Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
454 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 43 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 | #26032045 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #26022238 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #26019366 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #26018396 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C89×#1409 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19152404 | — | 2 | 2 | no | — |
| 7 | #19152233 | C22 | 1 | 1 | yes | characteristic |
| 8 | #19151885 | — | 2 | 1 | no | — |
| 9 | #19151858 | — | 2 | 2 | no | — |
| 10 | #19151786 | — | 2 | 2 | no | — |
| 11 | #19151780 | — | 2 | 1 | no | — |
| 12 | #19151765 | C22 | 1 | 2 | yes | — |
| 13 | #19151716 | — | 2 | 1 | no | — |
| 14 | #19151384 | — | 2 | 2 | no | — |
| 15 | #19151342 | C22 | 1 | 1 | yes | characteristic |
| 16 | #19151327 | — | 2 | 1 | no | — |
| 17 | #492×C89 | C22 | 1 | 1 | yes | characteristic |
| 18 | #353×C89 | C22 | 1 | 2 | yes | — |
| 19 | #15668060 | — | 2 | 1 | no | — |
| 20 | #15668057 | D4 | 1 | 2 | yes | complemented |
| 21 | C2×D4×D89 | — | 4 | 1 | no | — |
| 22 | #15668043 | — | 2 | 2 | no | — |
| 23 | #15668041 | — | 4 | 2 | no | — |
| 24 | #15668039 | — | 4 | 2 | no | — |
| 25 | #15668038 | — | 2 | 2 | no | — |
| 26 | #15668034 | — | 2 | 1 | no | — |
| 27 | #15668033 | — | 4 | 2 | no | — |
| 28 | SD16×D89 | — | 4 | 2 | no | — |
| 29 | #15668018 | — | 2 | 2 | no | — |
| 30 | #15667983 | — | 4 | 1 | no | — |
| 31 | #15667982 | — | 4 | 2 | no | — |
| 32 | #15667980 | — | 2 | 2 | no | — |
| 33 | C4×D356 | — | 2 | 1 | no | — |
| 34 | #15667948 | — | 2 | 1 | no | — |
| 35 | #15667943 | — | 4 | 1 | no | — |
| 36 | C4⋊#10623494 | — | 2 | 1 | no | — |
| 37 | #15667940 | — | 2 | 2 | no | — |
| 38 | #15667932 | — | 4 | 1 | no | — |
| 39 | #15667931 | — | 2 | 1 | no | — |
| 40 | #15667929 | D4 | 1 | 2 | yes | complemented |
| 41 | #15667926 | — | 4 | 2 | no | — |
| 42 | C2×SD16×C89 | — | 2 | 2 | no | — |
| 43 | C4⋊Q8×C89 | D4 | 1 | 1 | yes | characteristic, complemented |
| 44 | #127×C89 | C23 | 1 | 1 | yes | characteristic |
| 45 | #99×C89 | — | 2 | 2 | no | — |
| 46 | C4×C8×C89 | D4 | 1 | 1 | yes | characteristic, complemented |
| 47 | C23×D89 | — | 2 | 2 | no | — |
| 48 | #13083575 | — | 4 | 2 | no | — |
| 49 | #13083574 | — | 2 | 2 | no | — |
| 50 | #13083574 | — | 8 | 1 | no | — |
| 51 | #13083573 | — | 4 | 2 | no | — |
| 52 | Q8×D89 | — | 4 | 2 | no | — |
| 53 | C2×D356 | — | 2 | 1 | no | — |
| 54 | C2×D356 | — | 8 | 2 | no | — |
| 55 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 56 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 57 | D4×D89 | — | 4 | 2 | no | — |
| 58 | D4×D89 | — | 8 | 2 | no | — |
| 59 | D4×D89 | — | 8 | 2 | no | — |
| 60 | C2×C4×D89 | — | 4 | 1 | no | — |
| 61 | Dic356 | — | 2 | 1 | no | — |
| 62 | #13083564 | — | 4 | 2 | no | — |
| 63 | #13083563 | — | 2 | 2 | no | — |
| 64 | #13083562 | — | 4 | 2 | no | — |
| 65 | #13083561 | — | 4 | 2 | no | — |
| 66 | D712 | — | 2 | 1 | no | — |
| 67 | #13083559 | — | 4 | 2 | no | — |
| 68 | #13083558 | — | 2 | 2 | no | — |
| 69 | #13083557 | — | 4 | 2 | no | — |
| 70 | C2×#10623494 | — | 4 | 1 | no | — |
| 71 | C8×D89 | — | 4 | 2 | no | — |
| 72 | C4⋊Dic89 | — | 2 | 1 | no | — |
| 73 | C22⋊Dic89 | — | 2 | 2 | no | — |
| 74 | #13083551 | — | 4 | 1 | no | — |
| 75 | #13083551 | — | 8 | 2 | no | — |
| 76 | C2×Q8×C89 | — | 2 | 1 | no | — |
| 77 | C2×D4×C89 | — | 4 | 1 | no | — |
| 78 | C2×D4×C89 | C2×D4 | 1 | 2 | yes | — |
| 79 | SD16×C89 | — | 4 | 2 | no | — |
| 80 | SD16×C89 | — | 4 | 2 | no | — |
| 81 | C2×C8×C89 | — | 2 | 1 | no | — |
| 82 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 83 | C22⋊C4×C89 | — | 4 | 2 | no | — |
| 84 | C42×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C23×C89 | — | 2 | 2 | no | — |
| 86 | C22×D89 | — | 4 | 1 | no | — |
| 87 | C22×D89 | — | 4 | 2 | no | — |
| 88 | C22×D89 | — | 8 | 4 | no | — |
| 89 | C22×D89 | — | 8 | 2 | no | — |
| 90 | Q8×C89 | — | 4 | 2 | no | — |
| 91 | D4×C89 | — | 4 | 2 | no | — |
| 92 | D4×C89 | — | 8 | 1 | no | — |
| 93 | D4×C89 | — | 8 | 1 | no | — |
| 94 | C2×C4×C89 | — | 2 | 1 | no | — |
| 95 | C2×C4×C89 | — | 4 | 1 | no | — |
| 96 | C2×C4×C89 | — | 8 | 2 | no | — |
| 97 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 98 | C22⋊D89 | — | 4 | 2 | no | — |
| 99 | C22⋊D89 | — | 8 | 2 | no | — |
| 100 | C22⋊D89 | — | 8 | 4 | no | — |
| 101 | C2×Dic89 | — | 4 | 2 | no | — |
| 102 | C2×Dic89 | — | 4 | 1 | no | — |
| 103 | D356 | — | 2 | 2 | no | — |
| 104 | D356 | — | 2 | 2 | no | — |
| 105 | D356 | — | 4 | 1 | no | — |
| 106 | D356 | — | 8 | 1 | no | — |
| 107 | D356 | — | 16 | 4 | no | — |
| 108 | C4×D89 | — | 4 | 2 | no | — |
| 109 | C4×D89 | — | 8 | 2 | no | — |
| 110 | C4×D89 | — | 8 | 1 | no | — |
| 111 | C4×D89 | — | 8 | 1 | no | — |
| 112 | Dic178 | — | 2 | 2 | no | — |
| 113 | Dic178 | — | 8 | 1 | no | — |
| 114 | C8×C89 | — | 2 | 1 | no | — |
| 115 | C8×C89 | — | 2 | 1 | no | — |
| 116 | #10623494 | — | 4 | 2 | no | — |
| 117 | #10623494 | — | 8 | 1 | no | — |
| 118 | C22×C89 | — | 4 | 2 | no | — |
| 119 | C22×C89 | — | 8 | 2 | no | — |
| 120 | C22×C89 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 121 | C2×D89 | — | 4 | 2 | no | — |
| 122 | C2×D89 | — | 4 | 2 | no | — |
| 123 | C2×D89 | — | 8 | 1 | no | — |
| 124 | C2×D89 | — | 8 | 1 | no | — |
| 125 | C2×D89 | — | 8 | 4 | no | — |
| 126 | C2×D89 | — | 16 | 2 | no | — |
| 127 | C2×D89 | — | 16 | 2 | no | — |
| 128 | C4×C89 | — | 2 | 1 | no | — |
| 129 | C4×C89 | — | 2 | 1 | no | — |
| 130 | C4×C89 | — | 2 | 1 | no | — |
| 131 | C4×C89 | — | 8 | 1 | no | — |
| 132 | C4×C89 | — | 16 | 2 | no | — |
| 133 | Dic89 | — | 4 | 2 | no | — |
| 134 | Dic89 | — | 8 | 1 | no | — |
| 135 | #13674 | — | 89 | 1 | no | maximal, Sylow |
| 136 | C2×C89 | — | 2 | 1 | no | — |
| 137 | C2×C89 | — | 8 | 2 | no | — |
| 138 | C2×C89 | #2196 | 1 | 1 | yes | characteristic |
| 139 | D89 | — | 8 | 2 | no | — |
| 140 | D89 | — | 8 | 2 | no | — |
| 141 | #3291 | — | 89 | 1 | no | — |
| 142 | #2196 | — | 89 | 2 | no | — |
| 143 | #1629 | — | 89 | 1 | no | — |
| 144 | #1409 | D89 | 1 | 1 | yes | characteristic, complemented |
| 145 | #1402 | — | 89 | 2 | no | — |
| 146 | C89 | #13674 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 147 | #577 | — | 178 | 2 | no | — |
| 148 | D42 | — | 89 | 1 | no | — |
| 149 | #492 | C2×D89 | 1 | 1 | yes | characteristic |
| 150 | #476 | — | 178 | 1 | no | — |
| 151 | #471 | — | 178 | 2 | no | — |
| 152 | B(4,𝔽2) | — | 178 | 2 | no | — |
| 153 | #456 | — | 178 | 1 | no | — |
| 154 | #453 | — | 89 | 2 | no | — |
| 155 | #443 | — | 178 | 1 | no | — |
| 156 | #353 | C2×D89 | 1 | 2 | yes | — |
| 157 | AΣL(1,𝔽16) | — | 178 | 2 | no | — |
| 158 | #331 | — | 89 | 1 | no | — |
| 159 | D4⋊C8 | — | 178 | 1 | no | — |
| 160 | 2−1+4 | — | 178 | 1 | no | — |
| 161 | H(5,𝔽2) | — | 89 | 2 | no | — |
| 162 | C22×D4 | — | 356 | 1 | no | — |
| 163 | #137 | — | 356 | 2 | no | — |
| 164 | #137 | — | 178 | 2 | no | — |
| 165 | AGL(1,ℤ/8) | — | 356 | 2 | no | — |
| 166 | AGL(1,ℤ/8) | — | 178 | 2 | no | — |
| 167 | #135 | — | 356 | 2 | no | — |
| 168 | #135 | — | 178 | 1 | no | — |
| 169 | C2×SD16 | — | 2 | 2 | no | — |
| 170 | C2×SD16 | — | 356 | 2 | no | — |
| 171 | #131 | — | 178 | 2 | no | — |
| 172 | C4⋊Q8 | D356 | 1 | 1 | yes | characteristic, complemented |
| 173 | #127 | C22×D89 | 1 | 1 | yes | characteristic |
| 174 | #121 | — | 356 | 1 | no | — |
| 175 | #120 | — | 356 | 2 | no | — |
| 176 | #120 | — | 178 | 2 | no | — |
| 177 | C4×D4 | — | 178 | 1 | no | — |
| 178 | #108 | — | 178 | 1 | no | — |
| 179 | #106 | — | 356 | 1 | no | — |
| 180 | C4⋊C8 | — | 178 | 1 | no | — |
| 181 | #104 | — | 178 | 2 | no | — |
| 182 | #102 | — | 356 | 1 | no | — |
| 183 | #101 | — | 178 | 1 | no | — |
| 184 | #100 | — | 89 | 2 | no | — |
| 185 | #99 | — | 2 | 2 | no | — |
| 186 | #99 | — | 356 | 2 | no | — |
| 187 | C4×C8 | D356 | 1 | 1 | yes | characteristic, complemented |
| 188 | C24 | — | 178 | 2 | no | — |
| 189 | #41 | — | 356 | 2 | no | — |
| 190 | #41 | — | 356 | 2 | no | — |
| 191 | #41 | — | 178 | 2 | no | — |
| 192 | #41 | — | 712 | 1 | no | — |
| 193 | C2×Q8 | — | 2 | 1 | no | — |
| 194 | C2×Q8 | — | 356 | 2 | no | — |
| 195 | C2×D4 | — | 4 | 1 | no | — |
| 196 | C2×D4 | — | 356 | 2 | no | — |
| 197 | C2×D4 | — | 356 | 2 | no | — |
| 198 | C2×D4 | — | 356 | 2 | no | — |
| 199 | C2×D4 | — | 178 | 1 | no | — |
| 200 | C2×D4 | — | 712 | 2 | no | — |
| 201 | C2×D4 | — | 712 | 2 | no | — |
| 202 | C2×D4 | — | 712 | 2 | no | — |
| 203 | C2×D4 | D4×D89 | 1 | 2 | yes | — |
| 204 | C22×C4 | — | 356 | 1 | no | — |
| 205 | Q16 | — | 356 | 2 | no | — |
| 206 | Q16 | — | 178 | 1 | no | — |
| 207 | SD16 | — | 4 | 2 | no | — |
| 208 | SD16 | — | 4 | 2 | no | — |
| 209 | SD16 | — | 356 | 2 | no | — |
| 210 | SD16 | — | 356 | 2 | no | — |
| 211 | SD16 | — | 178 | 2 | no | — |
| 212 | D8 | — | 356 | 2 | no | — |
| 213 | D8 | — | 178 | 1 | no | — |
| 214 | M16 | — | 356 | 2 | no | — |
| 215 | M16 | — | 178 | 2 | no | — |
| 216 | C2×C8 | — | 2 | 1 | no | — |
| 217 | C2×C8 | — | 356 | 2 | no | — |
| 218 | C2×C8 | — | 356 | 1 | no | — |
| 219 | C4⋊C4 | — | 4 | 1 | no | — |
| 220 | C4⋊C4 | — | 178 | 1 | no | — |
| 221 | C22⋊C4 | — | 4 | 2 | no | — |
| 222 | C22⋊C4 | — | 356 | 1 | no | — |
| 223 | C22⋊C4 | — | 178 | 2 | no | — |
| 224 | C22⋊C4 | — | 712 | 2 | no | — |
| 225 | C42 | C2×D356 | 1 | 1 | yes | characteristic |
| 226 | C23 | — | 2 | 2 | no | — |
| 227 | C23 | — | 356 | 2 | no | — |
| 228 | C23 | — | 356 | 1 | no | — |
| 229 | C23 | — | 712 | 2 | no | — |
| 230 | C23 | — | 712 | 4 | no | — |
| 231 | Q8 | — | 4 | 2 | no | — |
| 232 | Q8 | — | 178 | 2 | no | — |
| 233 | Q8 | — | 712 | 1 | no | — |
| 234 | D4 | — | 4 | 2 | no | — |
| 235 | D4 | — | 8 | 1 | no | — |
| 236 | D4 | — | 8 | 1 | no | — |
| 237 | D4 | — | 356 | 1 | no | — |
| 238 | D4 | — | 356 | 2 | no | — |
| 239 | D4 | — | 1424 | 4 | no | — |
| 240 | D4 | — | 178 | 2 | no | — |
| 241 | D4 | — | 178 | 2 | no | — |
| 242 | D4 | — | 712 | 1 | no | — |
| 243 | D4 | — | 712 | 2 | no | — |
| 244 | D4 | — | 712 | 4 | no | — |
| 245 | C2×C4 | — | 2 | 1 | no | — |
| 246 | C2×C4 | — | 4 | 1 | no | — |
| 247 | C2×C4 | — | 8 | 2 | no | — |
| 248 | C2×C4 | — | 356 | 2 | no | — |
| 249 | C2×C4 | — | 356 | 2 | no | — |
| 250 | C2×C4 | — | 356 | 1 | no | — |
| 251 | C2×C4 | — | 712 | 1 | no | — |
| 252 | C2×C4 | — | 712 | 1 | no | — |
| 253 | C2×C4 | — | 712 | 2 | no | — |
| 254 | C2×C4 | #15667982 | 1 | 1 | yes | characteristic |
| 255 | C8 | — | 2 | 1 | no | — |
| 256 | C8 | — | 2 | 1 | no | — |
| 257 | C8 | — | 356 | 2 | no | — |
| 258 | C8 | — | 712 | 1 | no | — |
| 259 | C22 | — | 4 | 2 | no | — |
| 260 | C22 | — | 8 | 2 | no | — |
| 261 | C22 | — | 356 | 2 | no | — |
| 262 | C22 | — | 356 | 2 | no | — |
| 263 | C22 | — | 1424 | 2 | no | — |
| 264 | C22 | — | 1424 | 2 | no | — |
| 265 | C22 | — | 712 | 1 | no | — |
| 266 | C22 | — | 712 | 4 | no | — |
| 267 | C22 | — | 712 | 1 | no | — |
| 268 | C22 | #19151786 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 2 | 1 | no | — |
| 270 | C4 | — | 2 | 1 | no | — |
| 271 | C4 | — | 2 | 1 | no | — |
| 272 | C4 | — | 8 | 1 | no | — |
| 273 | C4 | — | 16 | 2 | no | — |
| 274 | C4 | — | 356 | 2 | no | — |
| 275 | C4 | — | 712 | 1 | no | — |
| 276 | C2 | — | 2 | 1 | no | — |
| 277 | C2 | — | 8 | 2 | no | — |
| 278 | C2 | — | 712 | 2 | no | — |
| 279 | C2 | — | 712 | 2 | no | — |
| 280 | C2 | #26022238 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26032045 | C2 | semidirect |
| 2. #26022238 | C2 | semidirect |
| 3. #26019366 | C2 | semidirect |
| 4. #26018396 | C2 | semidirect |
| 5. C89×#1409 | C2 | semidirect |
| 20. #15668057 | D4 | semidirect |
| 40. #15667929 | D4 | semidirect |
| 43. C4⋊Q8×C89 | D4 | semidirect |
| 46. C4×C8×C89 | D4 | semidirect |
| 144. #1409 | D89 | semidirect |
| 146. C89 | #13674 | semidirect |
| 172. C4⋊Q8 | D356 | semidirect |
| 187. C4×C8 | D356 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26032045 | 2 |
| #26022238 | 2 |
| #26019366 | 2 |
| #26018396 | 2 |
| C89×#1409 | 2 |
| #13674 | 89 |