Finite Group DatabasebetaFinite Group DatabasebetaOrder 18688.
356 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 29 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 | #22626957 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22625761 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #17989672 | — | 2 | 2 | no | — |
| 4 | C2×#14556649 | — | 2 | 2 | no | — |
| 5 | #17989530 | C4 | 1 | 1 | yes | characteristic |
| 6 | #17989530 | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×#11924027 | C4 | 1 | 1 | yes | characteristic |
| 8 | C22⋊#10596842 | — | 4 | 2 | no | — |
| 9 | C2×C22⋊#64408 | — | 2 | 1 | no | — |
| 10 | C2×C22⋊#64408 | D4 | 1 | 2 | yes | — |
| 11 | #14556649 | — | 4 | 4 | no | — |
| 12 | (C2×#64408)⋊C4 | — | 4 | 1 | no | — |
| 13 | (C2×#64408)⋊C4 | — | 4 | 1 | no | — |
| 14 | C2×#11924027 | — | 4 | 1 | no | — |
| 15 | C2×#11924027 | — | 4 | 1 | no | — |
| 16 | C2×#11924027 | — | 4 | 4 | no | — |
| 17 | C22×Dic146 | C8 | 1 | 1 | yes | characteristic, complemented |
| 18 | C22×C22⋊D73 | — | 2 | 1 | no | — |
| 19 | C23×Dic73 | — | 2 | 1 | no | — |
| 20 | C22×C4×D73 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C22×D4×C73 | C8 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×#10596842 | — | 8 | 2 | no | — |
| 23 | C22×#64408 | — | 2 | 2 | no | — |
| 24 | C22×#64408 | — | 4 | 1 | no | — |
| 25 | C22⋊#64408 | — | 4 | 2 | no | — |
| 26 | C22⋊#64408 | — | 4 | 2 | no | — |
| 27 | C23×D73 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 28 | #11924027 | — | 4 | 4 | no | — |
| 29 | #11924027 | — | 4 | 4 | no | — |
| 30 | #11924027 | — | 4 | 4 | no | — |
| 31 | #11924027 | — | 4 | 4 | no | — |
| 32 | C2×Dic146 | — | 4 | 1 | no | — |
| 33 | C2×Dic146 | — | 4 | 2 | no | — |
| 34 | C2×C22⋊D73 | — | 2 | 4 | no | — |
| 35 | C2×C22⋊D73 | — | 4 | 2 | no | — |
| 36 | C2×C22⋊D73 | — | 4 | 1 | no | — |
| 37 | C2×C22⋊D73 | — | 4 | 1 | no | — |
| 38 | C22×Dic73 | — | 2 | 1 | no | — |
| 39 | C22×Dic73 | — | 4 | 2 | no | — |
| 40 | C22×Dic73 | — | 4 | 2 | no | — |
| 41 | C22×Dic73 | — | 4 | 2 | no | — |
| 42 | C22×Dic73 | M16 | 1 | 1 | yes | characteristic |
| 43 | C2×C4×D73 | — | 2 | 1 | no | — |
| 44 | C2×C4×D73 | — | 2 | 1 | no | — |
| 45 | C2×C4×D73 | — | 4 | 2 | no | — |
| 46 | C24×C73 | — | 2 | 1 | no | — |
| 47 | C2×D4×C73 | — | 4 | 1 | no | — |
| 48 | C2×D4×C73 | — | 4 | 2 | no | — |
| 49 | C22×C4×C73 | C2×C8 | 1 | 1 | yes | characteristic |
| 50 | C23×C73 | — | 2 | 2 | no | — |
| 51 | C23×C73 | — | 4 | 1 | no | — |
| 52 | C23×C73 | — | 4 | 1 | no | — |
| 53 | C23×C73 | — | 8 | 2 | no | — |
| 54 | C23×C73 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 55 | C22×D73 | — | 2 | 2 | no | — |
| 56 | C22×D73 | — | 8 | 1 | no | — |
| 57 | C22×D73 | #99 | 1 | 1 | yes | characteristic |
| 58 | C22×D73 | #100 | 1 | 1 | yes | characteristic |
| 59 | C2×#64408 | — | 4 | 2 | no | — |
| 60 | C2×#64408 | — | 8 | 2 | no | — |
| 61 | C2×#64408 | — | 8 | 1 | no | — |
| 62 | C2×#64408 | — | 8 | 2 | no | — |
| 63 | #10596842 | — | 16 | 2 | no | — |
| 64 | D4×C73 | — | 4 | 2 | no | — |
| 65 | D4×C73 | — | 4 | 2 | no | — |
| 66 | C2×C4×C73 | — | 2 | 1 | no | — |
| 67 | C2×C4×C73 | — | 4 | 1 | no | — |
| 68 | C22⋊D73 | — | 4 | 4 | no | — |
| 69 | C22⋊D73 | — | 4 | 4 | no | — |
| 70 | C2×Dic73 | — | 2 | 2 | no | — |
| 71 | C2×Dic73 | — | 2 | 1 | no | — |
| 72 | C2×Dic73 | — | 4 | 1 | no | — |
| 73 | C2×Dic73 | — | 4 | 8 | no | — |
| 74 | C2×Dic73 | — | 4 | 1 | no | — |
| 75 | C2×Dic73 | — | 4 | 1 | no | — |
| 76 | C4×D73 | — | 4 | 2 | no | — |
| 77 | C4×D73 | — | 4 | 2 | no | — |
| 78 | Dic146 | — | 4 | 2 | no | — |
| 79 | Dic146 | — | 4 | 2 | no | — |
| 80 | C22×C73 | — | 2 | 1 | no | — |
| 81 | C22×C73 | — | 4 | 1 | no | — |
| 82 | C22×C73 | — | 4 | 2 | no | — |
| 83 | C22×C73 | — | 8 | 2 | no | — |
| 84 | C22×C73 | — | 8 | 2 | no | — |
| 85 | C22×C73 | — | 8 | 2 | no | — |
| 86 | C22×C73 | #323 | 1 | 1 | yes | characteristic |
| 87 | C2×D73 | — | 2 | 1 | no | — |
| 88 | C2×D73 | — | 4 | 1 | no | — |
| 89 | C2×D73 | — | 4 | 1 | no | — |
| 90 | C2×D73 | — | 8 | 2 | no | — |
| 91 | C2×D73 | AΣL(1,𝔽16) | 1 | 2 | yes | — |
| 92 | #64408 | — | 8 | 2 | no | — |
| 93 | #64408 | — | 16 | 1 | no | — |
| 94 | C4×C73 | — | 4 | 1 | no | — |
| 95 | Dic73 | — | 4 | 2 | no | — |
| 96 | Dic73 | — | 4 | 1 | no | — |
| 97 | #7339 | — | 73 | 1 | no | maximal, Sylow |
| 98 | C2×C73 | — | 2 | 1 | no | — |
| 99 | C2×C73 | — | 4 | 1 | no | — |
| 100 | C2×C73 | — | 8 | 2 | no | — |
| 101 | C2×C73 | #1316 | 1 | 1 | yes | characteristic |
| 102 | D73 | — | 2 | 2 | no | — |
| 103 | D73 | — | 4 | 1 | no | — |
| 104 | #1782 | — | 73 | 1 | no | — |
| 105 | #1317 | — | 73 | 2 | no | — |
| 106 | C73 | #7339 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 107 | C22×#41 | — | 73 | 1 | no | — |
| 108 | C2×#99 | — | 146 | 2 | no | — |
| 109 | #381 | — | 73 | 1 | no | — |
| 110 | #381 | — | 73 | 1 | no | — |
| 111 | #323 | — | 146 | 2 | no | — |
| 112 | C2×#41 | — | 292 | 4 | no | — |
| 113 | C2×#41 | — | 292 | 1 | no | — |
| 114 | C2×#41 | — | 292 | 1 | no | — |
| 115 | C22×Q8 | — | 73 | 1 | no | — |
| 116 | C22×D4 | — | 146 | 1 | no | — |
| 117 | C22×D4 | #10596842 | 1 | 1 | yes | characteristic, complemented |
| 118 | C23×C4 | — | 73 | 1 | no | — |
| 119 | C23×C4 | — | 146 | 1 | no | — |
| 120 | C2×C22⋊C4 | — | 73 | 2 | no | — |
| 121 | C2×C22⋊C4 | — | 146 | 1 | no | — |
| 122 | #99 | — | 292 | 4 | no | — |
| 123 | C22⋊C8 | — | 292 | 2 | no | — |
| 124 | #95 | — | 292 | 1 | no | — |
| 125 | #95 | — | 292 | 1 | no | — |
| 126 | C24 | — | 2 | 1 | no | — |
| 127 | C24 | — | 73 | 1 | no | — |
| 128 | #41 | — | 292 | 4 | no | — |
| 129 | #41 | — | 292 | 4 | no | — |
| 130 | #41 | — | 292 | 4 | no | — |
| 131 | #41 | — | 292 | 4 | no | — |
| 132 | C2×Q8 | — | 292 | 1 | no | — |
| 133 | C2×Q8 | — | 292 | 2 | no | — |
| 134 | C2×D4 | — | 4 | 1 | no | — |
| 135 | C2×D4 | — | 4 | 2 | no | — |
| 136 | C2×D4 | — | 292 | 1 | no | — |
| 137 | C2×D4 | — | 292 | 1 | no | — |
| 138 | C2×D4 | — | 292 | 2 | no | — |
| 139 | C2×D4 | — | 146 | 4 | no | — |
| 140 | C22×C4 | — | 292 | 2 | no | — |
| 141 | C22×C4 | — | 292 | 2 | no | — |
| 142 | C22×C4 | — | 292 | 2 | no | — |
| 143 | C22×C4 | — | 292 | 2 | no | — |
| 144 | C22×C4 | — | 292 | 1 | no | — |
| 145 | C22×C4 | — | 73 | 1 | no | — |
| 146 | C22×C4 | — | 146 | 1 | no | — |
| 147 | C22×C4 | — | 146 | 1 | no | — |
| 148 | C22×C4 | — | 146 | 1 | no | — |
| 149 | C22×C4 | — | 146 | 2 | no | — |
| 150 | C22×C4 | C2×#10596842 | 1 | 1 | yes | characteristic |
| 151 | C2×C8 | — | 584 | 2 | no | — |
| 152 | C22⋊C4 | — | 292 | 2 | no | — |
| 153 | C22⋊C4 | — | 292 | 2 | no | — |
| 154 | C23 | — | 2 | 2 | no | — |
| 155 | C23 | — | 4 | 1 | no | — |
| 156 | C23 | — | 4 | 1 | no | — |
| 157 | C23 | — | 8 | 2 | no | — |
| 158 | C23 | — | 584 | 1 | no | — |
| 159 | C23 | — | 73 | 1 | no | — |
| 160 | C23 | — | 73 | 1 | no | — |
| 161 | C23 | — | 146 | 2 | no | — |
| 162 | C23 | C22⋊#10596842 | 1 | 1 | yes | characteristic |
| 163 | Q8 | — | 292 | 2 | no | — |
| 164 | Q8 | — | 292 | 2 | no | — |
| 165 | D4 | — | 4 | 2 | no | — |
| 166 | D4 | — | 4 | 2 | no | — |
| 167 | D4 | — | 292 | 4 | no | — |
| 168 | D4 | — | 292 | 4 | no | — |
| 169 | C2×C4 | — | 2 | 1 | no | — |
| 170 | C2×C4 | — | 4 | 1 | no | — |
| 171 | C2×C4 | — | 292 | 2 | no | — |
| 172 | C2×C4 | — | 292 | 2 | no | — |
| 173 | C2×C4 | — | 292 | 1 | no | — |
| 174 | C2×C4 | — | 292 | 8 | no | — |
| 175 | C2×C4 | — | 292 | 1 | no | — |
| 176 | C2×C4 | — | 292 | 1 | no | — |
| 177 | C2×C4 | — | 292 | 2 | no | — |
| 178 | C2×C4 | — | 584 | 2 | no | — |
| 179 | C2×C4 | — | 584 | 2 | no | — |
| 180 | C2×C4 | — | 584 | 1 | no | — |
| 181 | C2×C4 | — | 146 | 2 | no | — |
| 182 | C2×C4 | — | 146 | 1 | no | — |
| 183 | C8 | — | 1168 | 2 | no | — |
| 184 | C22 | — | 2 | 1 | no | — |
| 185 | C22 | — | 4 | 1 | no | — |
| 186 | C22 | — | 4 | 2 | no | — |
| 187 | C22 | — | 8 | 2 | no | — |
| 188 | C22 | — | 8 | 2 | no | — |
| 189 | C22 | — | 8 | 2 | no | — |
| 190 | C22 | — | 292 | 1 | no | — |
| 191 | C22 | — | 292 | 1 | no | — |
| 192 | C22 | — | 584 | 2 | no | — |
| 193 | C22 | — | 73 | 2 | no | — |
| 194 | C22 | — | 146 | 1 | no | — |
| 195 | C22 | #17989672 | 1 | 1 | yes | characteristic |
| 196 | C4 | — | 4 | 1 | no | — |
| 197 | C4 | — | 292 | 1 | no | — |
| 198 | C4 | — | 292 | 2 | no | — |
| 199 | C4 | — | 584 | 2 | no | — |
| 200 | C4 | — | 1168 | 1 | no | — |
| 201 | C2 | — | 2 | 1 | no | — |
| 202 | C2 | — | 4 | 1 | no | — |
| 203 | C2 | — | 8 | 2 | no | — |
| 204 | C2 | — | 292 | 1 | no | — |
| 205 | C2 | — | 146 | 2 | no | — |
| 206 | C2 | #22626956 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22626957 | C2 | semidirect |
| 17. C22×Dic146 | C8 | semidirect |
| 21. C22×D4×C73 | C8 | semidirect |
| 106. C73 | #7339 | semidirect |
| 117. C22×D4 | #10596842 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22626957 | 2 |
| #22625761 | 2 |
| #7339 | 73 |