Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
356 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 356 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 | #26031691 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #26022198 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #26018236 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #26018205 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C89×#2053 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #19152203 | — | 2 | 1 | no | — |
| 7 | #19152033 | — | 2 | 2 | no | — |
| 8 | #19151899 | — | 2 | 2 | no | — |
| 9 | #19151851 | C22 | 1 | 2 | yes | — |
| 10 | #19151834 | — | 2 | 1 | no | — |
| 11 | #19151794 | — | 2 | 2 | no | — |
| 12 | #19151408 | — | 2 | 2 | no | — |
| 13 | #19151380 | C22 | 1 | 2 | yes | — |
| 14 | #575×C89 | C22 | 1 | 1 | yes | characteristic |
| 15 | #424×C89 | C22 | 1 | 1 | yes | characteristic |
| 16 | #416×C89 | — | 2 | 1 | no | — |
| 17 | #410×C89 | — | 2 | 1 | no | — |
| 18 | H(3,ℤ/4)×C89 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×D712 | — | 2 | 1 | no | — |
| 20 | C2×D712 | D4 | 1 | 2 | yes | — |
| 21 | #15668009 | — | 2 | 1 | no | — |
| 22 | #15668004 | — | 2 | 1 | no | — |
| 23 | #15667997 | — | 4 | 2 | no | — |
| 24 | #15667986 | — | 2 | 2 | no | — |
| 25 | #15667985 | — | 4 | 2 | no | — |
| 26 | C4×D356 | — | 2 | 2 | no | — |
| 27 | #15667955 | — | 4 | 2 | no | — |
| 28 | #15667953 | — | 4 | 2 | no | — |
| 29 | #15667952 | — | 2 | 2 | no | — |
| 30 | #15667946 | — | 2 | 1 | no | — |
| 31 | #15667944 | D4 | 1 | 2 | yes | — |
| 32 | #15667934 | — | 2 | 2 | no | — |
| 33 | #15667934 | — | 2 | 2 | no | — |
| 34 | C2×#41×C89 | — | 2 | 1 | no | — |
| 35 | #137×C89 | — | 4 | 1 | no | — |
| 36 | AGL(1,ℤ/8)×C89 | — | 4 | 1 | no | — |
| 37 | #135×C89 | — | 2 | 2 | no | — |
| 38 | C2×M16×C89 | C23 | 1 | 1 | yes | characteristic |
| 39 | C22×C8×C89 | D4 | 1 | 1 | yes | characteristic |
| 40 | #117×C89 | — | 2 | 1 | no | — |
| 41 | #117×C89 | D4 | 1 | 1 | yes | characteristic |
| 42 | C4⋊C8×C89 | — | 2 | 2 | no | — |
| 43 | #103×C89 | — | 4 | 1 | no | — |
| 44 | #102×C89 | — | 4 | 1 | no | — |
| 45 | #99×C89 | — | 4 | 2 | no | — |
| 46 | C2×D356 | — | 2 | 2 | no | — |
| 47 | C2×D356 | — | 4 | 2 | no | — |
| 48 | C2×C22⋊D89 | — | 4 | 2 | no | — |
| 49 | C2×C4×D89 | — | 4 | 2 | no | — |
| 50 | D712 | — | 4 | 2 | no | — |
| 51 | D712 | — | 8 | 1 | no | — |
| 52 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 53 | #13083553 | — | 4 | 2 | no | — |
| 54 | #13083551 | — | 4 | 2 | no | — |
| 55 | #13083551 | — | 4 | 2 | no | — |
| 56 | #13083549 | — | 4 | 2 | no | — |
| 57 | #41×C89 | — | 4 | 1 | no | — |
| 58 | #41×C89 | — | 4 | 1 | no | — |
| 59 | #41×C89 | — | 4 | 1 | no | — |
| 60 | #41×C89 | — | 4 | 1 | no | — |
| 61 | C2×Q8×C89 | — | 2 | 1 | no | — |
| 62 | C2×D4×C89 | — | 2 | 1 | no | — |
| 63 | C2×D4×C89 | — | 4 | 1 | no | — |
| 64 | C22×C4×C89 | — | 4 | 1 | no | — |
| 65 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | Q16×C89 | — | 2 | 2 | no | — |
| 67 | SD16×C89 | — | 4 | 2 | no | — |
| 68 | D8×C89 | — | 2 | 2 | no | — |
| 69 | M16×C89 | — | 4 | 1 | no | — |
| 70 | C2×C8×C89 | — | 2 | 1 | no | — |
| 71 | C2×C8×C89 | — | 4 | 1 | no | — |
| 72 | C2×C8×C89 | C2×D4 | 1 | 2 | yes | — |
| 73 | C4⋊C4×C89 | — | 2 | 1 | no | — |
| 74 | C4⋊C4×C89 | — | 4 | 1 | no | — |
| 75 | C22⋊C4×C89 | — | 4 | 1 | no | — |
| 76 | C22⋊C4×C89 | D8 | 1 | 2 | yes | — |
| 77 | C42×C89 | — | 2 | 1 | no | — |
| 78 | C42×C89 | — | 4 | 1 | no | — |
| 79 | C23×C89 | — | 4 | 1 | no | — |
| 80 | C23×C89 | C2×D8 | 1 | 1 | yes | characteristic |
| 81 | C22×D89 | — | 4 | 2 | no | — |
| 82 | Q8×C89 | — | 4 | 1 | no | — |
| 83 | Q8×C89 | — | 4 | 1 | no | — |
| 84 | D4×C89 | — | 4 | 1 | no | — |
| 85 | D4×C89 | — | 4 | 1 | no | — |
| 86 | D4×C89 | — | 8 | 1 | no | — |
| 87 | D4×C89 | — | 8 | 1 | no | — |
| 88 | C2×C4×C89 | — | 2 | 1 | no | — |
| 89 | C2×C4×C89 | — | 2 | 1 | no | — |
| 90 | C2×C4×C89 | — | 2 | 2 | no | — |
| 91 | C2×C4×C89 | — | 4 | 1 | no | — |
| 92 | C2×C4×C89 | — | 8 | 1 | no | — |
| 93 | C2×C4×C89 | — | 8 | 1 | no | — |
| 94 | C2×C4×C89 | — | 8 | 1 | no | — |
| 95 | C2×C4×C89 | #120 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 97 | C22⋊D89 | — | 16 | 2 | no | — |
| 98 | C2×Dic89 | — | 4 | 2 | no | — |
| 99 | D356 | — | 4 | 2 | no | — |
| 100 | D356 | — | 4 | 2 | no | — |
| 101 | D356 | — | 16 | 2 | no | — |
| 102 | C4×D89 | — | 16 | 2 | no | — |
| 103 | C8×C89 | — | 2 | 2 | no | — |
| 104 | C8×C89 | — | 4 | 1 | no | — |
| 105 | C22×C89 | — | 2 | 1 | no | — |
| 106 | C22×C89 | — | 4 | 1 | no | — |
| 107 | C22×C89 | — | 8 | 1 | no | — |
| 108 | C22×C89 | — | 16 | 1 | no | — |
| 109 | C22×C89 | #447 | 1 | 1 | yes | characteristic |
| 110 | C2×D89 | — | 8 | 2 | no | — |
| 111 | C2×D89 | — | 8 | 2 | no | — |
| 112 | C2×D89 | — | 8 | 2 | no | — |
| 113 | C4×C89 | — | 2 | 1 | no | — |
| 114 | C4×C89 | — | 4 | 2 | no | — |
| 115 | C4×C89 | — | 8 | 1 | no | — |
| 116 | C4×C89 | — | 16 | 1 | no | — |
| 117 | C4×C89 | #454 | 1 | 1 | yes | characteristic |
| 118 | C4×C89 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 119 | Dic89 | — | 8 | 2 | no | — |
| 120 | #12170 | — | 89 | 1 | no | maximal, Sylow |
| 121 | C2×C89 | — | 2 | 1 | no | — |
| 122 | C2×C89 | — | 4 | 1 | no | — |
| 123 | C2×C89 | — | 16 | 1 | no | — |
| 124 | C2×C89 | #1598 | 1 | 1 | yes | characteristic |
| 125 | D89 | — | 16 | 2 | no | — |
| 126 | #3259 | — | 89 | 1 | no | — |
| 127 | #2190 | — | 89 | 2 | no | — |
| 128 | #2053 | D89 | 1 | 1 | yes | characteristic, complemented |
| 129 | #1339 | — | 89 | 2 | no | — |
| 130 | #1328 | — | 89 | 1 | no | — |
| 131 | C89 | #12170 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 132 | #575 | C2×D89 | 1 | 1 | yes | characteristic |
| 133 | #540 | — | 178 | 1 | no | — |
| 134 | #509 | — | 178 | 2 | no | — |
| 135 | #478 | — | 178 | 2 | no | — |
| 136 | #469 | — | 89 | 2 | no | — |
| 137 | #466 | — | 178 | 1 | no | — |
| 138 | #459 | — | 178 | 2 | no | — |
| 139 | #424 | C2×D89 | 1 | 1 | yes | characteristic |
| 140 | #416 | — | 2 | 1 | no | — |
| 141 | #410 | — | 2 | 1 | no | — |
| 142 | #360 | — | 178 | 2 | no | — |
| 143 | #349 | — | 89 | 2 | no | — |
| 144 | H(3,ℤ/4) | C2×D89 | 1 | 1 | yes | characteristic |
| 145 | C2×#41 | — | 2 | 1 | no | — |
| 146 | #137 | — | 4 | 1 | no | — |
| 147 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 148 | #135 | — | 2 | 2 | no | — |
| 149 | C2×D8 | — | 89 | 2 | no | — |
| 150 | C2×D8 | — | 178 | 1 | no | — |
| 151 | C2×M16 | C22×D89 | 1 | 1 | yes | characteristic |
| 152 | C22×C8 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 153 | #127 | — | 178 | 1 | no | — |
| 154 | #125 | — | 178 | 1 | no | — |
| 155 | #123 | — | 356 | 2 | no | — |
| 156 | #121 | — | 356 | 2 | no | — |
| 157 | #121 | — | 178 | 2 | no | — |
| 158 | C4×D4 | — | 178 | 2 | no | — |
| 159 | #117 | — | 2 | 1 | no | — |
| 160 | #117 | C22⋊D89 | 1 | 1 | yes | characteristic |
| 161 | SD32 | — | 356 | 2 | no | — |
| 162 | D16 | — | 356 | 2 | no | — |
| 163 | M32 | — | 178 | 2 | no | — |
| 164 | #107 | — | 178 | 1 | no | — |
| 165 | #106 | — | 89 | 2 | no | — |
| 166 | C4⋊C8 | — | 2 | 2 | no | — |
| 167 | #103 | — | 4 | 1 | no | — |
| 168 | #102 | — | 4 | 1 | no | — |
| 169 | #102 | — | 178 | 2 | no | — |
| 170 | #102 | — | 178 | 2 | no | — |
| 171 | #99 | — | 4 | 2 | no | — |
| 172 | #41 | — | 4 | 1 | no | — |
| 173 | #41 | — | 4 | 1 | no | — |
| 174 | #41 | — | 4 | 1 | no | — |
| 175 | #41 | — | 4 | 1 | no | — |
| 176 | C2×Q8 | — | 2 | 1 | no | — |
| 177 | C2×D4 | — | 2 | 1 | no | — |
| 178 | C2×D4 | — | 4 | 1 | no | — |
| 179 | C2×D4 | — | 356 | 2 | no | — |
| 180 | C2×D4 | — | 356 | 2 | no | — |
| 181 | C2×D4 | — | 178 | 2 | no | — |
| 182 | C22×C4 | — | 4 | 1 | no | — |
| 183 | C22×C4 | — | 356 | 2 | no | — |
| 184 | C22×C4 | C2×C22⋊D89 | 1 | 1 | yes | characteristic |
| 185 | Q16 | — | 2 | 2 | no | — |
| 186 | SD16 | — | 4 | 2 | no | — |
| 187 | D8 | — | 2 | 2 | no | — |
| 188 | D8 | — | 356 | 2 | no | — |
| 189 | D8 | — | 712 | 1 | no | — |
| 190 | M16 | — | 4 | 1 | no | — |
| 191 | C2×C8 | — | 2 | 1 | no | — |
| 192 | C2×C8 | — | 4 | 1 | no | — |
| 193 | C2×C8 | D4×D89 | 1 | 2 | yes | — |
| 194 | C4⋊C4 | — | 2 | 1 | no | — |
| 195 | C4⋊C4 | — | 4 | 1 | no | — |
| 196 | C4⋊C4 | — | 356 | 2 | no | — |
| 197 | C4⋊C4 | — | 178 | 2 | no | — |
| 198 | C22⋊C4 | — | 4 | 1 | no | — |
| 199 | C22⋊C4 | — | 356 | 2 | no | — |
| 200 | C22⋊C4 | — | 356 | 2 | no | — |
| 201 | C22⋊C4 | #13083559 | 1 | 2 | yes | — |
| 202 | C42 | — | 2 | 1 | no | — |
| 203 | C42 | — | 4 | 1 | no | — |
| 204 | C16 | — | 356 | 2 | no | — |
| 205 | C23 | — | 4 | 1 | no | — |
| 206 | C23 | — | 356 | 2 | no | — |
| 207 | C23 | C2×#13083559 | 1 | 1 | yes | characteristic |
| 208 | Q8 | — | 4 | 1 | no | — |
| 209 | Q8 | — | 4 | 1 | no | — |
| 210 | D4 | — | 4 | 1 | no | — |
| 211 | D4 | — | 4 | 1 | no | — |
| 212 | D4 | — | 8 | 1 | no | — |
| 213 | D4 | — | 8 | 1 | no | — |
| 214 | D4 | — | 356 | 2 | no | — |
| 215 | D4 | — | 356 | 2 | no | — |
| 216 | D4 | — | 1424 | 2 | no | — |
| 217 | D4 | — | 1424 | 2 | no | — |
| 218 | C2×C4 | — | 2 | 1 | no | — |
| 219 | C2×C4 | — | 2 | 1 | no | — |
| 220 | C2×C4 | — | 2 | 2 | no | — |
| 221 | C2×C4 | — | 4 | 1 | no | — |
| 222 | C2×C4 | — | 8 | 1 | no | — |
| 223 | C2×C4 | — | 8 | 1 | no | — |
| 224 | C2×C4 | — | 8 | 1 | no | — |
| 225 | C2×C4 | — | 356 | 2 | no | — |
| 226 | C2×C4 | — | 1424 | 2 | no | — |
| 227 | C2×C4 | #15667980 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #15668037 | 1 | 1 | yes | characteristic |
| 229 | C8 | — | 2 | 2 | no | — |
| 230 | C8 | — | 4 | 1 | no | — |
| 231 | C22 | — | 2 | 1 | no | — |
| 232 | C22 | — | 4 | 1 | no | — |
| 233 | C22 | — | 8 | 1 | no | — |
| 234 | C22 | — | 16 | 1 | no | — |
| 235 | C22 | — | 712 | 2 | no | — |
| 236 | C22 | — | 712 | 2 | no | — |
| 237 | C22 | — | 712 | 2 | no | — |
| 238 | C22 | #19151731 | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 2 | 1 | no | — |
| 240 | C4 | — | 4 | 2 | no | — |
| 241 | C4 | — | 8 | 1 | no | — |
| 242 | C4 | — | 16 | 1 | no | — |
| 243 | C4 | — | 712 | 2 | no | — |
| 244 | C4 | #19151770 | 1 | 1 | yes | characteristic |
| 245 | C4 | #19151784 | 1 | 1 | yes | characteristic |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 4 | 1 | no | — |
| 248 | C2 | — | 16 | 1 | no | — |
| 249 | C2 | — | 1424 | 2 | no | — |
| 250 | C2 | #26019186 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #26031691 | C2 | semidirect |
| 2. #26022198 | C2 | semidirect |
| 3. #26018236 | C2 | semidirect |
| 4. #26018205 | C2 | semidirect |
| 5. C89×#2053 | C2 | semidirect |
| 128. #2053 | D89 | semidirect |
| 131. C89 | #12170 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26031691 | 2 |
| #26022198 | 2 |
| #26018236 | 2 |
| #26018205 | 2 |
| C89×#2053 | 2 |
| #12170 | 89 |