2×D712 | 1 | 2 | yes | — |
| 247 | C2×C4 | #15668040 | 1 | 2 | yes | — |
| 248 | C8 | — | 2 | 4 | no | — |
| 249 | C8 | — | 4 | 4 | no | — |
| 250 | C22 | — | 2 | 2 | no | — |
| 251 | C22 | — | 1424 | 2 | no | — |
| 252 | C22 | — | 1424 | 2 | no | — |
| 253 | C22 | — | 712 | 4 | no | — |
| 254 | C22 | — | 712 | 2 | no | — |
| 255 | C22 | — | 712 | 2 | no | — |
| 256 | C22 | — | 712 | 2 | no | — |
| 257 | C22 | — | 712 | 2 | no | — |
| 258 | C22 | #19151508 | 1 | 1 | yes | characteristic |
| 259 | C22 | #19151975 | 1 | 1 | yes | characteristic |
| 260 | C22 | #19151987 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 2 | 2 | no | — |
| 263 | C4 | — | 1424 | 1 | no | — |
| 264 | C4 | — | 712 | 2 | no | — |
| 265 | C4 | #19151734 | 1 | 2 | yes | — |
| 266 | C4 | #19151794 | 1 | 2 | yes | — |
| 267 | C2 | — | 2 | 1 | no | — |
| 268 | C2 | — | 2 | 1 | no | — |
| 269 | C2 | — | 1424 | 2 | no | — |
| 270 | C2 | — | 1424 | 2 | no | — |
| 271 | C2 | #26021308 | 1 | 2 | yes | — |
| 272 | C2 | #26021325 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26031172 | C2 | semidirect |
| 2. #26030698 | C2 | semidirect |
| 3. #26021076 | C2 | semidirect |
| 4. #26018813 | C2 | semidirect |
| 5. C89×#1281 | C2 | semidirect |
| 80. C2×C8×C89 | D8 | semidirect |
| 141. #1281 | D89 | semidirect |
| 142. C89 | #10218 | semidirect |
| 212. C2×C8 | D712 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26031172 | 2 |
| #26030698 | 2 |
| #26021076 | 2 |
| #26018813 | 2 |
| C89×#1281 | 2 |
| #10218 | 89 |