Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
814 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 151 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 | #22599268 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #22599254 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #22588005 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #22587978 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #22587952 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22586639 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C71×#1712 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #17986547 | C22 | 1 | 3 | yes | — |
| 9 | #127×D71 | C22 | 1 | 1 | yes | characteristic |
| 10 | #17986375 | C22 | 1 | 3 | yes | — |
| 11 | #17986371 | C22 | 1 | 3 | yes | complemented |
| 12 | #17986350 | C22 | 1 | 3 | yes | complemented |
| 13 | #17986346 | C22 | 1 | 3 | yes | — |
| 14 | #17986345 | C22 | 1 | 3 | yes | — |
| 15 | #17986345 | C22 | 1 | 3 | yes | complemented |
| 16 | #17986212 | — | 2 | 6 | no | — |
| 17 | #17986192 | — | 2 | 6 | no | — |
| 18 | #17986141 | — | 2 | 6 | no | — |
| 19 | #17986131 | — | 2 | 6 | no | — |
| 20 | #17986050 | C22 | 1 | 3 | yes | — |
| 21 | #17985721 | C22 | 1 | 3 | yes | complemented |
| 22 | #493×C71 | C22 | 1 | 3 | yes | — |
| 23 | #493×C71 | C22 | 1 | 3 | yes | complemented |
| 24 | C82×C71 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | C2×#11765864 | — | 2 | 3 | no | — |
| 26 | C2×D4×D71 | — | 2 | 3 | no | — |
| 27 | C2×#11765853 | — | 2 | 6 | no | — |
| 28 | C2×#11765850 | — | 2 | 6 | no | — |
| 29 | #14556128 | C23 | 1 | 3 | yes | — |
| 30 | #14556125 | C23 | 1 | 1 | yes | characteristic |
| 31 | #14556124 | — | 2 | 3 | no | — |
| 32 | #14556115 | — | 2 | 3 | no | — |
| 33 | #14556099 | — | 2 | 3 | no | — |
| 34 | #14556099 | — | 2 | 3 | no | — |
| 35 | D4×Dic71 | — | 2 | 3 | no | — |
| 36 | D4×Dic71 | — | 2 | 3 | no | — |
| 37 | C42×D71 | C23 | 1 | 1 | yes | characteristic |
| 38 | Dic71⋊C8 | D4 | 1 | 6 | yes | — |
| 39 | #14556053 | — | 2 | 6 | no | — |
| 40 | #14556050 | — | 2 | 6 | no | — |
| 41 | #14556049 | — | 2 | 6 | no | — |
| 42 | #14556049 | — | 2 | 6 | no | — |
| 43 | (C2×D71)⋊C8 | D4 | 1 | 6 | yes | — |
| 44 | C4×C71⋊C8 | C23 | 1 | 3 | yes | — |
| 45 | C2×D8×C71 | — | 2 | 6 | no | — |
| 46 | C2×D8×C71 | — | 2 | 6 | no | — |
| 47 | #127×C71 | C23 | 1 | 1 | yes | characteristic |
| 48 | #127×C71 | C23 | 1 | 3 | yes | — |
| 49 | C4×C8×C71 | C23 | 1 | 3 | yes | — |
| 50 | C23×D71 | — | 4 | 1 | no | — |
| 51 | #11765864 | — | 8 | 6 | no | — |
| 52 | C2×Dic142 | — | 2 | 3 | no | — |
| 53 | C2×D284 | — | 2 | 3 | no | — |
| 54 | C2×C22⋊D71 | — | 4 | 3 | no | — |
| 55 | C2×C22⋊D71 | — | 4 | 3 | no | — |
| 56 | D4×D71 | — | 8 | 6 | no | — |
| 57 | C22×Dic71 | — | 4 | 3 | no | — |
| 58 | C2×C4×D71 | C2×D4 | 1 | 3 | yes | — |
| 59 | #11765853 | — | 8 | 6 | no | — |
| 60 | #11765850 | — | 8 | 6 | no | — |
| 61 | C2×C71⋊C8 | C2×D4 | 1 | 6 | yes | — |
| 62 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 63 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 64 | C22⋊Dic71 | — | 4 | 3 | no | — |
| 65 | C22⋊Dic71 | — | 4 | 3 | no | — |
| 66 | C4×Dic71 | C2×D4 | 1 | 3 | yes | — |
| 67 | C2×D4×C71 | — | 2 | 3 | no | — |
| 68 | C2×D4×C71 | — | 2 | 3 | no | — |
| 69 | C2×D4×C71 | — | 2 | 3 | no | — |
| 70 | C2×D4×C71 | — | 2 | 3 | no | — |
| 71 | D8×C71 | — | 8 | 6 | no | — |
| 72 | D8×C71 | — | 8 | 6 | no | — |
| 73 | C2×C8×C71 | C2×D4 | 1 | 6 | yes | — |
| 74 | C42×C71 | C24 | 1 | 1 | yes | characteristic |
| 75 | C23×C71 | — | 4 | 3 | no | — |
| 76 | C23×C71 | — | 4 | 1 | no | — |
| 77 | C22×D71 | — | 4 | 1 | no | — |
| 78 | C22×D71 | — | 8 | 6 | no | — |
| 79 | C22×D71 | #127 | 1 | 1 | yes | characteristic |
| 80 | D4×C71 | — | 4 | 6 | no | — |
| 81 | D4×C71 | — | 4 | 6 | no | — |
| 82 | D4×C71 | — | 4 | 6 | no | — |
| 83 | D4×C71 | — | 4 | 6 | no | — |
| 84 | C2×C4×C71 | C22×D4 | 1 | 3 | yes | — |
| 85 | C22⋊D71 | — | 8 | 6 | no | — |
| 86 | C22⋊D71 | — | 8 | 6 | no | — |
| 87 | C2×Dic71 | — | 4 | 3 | no | — |
| 88 | C2×Dic71 | — | 8 | 6 | no | — |
| 89 | C2×Dic71 | #127 | 1 | 3 | yes | — |
| 90 | D284 | — | 4 | 6 | no | — |
| 91 | C4×D71 | — | 2 | 6 | no | — |
| 92 | Dic142 | — | 4 | 6 | no | — |
| 93 | C8×C71 | — | 2 | 6 | no | — |
| 94 | C71⋊C8 | — | 2 | 6 | no | — |
| 95 | C22×C71 | — | 8 | 3 | no | — |
| 96 | C22×C71 | — | 8 | 3 | no | — |
| 97 | C22×C71 | — | 8 | 3 | no | — |
| 98 | C22×C71 | — | 8 | 3 | no | — |
| 99 | C22×C71 | C2×#127 | 1 | 1 | yes | characteristic |
| 100 | C2×D71 | — | 2 | 3 | no | — |
| 101 | C2×D71 | — | 8 | 3 | no | — |
| 102 | C2×D71 | — | 16 | 4 | no | — |
| 103 | C4×C71 | #576 | 1 | 6 | yes | — |
| 104 | Dic71 | — | 2 | 3 | no | — |
| 105 | Dic71 | — | 8 | 3 | no | — |
| 106 | #19743 | — | 71 | 1 | no | maximal, Sylow |
| 107 | C2×C71 | — | 16 | 1 | no | — |
| 108 | C2×C71 | — | 16 | 3 | no | — |
| 109 | C2×C71 | #3155 | 1 | 3 | yes | — |
| 110 | D71 | — | 4 | 1 | no | — |
| 111 | D71 | — | 16 | 1 | no | — |
| 112 | #3207 | — | 71 | 3 | no | — |
| 113 | #3205 | — | 71 | 3 | no | — |
| 114 | #1721 | — | 71 | 3 | no | — |
| 115 | #1717 | — | 71 | 3 | no | — |
| 116 | #1712 | D71 | 1 | 1 | yes | characteristic, complemented |
| 117 | #1711 | — | 71 | 1 | no | — |
| 118 | #1454 | — | 71 | 1 | no | — |
| 119 | C71 | #19743 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 120 | #532 | — | 71 | 3 | no | — |
| 121 | C2×#127 | — | 71 | 1 | no | — |
| 122 | #493 | — | 71 | 3 | no | — |
| 123 | #493 | C2×D71 | 1 | 3 | yes | — |
| 124 | #493 | C2×D71 | 1 | 3 | yes | complemented |
| 125 | #492 | — | 71 | 3 | no | — |
| 126 | #487 | — | 71 | 3 | no | — |
| 127 | #486 | — | 71 | 3 | no | — |
| 128 | #486 | — | 71 | 3 | no | — |
| 129 | #486 | — | 71 | 3 | no | — |
| 130 | #463 | — | 142 | 6 | no | — |
| 131 | #459 | — | 142 | 6 | no | — |
| 132 | #449 | — | 142 | 6 | no | — |
| 133 | #447 | — | 142 | 6 | no | — |
| 134 | #431 | — | 71 | 3 | no | — |
| 135 | #322 | — | 71 | 3 | no | — |
| 136 | C82 | C2×D71 | 1 | 1 | yes | characteristic, complemented |
| 137 | C2×#41 | — | 142 | 3 | no | — |
| 138 | C22×D4 | — | 142 | 3 | no | — |
| 139 | C2×SD16 | — | 142 | 6 | no | — |
| 140 | C2×D8 | — | 2 | 6 | no | — |
| 141 | C2×D8 | — | 2 | 6 | no | — |
| 142 | C2×D8 | — | 142 | 6 | no | — |
| 143 | C4⋊Q8 | — | 71 | 3 | no | — |
| 144 | #127 | — | 71 | 1 | no | — |
| 145 | #127 | — | 142 | 3 | no | — |
| 146 | #127 | C22×D71 | 1 | 1 | yes | characteristic |
| 147 | #127 | C22×D71 | 1 | 3 | yes | — |
| 148 | #124 | — | 142 | 3 | no | — |
| 149 | #121 | — | 142 | 3 | no | — |
| 150 | #121 | — | 142 | 3 | no | — |
| 151 | C4×D4 | — | 142 | 3 | no | — |
| 152 | C4×D4 | — | 142 | 3 | no | — |
| 153 | C2×C42 | — | 71 | 1 | no | — |
| 154 | C4⋊C8 | — | 71 | 6 | no | — |
| 155 | #103 | — | 142 | 6 | no | — |
| 156 | #102 | — | 142 | 6 | no | — |
| 157 | #102 | — | 142 | 6 | no | — |
| 158 | #102 | — | 142 | 6 | no | — |
| 159 | C22⋊C8 | — | 71 | 6 | no | — |
| 160 | C4×C8 | — | 71 | 3 | no | — |
| 161 | C4×C8 | C22×D71 | 1 | 3 | yes | — |
| 162 | C24 | — | 284 | 1 | no | — |
| 163 | #41 | — | 568 | 6 | no | — |
| 164 | C2×Q8 | — | 142 | 3 | no | — |
| 165 | C2×D4 | — | 2 | 3 | no | — |
| 166 | C2×D4 | — | 2 | 3 | no | — |
| 167 | C2×D4 | — | 2 | 3 | no | — |
| 168 | C2×D4 | — | 2 | 3 | no | — |
| 169 | C2×D4 | — | 284 | 3 | no | — |
| 170 | C2×D4 | — | 284 | 3 | no | — |
| 171 | C2×D4 | — | 568 | 6 | no | — |
| 172 | C2×D4 | — | 142 | 3 | no | — |
| 173 | C22×C4 | — | 284 | 3 | no | — |
| 174 | C22×C4 | — | 71 | 3 | no | — |
| 175 | SD16 | — | 568 | 6 | no | — |
| 176 | D8 | — | 8 | 6 | no | — |
| 177 | D8 | — | 8 | 6 | no | — |
| 178 | D8 | — | 568 | 6 | no | — |
| 179 | C2×C8 | — | 71 | 6 | no | — |
| 180 | C2×C8 | D4×D71 | 1 | 6 | yes | — |
| 181 | C4⋊C4 | — | 142 | 3 | no | — |
| 182 | C4⋊C4 | — | 142 | 3 | no | — |
| 183 | C22⋊C4 | — | 284 | 3 | no | — |
| 184 | C22⋊C4 | — | 284 | 3 | no | — |
| 185 | C42 | — | 71 | 3 | no | — |
| 186 | C42 | C23×D71 | 1 | 1 | yes | characteristic |
| 187 | C23 | — | 4 | 3 | no | — |
| 188 | C23 | — | 4 | 1 | no | — |
| 189 | C23 | — | 284 | 1 | no | — |
| 190 | C23 | — | 568 | 6 | no | — |
| 191 | C23 | — | 71 | 1 | no | — |
| 192 | Q8 | — | 284 | 6 | no | — |
| 193 | D4 | — | 4 | 6 | no | — |
| 194 | D4 | — | 4 | 6 | no | — |
| 195 | D4 | — | 4 | 6 | no | — |
| 196 | D4 | — | 4 | 6 | no | — |
| 197 | D4 | — | 284 | 6 | no | — |
| 198 | D4 | — | 568 | 6 | no | — |
| 199 | D4 | — | 568 | 6 | no | — |
| 200 | C2×C4 | — | 284 | 3 | no | — |
| 201 | C2×C4 | — | 568 | 6 | no | — |
| 202 | C2×C4 | — | 71 | 3 | no | — |
| 203 | C2×C4 | — | 142 | 6 | no | — |
| 204 | C2×C4 | C2×D4×D71 | 1 | 3 | yes | — |
| 205 | C8 | — | 2 | 6 | no | — |
| 206 | C8 | — | 142 | 6 | no | — |
| 207 | C22 | — | 8 | 3 | no | — |
| 208 | C22 | — | 8 | 3 | no | — |
| 209 | C22 | — | 8 | 3 | no | — |
| 210 | C22 | — | 8 | 3 | no | — |
| 211 | C22 | — | 568 | 3 | no | — |
| 212 | C22 | — | 1136 | 4 | no | — |
| 213 | C22 | — | 142 | 3 | no | — |
| 214 | C22 | #127×D71 | 1 | 1 | yes | characteristic |
| 215 | C4 | — | 568 | 3 | no | — |
| 216 | C4 | — | 142 | 3 | no | — |
| 217 | C4 | #17986800 | 1 | 6 | yes | — |
| 218 | C2 | — | 16 | 1 | no | — |
| 219 | C2 | — | 16 | 3 | no | — |
| 220 | C2 | — | 284 | 1 | no | — |
| 221 | C2 | — | 1136 | 1 | no | — |
| 222 | C2 | #22598820 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22599268 | C2 | semidirect |
| 2. #22599254 | C2 | semidirect |
| 3. #22588005 | C2 | semidirect |
| 4. #22587978 | C2 | semidirect |
| 5. #22587952 | C2 | semidirect |
| 6. #22586639 | C2 | semidirect |
| 7. C71×#1712 | C2 | semidirect |
| 11. #17986371 | C22 | semidirect |
| 12. #17986350 | C22 | semidirect |
| 15. #17986345 | C22 | semidirect |
| 21. #17985721 | C22 | semidirect |
| 23. #493×C71 | C22 | semidirect |
| 24. C82×C71 | C22 | semidirect |
| 116. #1712 | D71 | semidirect |
| 119. C71 | #19743 | semidirect |
| 124. #493 | C2×D71 | semidirect |
| 136. C82 | C2×D71 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22599268 | 2 |
| #22599254 | 2 |
| #22588005 | 2 |
| #22587978 | 2 |
| #22587952 | 2 |
| #22586639 | 2 |
| C71×#1712 | 2 |
| #19743 | 71 |