Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
416 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 56 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 | #22626205 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22625064 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22625063 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22620360 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #17989568 | — | 2 | 2 | no | — |
| 6 | #17989493 | — | 2 | 2 | no | — |
| 7 | #17989492 | C22 | 1 | 4 | yes | — |
| 8 | C2×#14556614 | — | 2 | 2 | no | — |
| 9 | #17989138 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #17989138 | C22 | 1 | 1 | yes | characteristic |
| 11 | #17988972 | — | 2 | 2 | no | — |
| 12 | #17988964 | C22 | 1 | 1 | yes | characteristic |
| 13 | #17988961 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #17988961 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×C8×D73 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | #494×C73 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #14556670 | — | 4 | 2 | no | — |
| 18 | #14556650 | — | 2 | 4 | no | — |
| 19 | #14556650 | — | 4 | 2 | no | — |
| 20 | C2×#11924029 | — | 2 | 1 | no | — |
| 21 | C2×#11924029 | D4 | 1 | 2 | yes | — |
| 22 | #14556614 | — | 4 | 4 | no | — |
| 23 | #14556614 | — | 4 | 4 | no | — |
| 24 | C2×#11924015 | — | 2 | 2 | no | — |
| 25 | C2×#11924015 | — | 2 | 2 | no | — |
| 26 | C2×D584 | — | 2 | 2 | no | — |
| 27 | C2×C8×D73 | — | 2 | 1 | no | — |
| 28 | #14556588 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #14556588 | C23 | 1 | 1 | yes | characteristic |
| 30 | #14556580 | — | 2 | 1 | no | — |
| 31 | #14556580 | D4 | 1 | 2 | yes | — |
| 32 | #14556566 | — | 4 | 1 | no | — |
| 33 | #14556566 | — | 4 | 1 | no | — |
| 34 | #14556553 | — | 4 | 1 | no | — |
| 35 | #14556553 | — | 4 | 1 | no | — |
| 36 | C42×D73 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4×#10596831 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C8×Dic73 | — | 2 | 1 | no | — |
| 39 | C2×Q16×C73 | — | 2 | 2 | no | — |
| 40 | C4⋊Q8×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C4⋊Q8×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C4×C8×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C4⋊#64408 | — | 8 | 2 | no | — |
| 44 | C22⋊#64408 | — | 4 | 4 | no | — |
| 45 | C4×#64408 | — | 8 | 2 | no | — |
| 46 | #11924029 | — | 4 | 2 | no | — |
| 47 | #11924029 | — | 4 | 2 | no | — |
| 48 | #11924029 | — | 4 | 2 | no | — |
| 49 | #11924029 | — | 4 | 2 | no | — |
| 50 | C2×D292 | — | 2 | 1 | no | — |
| 51 | C2×D292 | — | 2 | 2 | no | — |
| 52 | C2×D292 | — | 4 | 1 | no | — |
| 53 | C2×D292 | — | 4 | 1 | no | — |
| 54 | C2×D292 | — | 4 | 1 | no | — |
| 55 | C2×D292 | C2×D4 | 1 | 2 | yes | — |
| 56 | C2×C4×D73 | — | 2 | 1 | no | — |
| 57 | C2×C4×D73 | — | 2 | 2 | no | — |
| 58 | C2×C4×D73 | — | 4 | 1 | no | — |
| 59 | C2×C4×D73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | #11924015 | — | 4 | 4 | no | — |
| 61 | #11924015 | — | 4 | 4 | no | — |
| 62 | D584 | — | 4 | 4 | no | — |
| 63 | C2×#10596831 | — | 2 | 1 | no | — |
| 64 | C8×D73 | — | 2 | 2 | no | — |
| 65 | C8×D73 | — | 2 | 2 | no | — |
| 66 | #11924005 | — | 4 | 2 | no | — |
| 67 | #11924005 | — | 4 | 2 | no | — |
| 68 | C4×Dic73 | — | 2 | 1 | no | — |
| 69 | C4×Dic73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×Q8×C73 | — | 2 | 1 | no | — |
| 71 | C2×Q8×C73 | C22⋊C4 | 1 | 2 | yes | — |
| 72 | Q16×C73 | — | 4 | 4 | no | — |
| 73 | C2×C8×C73 | — | 2 | 1 | no | — |
| 74 | C4⋊C4×C73 | — | 4 | 1 | no | — |
| 75 | C4⋊C4×C73 | — | 4 | 1 | no | — |
| 76 | C42×C73 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C22×D73 | — | 2 | 2 | no | — |
| 78 | C22×D73 | — | 4 | 1 | no | — |
| 79 | C22×D73 | #99 | 1 | 1 | yes | characteristic |
| 80 | C2×#64408 | — | 8 | 4 | no | — |
| 81 | Q8×C73 | — | 4 | 2 | no | — |
| 82 | Q8×C73 | — | 4 | 2 | no | — |
| 83 | C2×C4×C73 | — | 2 | 1 | no | — |
| 84 | C2×C4×C73 | — | 2 | 2 | no | — |
| 85 | C2×C4×C73 | — | 4 | 1 | no | — |
| 86 | C2×C4×C73 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 87 | C2×Dic73 | — | 2 | 1 | no | — |
| 88 | C2×Dic73 | #99 | 1 | 1 | yes | characteristic |
| 89 | D292 | — | 4 | 2 | no | — |
| 90 | D292 | — | 4 | 2 | no | — |
| 91 | D292 | — | 8 | 2 | no | — |
| 92 | D292 | — | 8 | 2 | no | — |
| 93 | D292 | — | 8 | 2 | no | — |
| 94 | D292 | — | 8 | 2 | no | — |
| 95 | C4×D73 | — | 2 | 1 | no | — |
| 96 | C4×D73 | — | 2 | 1 | no | — |
| 97 | C4×D73 | — | 4 | 2 | no | — |
| 98 | C4×D73 | — | 4 | 2 | no | — |
| 99 | C4×D73 | — | 4 | 2 | no | — |
| 100 | C4×D73 | — | 8 | 1 | no | — |
| 101 | C4×D73 | — | 8 | 1 | no | — |
| 102 | C8×C73 | — | 2 | 2 | no | — |
| 103 | #10596831 | — | 2 | 2 | no | — |
| 104 | C22×C73 | C2×#99 | 1 | 1 | yes | characteristic |
| 105 | C2×D73 | — | 2 | 1 | no | — |
| 106 | C2×D73 | — | 4 | 1 | no | — |
| 107 | C2×D73 | — | 4 | 2 | no | — |
| 108 | C2×D73 | — | 8 | 2 | no | — |
| 109 | C2×D73 | — | 8 | 2 | no | — |
| 110 | C2×D73 | — | 8 | 1 | no | — |
| 111 | #64408 | — | 16 | 4 | no | — |
| 112 | C4×C73 | — | 2 | 2 | no | — |
| 113 | C4×C73 | — | 2 | 1 | no | — |
| 114 | C4×C73 | — | 4 | 2 | no | — |
| 115 | C4×C73 | — | 8 | 1 | no | — |
| 116 | Dic73 | — | 4 | 1 | no | — |
| 117 | Dic73 | #353 | 1 | 1 | yes | characteristic |
| 118 | Dic73 | #353 | 1 | 1 | yes | characteristic |
| 119 | #13583 | — | 73 | 1 | no | maximal, Sylow |
| 120 | C2×C73 | — | 2 | 1 | no | — |
| 121 | C2×C73 | C2×#353 | 1 | 1 | yes | characteristic |
| 122 | D73 | — | 4 | 1 | no | — |
| 123 | D73 | — | 8 | 2 | no | — |
| 124 | D73 | — | 16 | 1 | no | — |
| 125 | #3147 | — | 73 | 1 | no | — |
| 126 | #2126 | — | 73 | 2 | no | — |
| 127 | #1409 | — | 73 | 2 | no | — |
| 128 | #1408 | — | 73 | 2 | no | — |
| 129 | C73 | #13583 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 130 | C2×#135 | — | 146 | 2 | no | — |
| 131 | #532 | — | 73 | 1 | no | — |
| 132 | #532 | — | 73 | 1 | no | — |
| 133 | #495 | — | 146 | 2 | no | — |
| 134 | #494 | #64408 | 1 | 1 | yes | characteristic, complemented |
| 135 | #493 | — | 73 | 1 | no | — |
| 136 | #492 | — | 73 | 1 | no | — |
| 137 | #492 | — | 73 | 1 | no | — |
| 138 | #410 | — | 146 | 2 | no | — |
| 139 | C2×C4×C8 | — | 73 | 1 | no | — |
| 140 | #354 | — | 146 | 2 | no | — |
| 141 | #353 | — | 73 | 4 | no | — |
| 142 | C2×#41 | — | 73 | 2 | no | — |
| 143 | C2×#41 | — | 146 | 1 | no | — |
| 144 | #135 | — | 292 | 4 | no | — |
| 145 | #135 | — | 292 | 4 | no | — |
| 146 | C2×Q16 | — | 2 | 2 | no | — |
| 147 | C2×SD16 | — | 146 | 2 | no | — |
| 148 | C2×SD16 | — | 146 | 2 | no | — |
| 149 | C2×D8 | — | 146 | 2 | no | — |
| 150 | C22×C8 | — | 146 | 1 | no | — |
| 151 | C4⋊Q8 | C2×#64408 | 1 | 1 | yes | characteristic |
| 152 | C4⋊Q8 | C2×#64408 | 1 | 1 | yes | characteristic |
| 153 | #127 | — | 73 | 1 | no | — |
| 154 | #127 | — | 73 | 1 | no | — |
| 155 | #124 | — | 73 | 2 | no | — |
| 156 | #124 | — | 146 | 1 | no | — |
| 157 | #121 | — | 292 | 1 | no | — |
| 158 | #121 | — | 292 | 1 | no | — |
| 159 | C4×D4 | — | 292 | 1 | no | — |
| 160 | C4×D4 | — | 292 | 1 | no | — |
| 161 | #117 | — | 292 | 2 | no | — |
| 162 | C2×C42 | — | 73 | 1 | no | — |
| 163 | #99 | — | 292 | 2 | no | — |
| 164 | #99 | — | 146 | 4 | no | — |
| 165 | C4×C8 | — | 73 | 1 | no | — |
| 166 | C4×C8 | — | 146 | 1 | no | — |
| 167 | C4×C8 | C2×#64408 | 1 | 1 | yes | characteristic |
| 168 | #41 | — | 292 | 2 | no | — |
| 169 | #41 | — | 292 | 2 | no | — |
| 170 | #41 | — | 292 | 2 | no | — |
| 171 | #41 | — | 292 | 2 | no | — |
| 172 | C2×Q8 | — | 2 | 1 | no | — |
| 173 | C2×Q8 | C22⋊#64408 | 1 | 2 | yes | — |
| 174 | C2×D4 | — | 292 | 1 | no | — |
| 175 | C2×D4 | — | 292 | 1 | no | — |
| 176 | C2×D4 | — | 292 | 1 | no | — |
| 177 | C2×D4 | — | 73 | 2 | no | — |
| 178 | C2×D4 | — | 146 | 2 | no | — |
| 179 | C2×D4 | — | 146 | 1 | no | — |
| 180 | C22×C4 | — | 292 | 1 | no | — |
| 181 | C22×C4 | — | 73 | 1 | no | — |
| 182 | C22×C4 | — | 146 | 2 | no | — |
| 183 | C22×C4 | — | 146 | 1 | no | — |
| 184 | Q16 | — | 4 | 4 | no | — |
| 185 | SD16 | — | 292 | 4 | no | — |
| 186 | SD16 | — | 292 | 4 | no | — |
| 187 | D8 | — | 292 | 4 | no | — |
| 188 | C2×C8 | — | 2 | 1 | no | — |
| 189 | C2×C8 | — | 146 | 2 | no | — |
| 190 | C2×C8 | — | 146 | 2 | no | — |
| 191 | C2×C8 | — | 146 | 1 | no | — |
| 192 | C4⋊C4 | — | 4 | 1 | no | — |
| 193 | C4⋊C4 | — | 4 | 1 | no | — |
| 194 | C4⋊C4 | — | 584 | 2 | no | — |
| 195 | C22⋊C4 | — | 292 | 2 | no | — |
| 196 | C22⋊C4 | — | 292 | 2 | no | — |
| 197 | C22⋊C4 | — | 292 | 4 | no | — |
| 198 | C42 | — | 584 | 2 | no | — |
| 199 | C42 | — | 73 | 1 | no | — |
| 200 | C42 | — | 146 | 1 | no | — |
| 201 | C42 | C22×#64408 | 1 | 1 | yes | characteristic |
| 202 | C23 | — | 292 | 1 | no | — |
| 203 | C23 | — | 73 | 1 | no | — |
| 204 | C23 | — | 146 | 2 | no | — |
| 205 | Q8 | — | 4 | 2 | no | — |
| 206 | Q8 | — | 4 | 2 | no | — |
| 207 | D4 | — | 292 | 2 | no | — |
| 208 | D4 | — | 292 | 2 | no | — |
| 209 | D4 | — | 584 | 2 | no | — |
| 210 | D4 | — | 584 | 2 | no | — |
| 211 | D4 | — | 584 | 2 | no | — |
| 212 | D4 | — | 584 | 2 | no | — |
| 213 | C2×C4 | — | 2 | 1 | no | — |
| 214 | C2×C4 | — | 2 | 2 | no | — |
| 215 | C2×C4 | — | 4 | 1 | no | — |
| 216 | C2×C4 | — | 292 | 2 | no | — |
| 217 | C2×C4 | — | 292 | 2 | no | — |
| 218 | C2×C4 | — | 292 | 2 | no | — |
| 219 | C2×C4 | — | 584 | 1 | no | — |
| 220 | C2×C4 | — | 584 | 1 | no | — |
| 221 | C2×C4 | — | 584 | 4 | no | — |
| 222 | C2×C4 | — | 73 | 1 | no | — |
| 223 | C2×C4 | — | 146 | 1 | no | — |
| 224 | C2×C4 | — | 146 | 1 | no | — |
| 225 | C2×C4 | — | 146 | 1 | no | — |
| 226 | C2×C4 | C2×C22⋊#64408 | 1 | 1 | yes | characteristic |
| 227 | C8 | — | 2 | 2 | no | — |
| 228 | C8 | — | 146 | 2 | no | — |
| 229 | C22 | — | 292 | 1 | no | — |
| 230 | C22 | — | 292 | 2 | no | — |
| 231 | C22 | — | 584 | 2 | no | — |
| 232 | C22 | — | 584 | 2 | no | — |
| 233 | C22 | — | 584 | 1 | no | — |
| 234 | C22 | — | 146 | 1 | no | — |
| 235 | C22 | D73⋊#99 | 1 | 1 | yes | characteristic |
| 236 | C4 | — | 2 | 2 | no | — |
| 237 | C4 | — | 2 | 1 | no | — |
| 238 | C4 | — | 4 | 2 | no | — |
| 239 | C4 | — | 8 | 1 | no | — |
| 240 | C4 | — | 292 | 1 | no | — |
| 241 | C4 | — | 73 | 1 | no | — |
| 242 | C4 | — | 73 | 1 | no | — |
| 243 | C4 | — | 1168 | 4 | no | — |
| 244 | C2 | — | 2 | 1 | no | — |
| 245 | C2 | — | 292 | 1 | no | — |
| 246 | C2 | — | 584 | 2 | no | — |
| 247 | C2 | — | 1168 | 1 | no | — |
| 248 | C2 | D73⋊#353 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22626205 | C2 | semidirect |
| 2. #22625064 | C2 | semidirect |
| 3. #22625063 | C2 | semidirect |
| 9. #17989138 | C4 | semidirect |
| 13. #17988961 | C4 | semidirect |
| 15. C4×C8×D73 | C4 | semidirect |
| 16. #494×C73 | C4 | semidirect |
| 129. C73 | #13583 | semidirect |
| 134. #494 | #64408 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22626205 | 2 |
| #22625064 | 2 |
| #22625063 | 2 |
| #22620360 | 2 |
| #13583 | 73 |