Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
384 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 384 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 | #22612007 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22609595 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22607972 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22607907 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C73×#3291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #17989022 | — | 2 | 2 | no | — |
| 7 | #17988964 | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | #17988963 | — | 2 | 1 | no | — |
| 9 | #17988935 | — | 2 | 1 | no | — |
| 10 | #17988408 | C22 | 1 | 2 | yes | complemented |
| 11 | #17988399 | — | 2 | 2 | no | — |
| 12 | #17988308 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #577×C73 | — | 2 | 2 | no | — |
| 14 | #492×C73 | C22 | 1 | 1 | yes | characteristic |
| 15 | #471×C73 | — | 2 | 2 | no | — |
| 16 | #456×C73 | — | 2 | 1 | no | — |
| 17 | #453×C73 | — | 2 | 1 | no | — |
| 18 | #443×C73 | C22 | 1 | 2 | yes | — |
| 19 | C2×D584 | — | 2 | 2 | no | — |
| 20 | C2×#11924013 | — | 4 | 2 | no | — |
| 21 | #14556588 | — | 2 | 1 | no | — |
| 22 | #14556534 | — | 4 | 2 | no | — |
| 23 | #14556532 | — | 4 | 2 | no | — |
| 24 | #14556531 | — | 2 | 2 | no | — |
| 25 | #14556511 | — | 4 | 2 | no | — |
| 26 | C4×#10596831 | — | 2 | 1 | no | — |
| 27 | 2−1+4×C73 | — | 2 | 1 | no | — |
| 28 | H(5,𝔽2)×C73 | — | 2 | 1 | no | — |
| 29 | #137×C73 | — | 2 | 2 | no | — |
| 30 | #137×C73 | — | 4 | 2 | no | — |
| 31 | AGL(1,ℤ/8)×C73 | — | 2 | 2 | no | — |
| 32 | AGL(1,ℤ/8)×C73 | — | 4 | 2 | no | — |
| 33 | #135×C73 | — | 4 | 2 | no | — |
| 34 | #135×C73 | D4 | 1 | 2 | yes | — |
| 35 | C2×SD16×C73 | — | 2 | 2 | no | — |
| 36 | C2×SD16×C73 | — | 4 | 2 | no | — |
| 37 | #131×C73 | — | 2 | 2 | no | — |
| 38 | C4⋊Q8×C73 | D4 | 1 | 1 | yes | characteristic |
| 39 | #127×C73 | D4 | 1 | 1 | yes | characteristic |
| 40 | #108×C73 | D4 | 1 | 2 | yes | — |
| 41 | #104×C73 | — | 2 | 2 | no | — |
| 42 | #101×C73 | — | 2 | 1 | no | — |
| 43 | #100×C73 | — | 2 | 1 | no | — |
| 44 | C4×C8×C73 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×D292 | — | 4 | 1 | no | — |
| 46 | C2×D292 | — | 4 | 1 | no | — |
| 47 | C2×D292 | — | 4 | 1 | no | — |
| 48 | D584 | — | 4 | 2 | no | — |
| 49 | D584 | — | 4 | 2 | no | — |
| 50 | #11924013 | — | 8 | 4 | no | — |
| 51 | C2×#10596831 | — | 2 | 2 | no | — |
| 52 | #11924003 | — | 4 | 2 | no | — |
| 53 | #41×C73 | — | 2 | 2 | no | — |
| 54 | #41×C73 | — | 4 | 2 | no | — |
| 55 | #41×C73 | — | 4 | 2 | no | — |
| 56 | #41×C73 | — | 8 | 1 | no | — |
| 57 | C2×Q8×C73 | — | 2 | 1 | no | — |
| 58 | C2×Q8×C73 | — | 4 | 2 | no | — |
| 59 | C2×D4×C73 | — | 2 | 1 | no | — |
| 60 | C2×D4×C73 | — | 4 | 1 | no | — |
| 61 | C2×D4×C73 | — | 4 | 2 | no | — |
| 62 | C2×D4×C73 | — | 8 | 1 | no | — |
| 63 | Q16×C73 | — | 4 | 2 | no | — |
| 64 | Q16×C73 | D8 | 1 | 2 | yes | complemented |
| 65 | SD16×C73 | — | 2 | 2 | no | — |
| 66 | SD16×C73 | — | 4 | 2 | no | — |
| 67 | SD16×C73 | — | 4 | 2 | no | — |
| 68 | SD16×C73 | — | 4 | 2 | no | — |
| 69 | SD16×C73 | — | 4 | 2 | no | — |
| 70 | D8×C73 | — | 4 | 2 | no | — |
| 71 | D8×C73 | D8 | 1 | 2 | yes | complemented |
| 72 | M16×C73 | — | 2 | 2 | no | — |
| 73 | M16×C73 | — | 4 | 2 | no | — |
| 74 | C2×C8×C73 | — | 4 | 2 | no | — |
| 75 | C2×C8×C73 | C2×D4 | 1 | 2 | yes | — |
| 76 | C4⋊C4×C73 | — | 4 | 1 | no | — |
| 77 | C42×C73 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C23×C73 | — | 4 | 1 | no | — |
| 79 | C23×C73 | — | 8 | 1 | no | — |
| 80 | C22×D73 | — | 8 | 1 | no | — |
| 81 | Q8×C73 | — | 2 | 2 | no | — |
| 82 | Q8×C73 | — | 4 | 2 | no | — |
| 83 | Q8×C73 | — | 8 | 1 | no | — |
| 84 | D4×C73 | — | 2 | 2 | no | — |
| 85 | D4×C73 | — | 4 | 2 | no | — |
| 86 | D4×C73 | — | 4 | 2 | no | — |
| 87 | D4×C73 | — | 8 | 2 | no | — |
| 88 | D4×C73 | — | 8 | 2 | no | — |
| 89 | D4×C73 | — | 8 | 1 | no | — |
| 90 | C2×C4×C73 | — | 2 | 1 | no | — |
| 91 | C2×C4×C73 | — | 4 | 2 | no | — |
| 92 | C2×C4×C73 | — | 4 | 1 | no | — |
| 93 | C2×C4×C73 | — | 8 | 1 | no | — |
| 94 | C2×C4×C73 | — | 8 | 2 | no | — |
| 95 | C2×C4×C73 | #120 | 1 | 1 | yes | characteristic |
| 96 | D292 | — | 8 | 2 | no | — |
| 97 | D292 | — | 8 | 2 | no | — |
| 98 | D292 | — | 8 | 2 | no | — |
| 99 | C8×C73 | — | 4 | 2 | no | — |
| 100 | C8×C73 | C2×D8 | 1 | 2 | yes | — |
| 101 | C8×C73 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 102 | #10596831 | — | 4 | 2 | no | — |
| 103 | C22×C73 | — | 4 | 2 | no | — |
| 104 | C22×C73 | — | 8 | 1 | no | — |
| 105 | C22×C73 | — | 16 | 1 | no | — |
| 106 | C22×C73 | — | 32 | 1 | no | — |
| 107 | C22×C73 | #453 | 1 | 1 | yes | characteristic |
| 108 | C2×D73 | — | 16 | 1 | no | — |
| 109 | C2×D73 | — | 16 | 1 | no | — |
| 110 | C2×D73 | — | 16 | 1 | no | — |
| 111 | C4×C73 | — | 2 | 2 | no | — |
| 112 | C4×C73 | — | 4 | 2 | no | — |
| 113 | C4×C73 | — | 8 | 1 | no | — |
| 114 | C4×C73 | #447 | 1 | 2 | yes | — |
| 115 | #12314 | — | 73 | 1 | no | maximal, Sylow |
| 116 | C2×C73 | — | 2 | 1 | no | — |
| 117 | C2×C73 | — | 8 | 2 | no | — |
| 118 | C2×C73 | — | 16 | 1 | no | — |
| 119 | C2×C73 | #1619 | 1 | 1 | yes | characteristic |
| 120 | D73 | — | 32 | 1 | no | — |
| 121 | #3291 | D73 | 1 | 1 | yes | characteristic, complemented |
| 122 | #2213 | — | 73 | 2 | no | — |
| 123 | #1721 | — | 73 | 1 | no | — |
| 124 | #1359 | — | 73 | 1 | no | — |
| 125 | #1336 | — | 73 | 2 | no | — |
| 126 | C73 | #12314 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 127 | #577 | — | 2 | 2 | no | — |
| 128 | #509 | — | 146 | 2 | no | — |
| 129 | #493 | — | 73 | 1 | no | — |
| 130 | #493 | — | 146 | 1 | no | — |
| 131 | #492 | C2×D73 | 1 | 1 | yes | characteristic |
| 132 | #486 | — | 146 | 1 | no | — |
| 133 | #471 | — | 2 | 2 | no | — |
| 134 | #456 | — | 2 | 1 | no | — |
| 135 | #453 | — | 2 | 1 | no | — |
| 136 | #443 | C2×D73 | 1 | 2 | yes | — |
| 137 | #364 | — | 73 | 2 | no | — |
| 138 | #361 | — | 146 | 2 | no | — |
| 139 | #322 | — | 73 | 1 | no | — |
| 140 | 2−1+4 | — | 2 | 1 | no | — |
| 141 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 142 | #137 | — | 2 | 2 | no | — |
| 143 | #137 | — | 4 | 2 | no | — |
| 144 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 145 | AGL(1,ℤ/8) | — | 4 | 2 | no | — |
| 146 | #135 | — | 4 | 2 | no | — |
| 147 | #135 | C22⋊D73 | 1 | 2 | yes | — |
| 148 | C2×SD16 | — | 2 | 2 | no | — |
| 149 | C2×SD16 | — | 4 | 2 | no | — |
| 150 | C2×D8 | — | 292 | 2 | no | — |
| 151 | C2×D8 | — | 146 | 2 | no | — |
| 152 | #131 | — | 2 | 2 | no | — |
| 153 | C4⋊Q8 | C22⋊D73 | 1 | 1 | yes | characteristic |
| 154 | #127 | — | 146 | 1 | no | — |
| 155 | #127 | C22⋊D73 | 1 | 1 | yes | characteristic |
| 156 | SD32 | — | 292 | 2 | no | — |
| 157 | D16 | — | 292 | 2 | no | — |
| 158 | M32 | — | 146 | 2 | no | — |
| 159 | #108 | C22⋊D73 | 1 | 2 | yes | — |
| 160 | #104 | — | 2 | 2 | no | — |
| 161 | #102 | — | 292 | 2 | no | — |
| 162 | #101 | — | 2 | 1 | no | — |
| 163 | #100 | — | 2 | 1 | no | — |
| 164 | C4×C8 | — | 146 | 1 | no | — |
| 165 | C4×C8 | C22×D73 | 1 | 1 | yes | characteristic |
| 166 | #41 | — | 2 | 2 | no | — |
| 167 | #41 | — | 4 | 2 | no | — |
| 168 | #41 | — | 4 | 2 | no | — |
| 169 | #41 | — | 8 | 1 | no | — |
| 170 | C2×Q8 | — | 2 | 1 | no | — |
| 171 | C2×Q8 | — | 4 | 2 | no | — |
| 172 | C2×D4 | — | 2 | 1 | no | — |
| 173 | C2×D4 | — | 4 | 1 | no | — |
| 174 | C2×D4 | — | 4 | 2 | no | — |
| 175 | C2×D4 | — | 8 | 1 | no | — |
| 176 | C2×D4 | — | 292 | 1 | no | — |
| 177 | C2×D4 | — | 292 | 1 | no | — |
| 178 | C2×D4 | — | 292 | 1 | no | — |
| 179 | Q16 | — | 4 | 2 | no | — |
| 180 | Q16 | #11924013 | 1 | 2 | yes | complemented |
| 181 | SD16 | — | 2 | 2 | no | — |
| 182 | SD16 | — | 4 | 2 | no | — |
| 183 | SD16 | — | 4 | 2 | no | — |
| 184 | SD16 | — | 4 | 2 | no | — |
| 185 | SD16 | — | 4 | 2 | no | — |
| 186 | D8 | — | 4 | 2 | no | — |
| 187 | D8 | — | 292 | 2 | no | — |
| 188 | D8 | — | 292 | 2 | no | — |
| 189 | D8 | — | 584 | 4 | no | — |
| 190 | D8 | #11924013 | 1 | 2 | yes | complemented |
| 191 | M16 | — | 2 | 2 | no | — |
| 192 | M16 | — | 4 | 2 | no | — |
| 193 | C2×C8 | — | 4 | 2 | no | — |
| 194 | C2×C8 | — | 146 | 2 | no | — |
| 195 | C2×C8 | C2×C22⋊D73 | 1 | 2 | yes | — |
| 196 | C4⋊C4 | — | 4 | 1 | no | — |
| 197 | C42 | C2×C22⋊D73 | 1 | 1 | yes | characteristic |
| 198 | C16 | — | 292 | 2 | no | — |
| 199 | C23 | — | 4 | 1 | no | — |
| 200 | C23 | — | 8 | 1 | no | — |
| 201 | C23 | — | 584 | 1 | no | — |
| 202 | Q8 | — | 2 | 2 | no | — |
| 203 | Q8 | — | 4 | 2 | no | — |
| 204 | Q8 | — | 8 | 1 | no | — |
| 205 | D4 | — | 2 | 2 | no | — |
| 206 | D4 | — | 4 | 2 | no | — |
| 207 | D4 | — | 4 | 2 | no | — |
| 208 | D4 | — | 8 | 2 | no | — |
| 209 | D4 | — | 8 | 2 | no | — |
| 210 | D4 | — | 8 | 1 | no | — |
| 211 | D4 | — | 584 | 2 | no | — |
| 212 | D4 | — | 584 | 2 | no | — |
| 213 | D4 | — | 584 | 2 | no | — |
| 214 | C2×C4 | — | 2 | 1 | no | — |
| 215 | C2×C4 | — | 4 | 2 | no | — |
| 216 | C2×C4 | — | 4 | 1 | no | — |
| 217 | C2×C4 | — | 8 | 1 | no | — |
| 218 | C2×C4 | — | 8 | 2 | no | — |
| 219 | C2×C4 | #14556560 | 1 | 1 | yes | characteristic |
| 220 | C8 | — | 4 | 2 | no | — |
| 221 | C8 | — | 292 | 2 | no | — |
| 222 | C8 | C2×#11924013 | 1 | 2 | yes | — |
| 223 | C8 | #14556617 | 1 | 2 | yes | — |
| 224 | C22 | — | 4 | 2 | no | — |
| 225 | C22 | — | 8 | 1 | no | — |
| 226 | C22 | — | 16 | 1 | no | — |
| 227 | C22 | — | 32 | 1 | no | — |
| 228 | C22 | — | 1168 | 1 | no | — |
| 229 | C22 | — | 1168 | 1 | no | — |
| 230 | C22 | — | 1168 | 1 | no | — |
| 231 | C22 | #17988755 | 1 | 1 | yes | characteristic |
| 232 | C4 | — | 2 | 2 | no | — |
| 233 | C4 | — | 4 | 2 | no | — |
| 234 | C4 | — | 8 | 1 | no | — |
| 235 | C4 | #17988721 | 1 | 2 | yes | — |
| 236 | C2 | — | 2 | 1 | no | — |
| 237 | C2 | — | 8 | 2 | no | — |
| 238 | C2 | — | 16 | 1 | no | — |
| 239 | C2 | — | 2336 | 1 | no | — |
| 240 | C2 | #22608985 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22612007 | C2 | semidirect |
| 2. #22609595 | C2 | semidirect |
| 3. #22607972 | C2 | semidirect |
| 4. #22607907 | C2 | semidirect |
| 5. C73×#3291 | C2 | semidirect |
| 7. #17988964 | C22 | semidirect |
| 10. #17988408 | C22 | semidirect |
| 12. #17988308 | C22 | semidirect |
| 64. Q16×C73 | D8 | semidirect |
| 71. D8×C73 | D8 | semidirect |
| 121. #3291 | D73 | semidirect |
| 126. C73 | #12314 | semidirect |
| 180. Q16 | #11924013 | semidirect |
| 190. D8 | #11924013 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22612007 | 2 |
| #22609595 | 2 |
| #22607972 | 2 |
| #22607907 | 2 |
| C73×#3291 | 2 |
| #12314 | 73 |