Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
784 conjugacy classes of subgroups; only the 155 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 155 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 | #26034747 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26031468 | C2 | 1 | 2 | yes | maximal |
| 3 | #26030774 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #26019582 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #26018868 | C2 | 1 | 2 | yes | maximal |
| 6 | C89×#1537 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C89⋊#564 | C22 | 1 | 4 | yes | — |
| 8 | #19152327 | — | 2 | 2 | no | — |
| 9 | #19152326 | — | 2 | 2 | no | — |
| 10 | #19152323 | — | 2 | 2 | no | — |
| 11 | #19152210 | — | 2 | 2 | no | — |
| 12 | #19152208 | — | 2 | 4 | no | — |
| 13 | #19152155 | C22 | 1 | 1 | yes | characteristic |
| 14 | #19152155 | C22 | 1 | 2 | yes | — |
| 15 | #19152008 | C22 | 1 | 4 | yes | — |
| 16 | #19152007 | C22 | 1 | 4 | yes | — |
| 17 | #19152004 | — | 2 | 2 | no | — |
| 18 | #19151967 | C22 | 1 | 4 | yes | — |
| 19 | #19151940 | — | 2 | 4 | no | — |
| 20 | #19151894 | — | 2 | 2 | no | — |
| 21 | #19151884 | — | 2 | 2 | no | — |
| 22 | #19151634 | — | 2 | 2 | no | — |
| 23 | C2×#15667923 | C22 | 1 | 1 | yes | characteristic |
| 24 | #19151339 | C22 | 1 | 8 | yes | — |
| 25 | #533×C89 | C22 | 1 | 1 | yes | characteristic |
| 26 | #432×C89 | C22 | 1 | 2 | yes | — |
| 27 | #330×C89 | C22 | 1 | 4 | yes | — |
| 28 | C22×#10623494 | — | 2 | 1 | no | — |
| 29 | #15668012 | — | 2 | 2 | no | — |
| 30 | #15668012 | D4 | 1 | 2 | yes | — |
| 31 | #15668012 | D4 | 1 | 2 | yes | — |
| 32 | #15668012 | C23 | 1 | 2 | yes | — |
| 33 | #15668010 | — | 2 | 4 | no | — |
| 34 | #15668005 | — | 2 | 8 | no | — |
| 35 | #15668004 | — | 2 | 2 | no | — |
| 36 | #15668004 | — | 2 | 1 | no | — |
| 37 | #15668004 | C23 | 1 | 4 | yes | — |
| 38 | #15668003 | — | 2 | 4 | no | — |
| 39 | #15668003 | — | 2 | 8 | no | — |
| 40 | #15668002 | — | 2 | 8 | no | — |
| 41 | #15668002 | — | 2 | 8 | no | — |
| 42 | #15667991 | — | 2 | 2 | no | — |
| 43 | #15667991 | — | 2 | 2 | no | — |
| 44 | #15667991 | — | 2 | 2 | no | — |
| 45 | Q8×Dic89 | — | 2 | 4 | no | — |
| 46 | C4×Dic178 | — | 2 | 4 | no | — |
| 47 | C4×Dic178 | — | 2 | 2 | no | — |
| 48 | C4×Dic178 | — | 2 | 2 | no | — |
| 49 | C4×Dic178 | — | 2 | 2 | no | — |
| 50 | #15667977 | — | 2 | 4 | no | — |
| 51 | #15667971 | — | 2 | 1 | no | — |
| 52 | #15667971 | D4 | 1 | 4 | yes | — |
| 53 | #15667946 | — | 2 | 4 | no | — |
| 54 | #15667945 | — | 2 | 4 | no | — |
| 55 | #15667944 | — | 2 | 4 | no | — |
| 56 | #15667943 | — | 2 | 4 | no | — |
| 57 | #15667937 | — | 2 | 4 | no | — |
| 58 | #15667937 | — | 2 | 4 | no | — |
| 59 | #15667923 | — | 2 | 1 | no | — |
| 60 | #15667923 | C23 | 1 | 2 | yes | — |
| 61 | C4⋊Q8×C89 | D4 | 1 | 2 | yes | — |
| 62 | #125×C89 | C23 | 1 | 2 | yes | — |
| 63 | C22⋊Q8×C89 | — | 2 | 2 | no | — |
| 64 | C4×Q8×C89 | — | 2 | 2 | no | — |
| 65 | #117×C89 | D4 | 1 | 2 | yes | — |
| 66 | C2×C42×C89 | C23 | 1 | 1 | yes | characteristic |
| 67 | C4⋊C8×C89 | — | 2 | 2 | no | — |
| 68 | C22⋊C8×C89 | — | 2 | 2 | no | — |
| 69 | #97×C89 | C23 | 1 | 4 | yes | — |
| 70 | C2×Dic178 | — | 2 | 2 | no | — |
| 71 | C2×Dic178 | — | 2 | 2 | no | — |
| 72 | C2×Dic178 | — | 2 | 2 | no | — |
| 73 | C2×#10623494 | — | 2 | 4 | no | — |
| 74 | C2×#10623494 | — | 2 | 2 | no | — |
| 75 | C4⋊Dic89 | — | 2 | 8 | no | — |
| 76 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 77 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 78 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 79 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 80 | C4⋊Dic89 | C2×D4 | 1 | 4 | yes | — |
| 81 | C4⋊Dic89 | C2×D4 | 1 | 4 | yes | — |
| 82 | #13083553 | — | 2 | 8 | no | — |
| 83 | #13083553 | — | 2 | 8 | no | — |
| 84 | #13083553 | — | 2 | 4 | no | — |
| 85 | #13083553 | — | 2 | 4 | no | — |
| 86 | #13083553 | — | 4 | 2 | no | — |
| 87 | C22⋊Dic89 | — | 2 | 4 | no | — |
| 88 | C22⋊Dic89 | — | 4 | 1 | no | — |
| 89 | C4×Dic89 | — | 2 | 8 | no | — |
| 90 | C4×Dic89 | — | 2 | 4 | no | — |
| 91 | C4×Dic89 | — | 4 | 1 | no | — |
| 92 | C2×Q8×C89 | — | 2 | 2 | no | — |
| 93 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C22×C4×C89 | C2×D4 | 1 | 2 | yes | — |
| 95 | C2×C8×C89 | — | 2 | 4 | no | — |
| 96 | C4⋊C4×C89 | — | 2 | 2 | no | — |
| 97 | C4⋊C4×C89 | — | 2 | 4 | no | — |
| 98 | C4⋊C4×C89 | C2×D4 | 1 | 4 | yes | — |
| 99 | C22⋊C4×C89 | — | 2 | 2 | no | — |
| 100 | C42×C89 | — | 2 | 2 | no | — |
| 101 | C42×C89 | C2×D4 | 1 | 2 | yes | — |
| 102 | C42×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C42×C89 | C24 | 1 | 1 | yes | characteristic |
| 104 | C23×C89 | #120 | 1 | 1 | yes | characteristic |
| 105 | Q8×C89 | — | 4 | 4 | no | — |
| 106 | C2×C4×C89 | — | 2 | 2 | no | — |
| 107 | C2×C4×C89 | — | 2 | 4 | no | — |
| 108 | C2×C4×C89 | — | 4 | 2 | no | — |
| 109 | C2×C4×C89 | #120 | 1 | 2 | yes | — |
| 110 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C89 | C22×D4 | 1 | 2 | yes | — |
| 112 | C2×C4×C89 | C22×D4 | 1 | 1 | yes | characteristic |
| 113 | C2×Dic89 | — | 2 | 4 | no | — |
| 114 | C2×Dic89 | — | 2 | 4 | no | — |
| 115 | C2×Dic89 | — | 4 | 2 | no | — |
| 116 | Dic178 | — | 4 | 4 | no | — |
| 117 | Dic178 | — | 4 | 4 | no | — |
| 118 | Dic178 | — | 4 | 4 | no | — |
| 119 | C8×C89 | — | 4 | 4 | no | — |
| 120 | #10623494 | — | 2 | 4 | no | — |
| 121 | C22×C89 | — | 2 | 1 | no | — |
| 122 | C22×C89 | — | 2 | 2 | no | — |
| 123 | C22×C89 | C2×#120 | 1 | 1 | yes | characteristic |
| 124 | C4×C89 | — | 2 | 1 | no | — |
| 125 | C4×C89 | — | 2 | 2 | no | — |
| 126 | C4×C89 | — | 2 | 2 | no | — |
| 127 | C4×C89 | — | 4 | 4 | no | — |
| 128 | C4×C89 | #456 | 1 | 2 | yes | — |
| 129 | Dic89 | — | 4 | 4 | no | — |
| 130 | Dic89 | — | 4 | 4 | no | — |
| 131 | Dic89 | — | 8 | 2 | no | — |
| 132 | #19067 | — | 89 | 1 | no | maximal, Sylow |
| 133 | C2×C89 | — | 4 | 1 | no | — |
| 134 | C2×C89 | C2×#456 | 1 | 1 | yes | characteristic |
| 135 | C2×C89 | #3020 | 1 | 2 | yes | — |
| 136 | #3556 | — | 89 | 1 | no | — |
| 137 | #3237 | — | 89 | 2 | no | — |
| 138 | #3162 | — | 89 | 1 | no | — |
| 139 | #1663 | — | 89 | 8 | no | — |
| 140 | #1537 | D89 | 1 | 1 | yes | characteristic |
| 141 | #1531 | — | 89 | 2 | no | — |
| 142 | C89 | #19067 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 143 | #564 | — | 89 | 4 | no | — |
| 144 | #557 | — | 178 | 2 | no | — |
| 145 | #557 | — | 178 | 2 | no | — |
| 146 | #557 | — | 178 | 2 | no | — |
| 147 | #541 | — | 178 | 4 | no | — |
| 148 | #541 | — | 178 | 2 | no | — |
| 149 | #533 | — | 89 | 2 | no | — |
| 150 | #533 | — | 89 | 1 | no | — |
| 151 | #533 | C2×D89 | 1 | 1 | yes | characteristic |
| 152 | #501 | — | 89 | 4 | no | — |
| 153 | #501 | — | 89 | 4 | no | — |
| 154 | #501 | — | 178 | 2 | no | — |
| 155 | #491 | — | 89 | 4 | no | — |
| 156 | #485 | — | 178 | 4 | no | — |
| 157 | #477 | — | 178 | 2 | no | — |
| 158 | #475 | — | 178 | 2 | no | — |
| 159 | #432 | C2×D89 | 1 | 2 | yes | — |
| 160 | #427 | — | 178 | 2 | no | — |
| 161 | C2×#97 | — | 89 | 1 | no | — |
| 162 | #330 | — | 89 | 8 | no | — |
| 163 | #330 | C2×D89 | 1 | 4 | yes | — |
| 164 | C22×C8 | — | 178 | 1 | no | — |
| 165 | C4⋊Q8 | — | 89 | 2 | no | — |
| 166 | C4⋊Q8 | — | 89 | 2 | no | — |
| 167 | C4⋊Q8 | — | 89 | 2 | no | — |
| 168 | C4⋊Q8 | — | 178 | 2 | no | — |
| 169 | C4⋊Q8 | — | 178 | 4 | no | — |
| 170 | C4⋊Q8 | D356 | 1 | 2 | yes | — |
| 171 | #126 | — | 178 | 8 | no | — |
| 172 | #125 | — | 89 | 4 | no | — |
| 173 | #125 | — | 178 | 2 | no | — |
| 174 | #125 | — | 178 | 8 | no | — |
| 175 | #125 | — | 178 | 8 | no | — |
| 176 | #125 | — | 178 | 4 | no | — |
| 177 | #125 | — | 178 | 8 | no | — |
| 178 | #125 | — | 178 | 1 | no | — |
| 179 | #125 | C22×D89 | 1 | 2 | yes | — |
| 180 | C22⋊Q8 | — | 2 | 2 | no | — |
| 181 | C22⋊Q8 | — | 178 | 2 | no | — |
| 182 | C22⋊Q8 | — | 178 | 2 | no | — |
| 183 | C22⋊Q8 | — | 178 | 2 | no | — |
| 184 | C4×Q8 | — | 2 | 2 | no | — |
| 185 | C4×Q8 | — | 178 | 4 | no | — |
| 186 | C4×Q8 | — | 178 | 2 | no | — |
| 187 | C4×Q8 | — | 178 | 4 | no | — |
| 188 | C4×Q8 | — | 178 | 4 | no | — |
| 189 | C4×Q8 | — | 178 | 2 | no | — |
| 190 | C4×Q8 | — | 178 | 2 | no | — |
| 191 | #117 | — | 89 | 4 | no | — |
| 192 | #117 | — | 178 | 1 | no | — |
| 193 | #117 | D356 | 1 | 2 | yes | — |
| 194 | C2×C42 | C22×D89 | 1 | 1 | yes | characteristic |
| 195 | #107 | — | 178 | 4 | no | — |
| 196 | #107 | — | 178 | 4 | no | — |
| 197 | #106 | — | 178 | 4 | no | — |
| 198 | #106 | — | 178 | 4 | no | — |
| 199 | C4⋊C8 | — | 2 | 2 | no | — |
| 200 | #103 | — | 178 | 4 | no | — |
| 201 | #103 | — | 178 | 4 | no | — |
| 202 | C22⋊C8 | — | 2 | 2 | no | — |
| 203 | #97 | — | 89 | 2 | no | — |
| 204 | #97 | — | 178 | 1 | no | — |
| 205 | #97 | C22×D89 | 1 | 4 | yes | — |
| 206 | C2×Q8 | — | 2 | 2 | no | — |
| 207 | C2×Q8 | — | 178 | 2 | no | — |
| 208 | C2×Q8 | — | 178 | 2 | no | — |
| 209 | C2×Q8 | — | 178 | 2 | no | — |
| 210 | C22×C4 | D4×D89 | 1 | 2 | yes | — |
| 211 | C22×C4 | C2×D356 | 1 | 1 | yes | characteristic |
| 212 | C2×C8 | — | 2 | 4 | no | — |
| 213 | C2×C8 | — | 178 | 4 | no | — |
| 214 | C2×C8 | — | 178 | 2 | no | — |
| 215 | C4⋊C4 | — | 2 | 2 | no | — |
| 216 | C4⋊C4 | — | 2 | 4 | no | — |
| 217 | C4⋊C4 | — | 89 | 4 | no | — |
| 218 | C4⋊C4 | — | 89 | 4 | no | — |
| 219 | C4⋊C4 | — | 356 | 2 | no | — |
| 220 | C4⋊C4 | — | 178 | 8 | no | — |
| 221 | C4⋊C4 | — | 178 | 8 | no | — |
| 222 | C4⋊C4 | — | 178 | 8 | no | — |
| 223 | C4⋊C4 | — | 178 | 2 | no | — |
| 224 | C4⋊C4 | — | 178 | 2 | no | — |
| 225 | C4⋊C4 | — | 178 | 2 | no | — |
| 226 | C4⋊C4 | — | 178 | 2 | no | — |
| 227 | C4⋊C4 | — | 178 | 4 | no | — |
| 228 | C4⋊C4 | — | 178 | 4 | no | — |
| 229 | C4⋊C4 | C2×D356 | 1 | 4 | yes | — |
| 230 | C22⋊C4 | — | 2 | 2 | no | — |
| 231 | C22⋊C4 | — | 356 | 1 | no | — |
| 232 | C22⋊C4 | — | 178 | 4 | no | — |
| 233 | C42 | — | 2 | 2 | no | — |
| 234 | C42 | — | 356 | 1 | no | — |
| 235 | C42 | — | 178 | 8 | no | — |
| 236 | C42 | — | 178 | 4 | no | — |
| 237 | C42 | D4×D89 | 1 | 2 | yes | — |
| 238 | C42 | C2×D356 | 1 | 1 | yes | characteristic |
| 239 | C42 | C23×D89 | 1 | 1 | yes | characteristic |
| 240 | C23 | #15667982 | 1 | 1 | yes | characteristic |
| 241 | Q8 | — | 4 | 4 | no | — |
| 242 | Q8 | — | 356 | 4 | no | — |
| 243 | Q8 | — | 356 | 4 | no | — |
| 244 | Q8 | — | 356 | 4 | no | — |
| 245 | C2×C4 | — | 2 | 2 | no | — |
| 246 | C2×C4 | — | 2 | 4 | no | — |
| 247 | C2×C4 | — | 4 | 2 | no | — |
| 248 | C2×C4 | — | 356 | 2 | no | — |
| 249 | C2×C4 | — | 178 | 4 | no | — |
| 250 | C2×C4 | — | 178 | 4 | no | — |
| 251 | C2×C4 | #15667982 | 1 | 2 | yes | — |
| 252 | C2×C4 | #15667982 | 1 | 1 | yes | characteristic |
| 253 | C2×C4 | C2×D4×D89 | 1 | 2 | yes | — |
| 254 | C2×C4 | C22×D356 | 1 | 1 | yes | characteristic |
| 255 | C8 | — | 4 | 4 | no | — |
| 256 | C8 | — | 178 | 4 | no | — |
| 257 | C22 | — | 2 | 1 | no | — |
| 258 | C22 | — | 2 | 2 | no | — |
| 259 | C22 | C2×#15667982 | 1 | 1 | yes | characteristic |
| 260 | C4 | — | 2 | 1 | no | — |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 2 | 2 | no | — |
| 263 | C4 | — | 4 | 4 | no | — |
| 264 | C4 | — | 356 | 4 | no | — |
| 265 | C4 | — | 356 | 4 | no | — |
| 266 | C4 | — | 712 | 2 | no | — |
| 267 | C4 | #19151783 | 1 | 2 | yes | — |
| 268 | C2 | — | 4 | 1 | no | — |
| 269 | C2 | C2×#19151783 | 1 | 1 | yes | characteristic |
| 270 | C2 | #26029618 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #26019582 | C2 | semidirect |
| 142. C89 | #19067 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26034747 | 2 |
| #26031468 | 2 |
| #26030774 | 2 |
| #26019582 | 2 |
| #26018868 | 2 |
| C89×#1537 | 2 |
| #19067 | 89 |