Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
620 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 67 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 | #26020167 | C2 | 1 | 2 | yes | maximal |
| 2 | C89×#1783 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#15667927 | — | 2 | 2 | no | — |
| 4 | C2×#15667927 | C4 | 1 | 4 | yes | complemented |
| 5 | #19151454 | — | 2 | 2 | no | — |
| 6 | C23×D4×C89 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#100×C89 | — | 2 | 2 | no | — |
| 8 | #407×C89 | C4 | 1 | 2 | yes | complemented |
| 9 | C2×C22⋊Dic89 | — | 2 | 4 | no | — |
| 10 | C2×C22⋊Dic89 | — | 4 | 4 | no | — |
| 11 | #15667927 | — | 4 | 4 | no | — |
| 12 | #15667927 | — | 4 | 4 | no | — |
| 13 | C25×C89 | D4 | 1 | 1 | yes | characteristic |
| 14 | C25×C89 | Q8 | 1 | 1 | yes | characteristic |
| 15 | C22×D4×C89 | — | 4 | 2 | no | — |
| 16 | C22×D4×C89 | — | 4 | 4 | no | — |
| 17 | C22×D4×C89 | C2×C4 | 1 | 2 | yes | — |
| 18 | C22×D4×C89 | D4 | 1 | 2 | yes | — |
| 19 | C23×C4×C89 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | C2×M16×C89 | — | 2 | 2 | no | — |
| 21 | #100×C89 | — | 4 | 4 | no | — |
| 22 | C22⋊C8×C89 | — | 4 | 2 | no | — |
| 23 | C22×Dic89 | — | 4 | 4 | no | — |
| 24 | C22⋊Dic89 | — | 4 | 4 | no | — |
| 25 | C22⋊Dic89 | — | 4 | 4 | no | — |
| 26 | C22⋊Dic89 | — | 8 | 8 | no | — |
| 27 | C24×C89 | — | 2 | 4 | no | — |
| 28 | C24×C89 | — | 4 | 1 | no | — |
| 29 | C24×C89 | — | 4 | 1 | no | — |
| 30 | C24×C89 | — | 4 | 2 | no | — |
| 31 | C24×C89 | — | 8 | 2 | no | — |
| 32 | C24×C89 | — | 8 | 2 | no | — |
| 33 | C24×C89 | C22⋊C4 | 1 | 2 | yes | — |
| 34 | C24×C89 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 35 | C24×C89 | C4⋊C4 | 1 | 2 | yes | — |
| 36 | C2×D4×C89 | — | 2 | 4 | no | — |
| 37 | C2×D4×C89 | — | 2 | 4 | no | — |
| 38 | C2×D4×C89 | — | 2 | 4 | no | — |
| 39 | C2×D4×C89 | — | 4 | 2 | no | — |
| 40 | C2×D4×C89 | — | 4 | 2 | no | — |
| 41 | C2×D4×C89 | — | 4 | 2 | no | — |
| 42 | C2×D4×C89 | — | 4 | 16 | no | — |
| 43 | C22×C4×C89 | — | 2 | 2 | no | — |
| 44 | C22×C4×C89 | — | 4 | 2 | no | — |
| 45 | C22×C4×C89 | C42 | 1 | 1 | yes | characteristic |
| 46 | C22×C4×C89 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | M16×C89 | — | 4 | 4 | no | — |
| 48 | C2×C8×C89 | — | 4 | 2 | no | — |
| 49 | C23×C89 | — | 2 | 1 | no | — |
| 50 | C23×C89 | — | 2 | 2 | no | — |
| 51 | C23×C89 | — | 2 | 2 | no | — |
| 52 | C23×C89 | — | 2 | 2 | no | — |
| 53 | C23×C89 | — | 2 | 2 | no | — |
| 54 | C23×C89 | — | 2 | 2 | no | — |
| 55 | C23×C89 | — | 2 | 2 | no | — |
| 56 | C23×C89 | — | 4 | 2 | no | — |
| 57 | C23×C89 | — | 4 | 4 | no | — |
| 58 | C23×C89 | — | 4 | 8 | no | — |
| 59 | C23×C89 | — | 4 | 4 | no | — |
| 60 | C23×C89 | — | 8 | 4 | no | — |
| 61 | C23×C89 | — | 8 | 2 | no | — |
| 62 | C23×C89 | — | 8 | 2 | no | — |
| 63 | C23×C89 | — | 8 | 2 | no | — |
| 64 | C23×C89 | — | 8 | 4 | no | — |
| 65 | C23×C89 | — | 8 | 2 | no | — |
| 66 | C23×C89 | — | 8 | 2 | no | — |
| 67 | C23×C89 | — | 8 | 4 | no | — |
| 68 | C23×C89 | — | 8 | 2 | no | — |
| 69 | C23×C89 | #95 | 1 | 1 | yes | characteristic |
| 70 | C23×C89 | #99 | 1 | 4 | yes | — |
| 71 | D4×C89 | — | 4 | 8 | no | — |
| 72 | D4×C89 | — | 4 | 8 | no | — |
| 73 | C2×C4×C89 | — | 2 | 2 | no | — |
| 74 | C2×C4×C89 | — | 2 | 1 | no | — |
| 75 | C2×C4×C89 | — | 2 | 2 | no | — |
| 76 | C2×C4×C89 | — | 4 | 4 | no | — |
| 77 | C2×C4×C89 | #99 | 1 | 2 | yes | — |
| 78 | C2×Dic89 | — | 8 | 4 | no | — |
| 79 | C2×Dic89 | — | 8 | 8 | no | — |
| 80 | C8×C89 | — | 8 | 2 | no | — |
| 81 | C22×C89 | — | 2 | 4 | no | — |
| 82 | C22×C89 | — | 2 | 2 | no | — |
| 83 | C22×C89 | — | 2 | 4 | no | — |
| 84 | C22×C89 | — | 4 | 1 | no | — |
| 85 | C22×C89 | — | 4 | 4 | no | — |
| 86 | C22×C89 | — | 4 | 2 | no | — |
| 87 | C22×C89 | — | 4 | 1 | no | — |
| 88 | C22×C89 | — | 4 | 4 | no | — |
| 89 | C22×C89 | — | 4 | 1 | no | — |
| 90 | C22×C89 | — | 4 | 2 | no | — |
| 91 | C22×C89 | — | 8 | 2 | no | — |
| 92 | C22×C89 | — | 8 | 2 | no | — |
| 93 | C22×C89 | — | 8 | 2 | no | — |
| 94 | C22×C89 | — | 8 | 2 | no | — |
| 95 | C22×C89 | — | 8 | 8 | no | — |
| 96 | C22×C89 | — | 8 | 8 | no | — |
| 97 | C22×C89 | #342 | 1 | 2 | yes | — |
| 98 | C22×C89 | #342 | 1 | 1 | yes | characteristic |
| 99 | C4×C89 | — | 2 | 2 | no | — |
| 100 | C4×C89 | — | 4 | 1 | no | — |
| 101 | Dic89 | — | 16 | 4 | no | — |
| 102 | #7314 | — | 89 | 1 | no | maximal, Sylow |
| 103 | C2×C89 | — | 2 | 2 | no | — |
| 104 | C2×C89 | — | 2 | 1 | no | — |
| 105 | C2×C89 | — | 4 | 4 | no | — |
| 106 | C2×C89 | — | 4 | 2 | no | — |
| 107 | C2×C89 | — | 8 | 2 | no | — |
| 108 | C2×C89 | #1304 | 1 | 1 | yes | characteristic |
| 109 | #1783 | D89 | 1 | 1 | yes | characteristic |
| 110 | #1781 | — | 89 | 2 | no | — |
| 111 | C89 | #7314 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 112 | C23×D4 | C2×D89 | 1 | 1 | yes | characteristic |
| 113 | C2×#100 | — | 2 | 2 | no | — |
| 114 | C2×#99 | — | 89 | 4 | no | — |
| 115 | C2×#99 | — | 178 | 2 | no | — |
| 116 | #407 | Dic89 | 1 | 2 | yes | complemented |
| 117 | #379 | — | 178 | 2 | no | — |
| 118 | C25 | Dic178 | 1 | 1 | yes | characteristic |
| 119 | C25 | D356 | 1 | 1 | yes | characteristic |
| 120 | C22×D4 | — | 4 | 2 | no | — |
| 121 | C22×D4 | — | 4 | 4 | no | — |
| 122 | C22×D4 | C4×D89 | 1 | 2 | yes | — |
| 123 | C22×D4 | C22⋊D89 | 1 | 2 | yes | — |
| 124 | C23×C4 | C2×Dic89 | 1 | 1 | yes | characteristic |
| 125 | C2×M16 | — | 2 | 2 | no | — |
| 126 | C2×C22⋊C4 | — | 356 | 4 | no | — |
| 127 | C2×C22⋊C4 | — | 178 | 4 | no | — |
| 128 | #100 | — | 4 | 4 | no | — |
| 129 | #99 | — | 356 | 4 | no | — |
| 130 | #99 | — | 356 | 4 | no | — |
| 131 | C22⋊C8 | — | 4 | 2 | no | — |
| 132 | C24 | — | 2 | 4 | no | — |
| 133 | C24 | — | 4 | 2 | no | — |
| 134 | C24 | — | 4 | 1 | no | — |
| 135 | C24 | — | 4 | 1 | no | — |
| 136 | C24 | — | 8 | 2 | no | — |
| 137 | C24 | — | 8 | 2 | no | — |
| 138 | C24 | #13083551 | 1 | 2 | yes | — |
| 139 | C24 | #13083553 | 1 | 2 | yes | — |
| 140 | C24 | C4⋊Dic89 | 1 | 1 | yes | characteristic |
| 141 | C2×D4 | — | 2 | 4 | no | — |
| 142 | C2×D4 | — | 2 | 4 | no | — |
| 143 | C2×D4 | — | 2 | 4 | no | — |
| 144 | C2×D4 | — | 4 | 2 | no | — |
| 145 | C2×D4 | — | 4 | 2 | no | — |
| 146 | C2×D4 | — | 4 | 2 | no | — |
| 147 | C2×D4 | — | 4 | 16 | no | — |
| 148 | C22×C4 | — | 2 | 2 | no | — |
| 149 | C22×C4 | — | 4 | 2 | no | — |
| 150 | C22×C4 | — | 356 | 4 | no | — |
| 151 | C22×C4 | C4×Dic89 | 1 | 1 | yes | characteristic |
| 152 | C22×C4 | C22⋊Dic89 | 1 | 1 | yes | characteristic |
| 153 | M16 | — | 4 | 4 | no | — |
| 154 | C2×C8 | — | 4 | 2 | no | — |
| 155 | C22⋊C4 | — | 356 | 4 | no | — |
| 156 | C22⋊C4 | — | 356 | 4 | no | — |
| 157 | C22⋊C4 | — | 712 | 8 | no | — |
| 158 | C23 | — | 2 | 2 | no | — |
| 159 | C23 | — | 2 | 2 | no | — |
| 160 | C23 | — | 2 | 1 | no | — |
| 161 | C23 | — | 2 | 2 | no | — |
| 162 | C23 | — | 2 | 2 | no | — |
| 163 | C23 | — | 2 | 2 | no | — |
| 164 | C23 | — | 2 | 2 | no | — |
| 165 | C23 | — | 4 | 2 | no | — |
| 166 | C23 | — | 4 | 4 | no | — |
| 167 | C23 | — | 4 | 8 | no | — |
| 168 | C23 | — | 4 | 4 | no | — |
| 169 | C23 | — | 8 | 4 | no | — |
| 170 | C23 | — | 8 | 2 | no | — |
| 171 | C23 | — | 8 | 2 | no | — |
| 172 | C23 | — | 8 | 2 | no | — |
| 173 | C23 | — | 8 | 4 | no | — |
| 174 | C23 | — | 8 | 2 | no | — |
| 175 | C23 | — | 8 | 2 | no | — |
| 176 | C23 | — | 8 | 4 | no | — |
| 177 | C23 | — | 8 | 2 | no | — |
| 178 | C23 | #15667919 | 1 | 1 | yes | characteristic |
| 179 | C23 | #15667926 | 1 | 4 | yes | — |
| 180 | D4 | — | 4 | 8 | no | — |
| 181 | D4 | — | 4 | 8 | no | — |
| 182 | C2×C4 | — | 2 | 2 | no | — |
| 183 | C2×C4 | — | 2 | 2 | no | — |
| 184 | C2×C4 | — | 2 | 1 | no | — |
| 185 | C2×C4 | — | 4 | 4 | no | — |
| 186 | C2×C4 | — | 712 | 4 | no | — |
| 187 | C2×C4 | — | 712 | 8 | no | — |
| 188 | C2×C4 | #15667927 | 1 | 2 | yes | — |
| 189 | C8 | — | 8 | 2 | no | — |
| 190 | C22 | — | 2 | 4 | no | — |
| 191 | C22 | — | 2 | 2 | no | — |
| 192 | C22 | — | 2 | 4 | no | — |
| 193 | C22 | — | 4 | 1 | no | — |
| 194 | C22 | — | 4 | 4 | no | — |
| 195 | C22 | — | 4 | 4 | no | — |
| 196 | C22 | — | 4 | 2 | no | — |
| 197 | C22 | — | 4 | 1 | no | — |
| 198 | C22 | — | 4 | 1 | no | — |
| 199 | C22 | — | 4 | 2 | no | — |
| 200 | C22 | — | 8 | 2 | no | — |
| 201 | C22 | — | 8 | 2 | no | — |
| 202 | C22 | — | 8 | 2 | no | — |
| 203 | C22 | — | 8 | 2 | no | — |
| 204 | C22 | — | 8 | 8 | no | — |
| 205 | C22 | — | 8 | 8 | no | — |
| 206 | C22 | #19151364 | 1 | 2 | yes | — |
| 207 | C22 | #19151365 | 1 | 1 | yes | characteristic |
| 208 | C4 | — | 2 | 2 | no | — |
| 209 | C4 | — | 4 | 1 | no | — |
| 210 | C4 | — | 1424 | 4 | no | — |
| 211 | C2 | — | 2 | 2 | no | — |
| 212 | C2 | — | 2 | 1 | no | — |
| 213 | C2 | — | 4 | 4 | no | — |
| 214 | C2 | — | 4 | 2 | no | — |
| 215 | C2 | — | 8 | 2 | no | — |
| 216 | C2 | #26018156 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×#15667927 | C4 | semidirect |
| 8. #407×C89 | C4 | semidirect |
| 111. C89 | #7314 | semidirect |
| 116. #407 | Dic89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #26020167 | 2 |
| C89×#1783 | 2 |
| #7314 | 89 |