Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
254 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 32 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 | #22625022 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22624866 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22611041 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #17989573 | — | 2 | 1 | no | — |
| 5 | #17989483 | — | 2 | 1 | no | — |
| 6 | C2×#14556619 | — | 2 | 1 | no | — |
| 7 | C22×D584 | C4 | 1 | 1 | yes | characteristic |
| 8 | C2×#14556507 | — | 2 | 1 | no | — |
| 9 | #17988553 | C22 | 1 | 1 | yes | characteristic |
| 10 | #341×C73 | C4 | 1 | 1 | yes | characteristic |
| 11 | D73⋊M16 | — | 4 | 1 | no | — |
| 12 | #14556665 | — | 4 | 1 | no | — |
| 13 | #14556652 | — | 4 | 1 | no | — |
| 14 | #14556651 | — | 2 | 2 | no | — |
| 15 | C2×#11924028 | D4 | 1 | 1 | yes | characteristic |
| 16 | C22×D292 | — | 2 | 1 | no | — |
| 17 | #14556619 | — | 4 | 2 | no | — |
| 18 | #14556619 | — | 4 | 2 | no | — |
| 19 | C2×#11924017 | — | 4 | 1 | no | — |
| 20 | C2×D584 | — | 4 | 1 | no | — |
| 21 | C2×D584 | — | 4 | 1 | no | — |
| 22 | C2×D584 | — | 4 | 2 | no | — |
| 23 | C2×#11924012 | D4 | 1 | 1 | yes | characteristic |
| 24 | #14556507 | — | 4 | 2 | no | — |
| 25 | #14556503 | — | 4 | 1 | no | — |
| 26 | C2×M16×C73 | — | 2 | 1 | no | — |
| 27 | C22×C8×C73 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #11924038 | — | 8 | 1 | no | — |
| 29 | #11924037 | — | 4 | 2 | no | — |
| 30 | #11924036 | — | 8 | 1 | no | — |
| 31 | #11924031 | — | 8 | 1 | no | — |
| 32 | C23×D73 | — | 4 | 1 | no | — |
| 33 | #11924028 | — | 4 | 1 | no | — |
| 34 | #11924028 | — | 4 | 1 | no | — |
| 35 | C2×Dic146 | D8 | 1 | 1 | yes | characteristic |
| 36 | C2×D292 | — | 2 | 2 | no | — |
| 37 | C2×D292 | — | 4 | 1 | no | — |
| 38 | C2×D292 | — | 4 | 1 | no | — |
| 39 | C2×D292 | — | 4 | 1 | no | — |
| 40 | C2×D292 | — | 4 | 1 | no | — |
| 41 | C2×D292 | — | 4 | 1 | no | — |
| 42 | C2×D292 | SD16 | 1 | 1 | yes | characteristic |
| 43 | C2×C22⋊D73 | — | 2 | 1 | no | — |
| 44 | C2×C4×D73 | — | 2 | 1 | no | — |
| 45 | #11924017 | — | 4 | 4 | no | — |
| 46 | D584 | — | 4 | 2 | no | — |
| 47 | D584 | — | 4 | 2 | no | — |
| 48 | D584 | — | 4 | 4 | no | — |
| 49 | #11924012 | — | 4 | 1 | no | — |
| 50 | C2×#10596831 | — | 2 | 1 | no | — |
| 51 | C22×C4×C73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 52 | M16×C73 | — | 2 | 2 | no | — |
| 53 | M16×C73 | — | 4 | 1 | no | — |
| 54 | C2×C8×C73 | — | 2 | 1 | no | — |
| 55 | C2×C8×C73 | — | 4 | 1 | no | — |
| 56 | C2×C8×C73 | — | 4 | 1 | no | — |
| 57 | C23×C73 | #104 | 1 | 1 | yes | characteristic |
| 58 | C22×D73 | — | 2 | 1 | no | — |
| 59 | C22×D73 | — | 4 | 2 | no | — |
| 60 | C22×D73 | — | 8 | 1 | no | — |
| 61 | C22×D73 | — | 8 | 1 | no | — |
| 62 | C22×D73 | — | 16 | 2 | no | — |
| 63 | C2×C4×C73 | — | 2 | 1 | no | — |
| 64 | C2×C4×C73 | — | 2 | 1 | no | — |
| 65 | C2×C4×C73 | #99 | 1 | 1 | yes | characteristic |
| 66 | C2×C4×C73 | #102 | 1 | 1 | yes | characteristic |
| 67 | C22⋊D73 | — | 8 | 1 | no | — |
| 68 | C2×Dic73 | — | 2 | 1 | no | — |
| 69 | D292 | — | 4 | 1 | no | — |
| 70 | D292 | — | 4 | 2 | no | — |
| 71 | D292 | — | 4 | 2 | no | — |
| 72 | D292 | — | 8 | 1 | no | — |
| 73 | D292 | — | 8 | 1 | no | — |
| 74 | C4×D73 | — | 4 | 1 | no | — |
| 75 | C4×D73 | — | 4 | 1 | no | — |
| 76 | Dic146 | — | 4 | 1 | no | — |
| 77 | #10596833 | — | 8 | 2 | no | — |
| 78 | C8×C73 | — | 4 | 2 | no | — |
| 79 | C8×C73 | — | 4 | 1 | no | — |
| 80 | #10596831 | — | 4 | 1 | no | — |
| 81 | C22×C73 | — | 2 | 1 | no | — |
| 82 | C22×C73 | — | 4 | 1 | no | — |
| 83 | C22×C73 | #327 | 1 | 1 | yes | characteristic |
| 84 | C2×D73 | — | 4 | 1 | no | — |
| 85 | C2×D73 | — | 8 | 1 | no | — |
| 86 | C2×D73 | — | 8 | 2 | no | — |
| 87 | C2×D73 | — | 16 | 2 | no | — |
| 88 | C2×D73 | — | 16 | 2 | no | — |
| 89 | C2×D73 | — | 16 | 2 | no | — |
| 90 | C4×C73 | — | 2 | 1 | no | — |
| 91 | C4×C73 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 92 | C4×C73 | #353 | 1 | 1 | yes | characteristic |
| 93 | Dic73 | — | 4 | 1 | no | — |
| 94 | #7400 | — | 73 | 1 | no | maximal, Sylow |
| 95 | C2×C73 | — | 2 | 1 | no | — |
| 96 | C2×C73 | — | 4 | 1 | no | — |
| 97 | C2×C73 | #1343 | 1 | 1 | yes | characteristic |
| 98 | D73 | — | 8 | 1 | no | — |
| 99 | D73 | — | 16 | 2 | no | — |
| 100 | #2018 | — | 73 | 1 | no | — |
| 101 | #1392 | — | 73 | 1 | no | — |
| 102 | #1321 | — | 73 | 1 | no | — |
| 103 | C73 | #7400 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 104 | C2×AGL(1,ℤ/8) | — | 146 | 1 | no | — |
| 105 | C22×D8 | — | 73 | 1 | no | — |
| 106 | #413 | — | 146 | 1 | no | — |
| 107 | C2×#100 | — | 146 | 1 | no | — |
| 108 | #408 | — | 73 | 1 | no | — |
| 109 | #349 | — | 146 | 1 | no | — |
| 110 | #341 | #64408 | 1 | 1 | yes | characteristic |
| 111 | C2×#41 | — | 73 | 1 | no | — |
| 112 | C22×D4 | — | 146 | 1 | no | — |
| 113 | AGL(1,ℤ/8) | — | 292 | 2 | no | — |
| 114 | AGL(1,ℤ/8) | — | 292 | 2 | no | — |
| 115 | C2×SD16 | — | 292 | 1 | no | — |
| 116 | C2×D8 | — | 292 | 1 | no | — |
| 117 | C2×D8 | — | 292 | 1 | no | — |
| 118 | C2×D8 | — | 292 | 2 | no | — |
| 119 | C2×M16 | — | 2 | 1 | no | — |
| 120 | C2×M16 | — | 292 | 1 | no | — |
| 121 | C2×M16 | — | 73 | 1 | no | — |
| 122 | C22×C8 | C2×#64408 | 1 | 1 | yes | characteristic |
| 123 | M32 | — | 292 | 1 | no | — |
| 124 | #101 | — | 292 | 1 | no | — |
| 125 | #100 | — | 292 | 2 | no | — |
| 126 | #100 | — | 146 | 2 | no | — |
| 127 | C22⋊C8 | — | 292 | 1 | no | — |
| 128 | C24 | — | 292 | 1 | no | — |
| 129 | #41 | — | 292 | 1 | no | — |
| 130 | #41 | — | 292 | 1 | no | — |
| 131 | C2×Q8 | — | 73 | 1 | no | — |
| 132 | C2×D4 | — | 292 | 1 | no | — |
| 133 | C2×D4 | — | 292 | 1 | no | — |
| 134 | C2×D4 | — | 292 | 1 | no | — |
| 135 | C2×D4 | — | 292 | 1 | no | — |
| 136 | C2×D4 | — | 292 | 1 | no | — |
| 137 | C2×D4 | — | 73 | 1 | no | — |
| 138 | C2×D4 | — | 146 | 2 | no | — |
| 139 | C2×D4 | — | 146 | 1 | no | — |
| 140 | C22×C4 | — | 146 | 1 | no | — |
| 141 | C22×C4 | C22⋊#64408 | 1 | 1 | yes | characteristic |
| 142 | SD16 | — | 292 | 4 | no | — |
| 143 | D8 | — | 292 | 2 | no | — |
| 144 | D8 | — | 292 | 2 | no | — |
| 145 | D8 | — | 292 | 4 | no | — |
| 146 | M16 | — | 2 | 2 | no | — |
| 147 | M16 | — | 4 | 1 | no | — |
| 148 | M16 | — | 292 | 1 | no | — |
| 149 | M16 | — | 292 | 2 | no | — |
| 150 | M16 | — | 584 | 1 | no | — |
| 151 | C2×C8 | — | 2 | 1 | no | — |
| 152 | C2×C8 | — | 4 | 1 | no | — |
| 153 | C2×C8 | — | 4 | 1 | no | — |
| 154 | C2×C8 | — | 584 | 1 | no | — |
| 155 | C2×C8 | — | 146 | 1 | no | — |
| 156 | C16 | — | 584 | 1 | no | — |
| 157 | C23 | — | 292 | 2 | no | — |
| 158 | C23 | — | 584 | 1 | no | — |
| 159 | C23 | — | 584 | 1 | no | — |
| 160 | C23 | — | 1168 | 2 | no | — |
| 161 | C23 | — | 146 | 1 | no | — |
| 162 | C23 | #14556656 | 1 | 1 | yes | characteristic |
| 163 | Q8 | — | 292 | 1 | no | — |
| 164 | D4 | — | 292 | 1 | no | — |
| 165 | D4 | — | 292 | 2 | no | — |
| 166 | D4 | — | 292 | 2 | no | — |
| 167 | D4 | — | 584 | 1 | no | — |
| 168 | D4 | — | 584 | 1 | no | — |
| 169 | D4 | — | 584 | 1 | no | — |
| 170 | C2×C4 | — | 2 | 1 | no | — |
| 171 | C2×C4 | — | 2 | 1 | no | — |
| 172 | C2×C4 | — | 292 | 1 | no | — |
| 173 | C2×C4 | — | 292 | 1 | no | — |
| 174 | C2×C4 | — | 146 | 1 | no | — |
| 175 | C2×C4 | #14556650 | 1 | 1 | yes | characteristic |
| 176 | C2×C4 | #14556653 | 1 | 1 | yes | characteristic |
| 177 | C8 | — | 4 | 2 | no | — |
| 178 | C8 | — | 4 | 1 | no | — |
| 179 | C8 | — | 292 | 1 | no | — |
| 180 | C8 | — | 584 | 2 | no | — |
| 181 | C22 | — | 2 | 1 | no | — |
| 182 | C22 | — | 4 | 1 | no | — |
| 183 | C22 | — | 292 | 1 | no | — |
| 184 | C22 | — | 584 | 1 | no | — |
| 185 | C22 | — | 584 | 2 | no | — |
| 186 | C22 | — | 1168 | 2 | no | — |
| 187 | C22 | — | 1168 | 2 | no | — |
| 188 | C22 | — | 1168 | 2 | no | — |
| 189 | C22 | #17989439 | 1 | 1 | yes | characteristic |
| 190 | C4 | — | 2 | 1 | no | — |
| 191 | C4 | — | 292 | 1 | no | — |
| 192 | C4 | #17989488 | 1 | 1 | yes | characteristic |
| 193 | C4 | #17989492 | 1 | 1 | yes | characteristic |
| 194 | C2 | — | 2 | 1 | no | — |
| 195 | C2 | — | 4 | 1 | no | — |
| 196 | C2 | — | 584 | 1 | no | — |
| 197 | C2 | — | 1168 | 2 | no | — |
| 198 | C2 | #22624910 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22625022 | C2 | semidirect |
| 2. #22624866 | C2 | semidirect |
| 103. C73 | #7400 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22625022 | 2 |
| #22624866 | 2 |
| #22611041 | 2 |
| #7400 | 73 |