Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
258 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 37 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 | #11727760 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#1391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11725496 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#6512 | — | 2 | 1 | no | — |
| 5 | C22×D56 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | C7×#410 | — | 2 | 1 | no | — |
| 7 | C7×#408 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#6397 | — | 2 | 1 | no | — |
| 9 | #89348 | — | 2 | 1 | no | — |
| 10 | #89347 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | #7398 | — | 7 | 1 | no | maximal, Sylow |
| 12 | C2×C7×#41 | D4 | 1 | 1 | yes | characteristic |
| 13 | C22×D28 | — | 2 | 1 | no | — |
| 14 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 15 | C22×C7×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C7×#117 | — | 4 | 1 | no | — |
| 17 | #6512 | — | 4 | 2 | no | — |
| 18 | #6512 | — | 4 | 2 | no | — |
| 19 | C2×#1143 | — | 4 | 1 | no | — |
| 20 | C2×#1141 | — | 4 | 1 | no | — |
| 21 | C2×#1136 | — | 2 | 1 | no | — |
| 22 | C2×D56 | — | 4 | 1 | no | — |
| 23 | C2×D56 | — | 4 | 2 | no | — |
| 24 | C7×#99 | — | 2 | 2 | no | — |
| 25 | C7×#99 | — | 4 | 1 | no | — |
| 26 | C7×C22⋊C8 | — | 4 | 1 | no | — |
| 27 | #6397 | — | 4 | 2 | no | — |
| 28 | #6386 | — | 4 | 1 | no | — |
| 29 | #2018 | — | 7 | 1 | no | — |
| 30 | #1391 | D7 | 1 | 1 | yes | characteristic, complemented |
| 31 | #1321 | — | 7 | 1 | no | — |
| 32 | C23×D7 | — | 4 | 1 | no | — |
| 33 | C7×#41 | — | 4 | 1 | no | — |
| 34 | C7×#41 | — | 4 | 1 | no | — |
| 35 | C2×C7×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 36 | C2×C7×D4 | — | 2 | 1 | no | — |
| 37 | C2×C7×D4 | D8 | 1 | 1 | yes | characteristic |
| 38 | C22×C4×C7 | — | 2 | 1 | no | — |
| 39 | C22×C4×C7 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 40 | C2×D28 | — | 2 | 2 | no | — |
| 41 | C2×D28 | — | 4 | 1 | no | — |
| 42 | C2×D28 | — | 4 | 1 | no | — |
| 43 | C2×D28 | — | 4 | 1 | no | — |
| 44 | C2×D28 | — | 4 | 1 | no | — |
| 45 | C2×D28 | — | 4 | 1 | no | — |
| 46 | C7×M16 | — | 4 | 1 | no | — |
| 47 | C2×C7×C8 | — | 2 | 1 | no | — |
| 48 | C2×C7×C8 | — | 2 | 1 | no | — |
| 49 | C2×C7×C8 | — | 4 | 1 | no | — |
| 50 | C7×C4⋊C4 | — | 8 | 1 | no | — |
| 51 | C7×C22⋊C4 | — | 4 | 2 | no | — |
| 52 | C42×C7 | — | 8 | 1 | no | — |
| 53 | #1143 | — | 4 | 4 | no | — |
| 54 | #1141 | — | 4 | 4 | no | — |
| 55 | #1136 | — | 2 | 2 | no | — |
| 56 | #1136 | — | 4 | 1 | no | — |
| 57 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 58 | D56 | — | 4 | 2 | no | — |
| 59 | D56 | — | 4 | 2 | no | — |
| 60 | C7⋊C16 | — | 8 | 1 | no | — |
| 61 | C2×AGL(1,ℤ/8) | — | 14 | 1 | no | — |
| 62 | C22×D8 | — | 7 | 1 | no | — |
| 63 | C2×#100 | — | 14 | 1 | no | — |
| 64 | #410 | — | 2 | 1 | no | — |
| 65 | #408 | C2×D7 | 1 | 1 | yes | characteristic |
| 66 | #349 | — | 14 | 1 | no | — |
| 67 | #341 | — | 7 | 1 | no | — |
| 68 | C23×C7 | — | 2 | 1 | no | — |
| 69 | C23×C7 | #104 | 1 | 1 | yes | characteristic |
| 70 | C22×D7 | — | 4 | 2 | no | — |
| 71 | C22×D7 | — | 8 | 1 | no | — |
| 72 | C22×D7 | — | 8 | 1 | no | — |
| 73 | C22×D7 | — | 16 | 2 | no | — |
| 74 | C7×Q8 | — | 4 | 1 | no | — |
| 75 | C7×D4 | — | 4 | 1 | no | — |
| 76 | C7×D4 | — | 8 | 1 | no | — |
| 77 | C2×C4×C7 | — | 2 | 1 | no | — |
| 78 | C2×C4×C7 | — | 2 | 1 | no | — |
| 79 | C2×C4×C7 | — | 2 | 1 | no | — |
| 80 | C2×C4×C7 | — | 4 | 1 | no | — |
| 81 | C2×C4×C7 | — | 4 | 1 | no | — |
| 82 | C2×C4×C7 | — | 8 | 2 | no | — |
| 83 | C2×C4×C7 | #99 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C7 | #102 | 1 | 1 | yes | characteristic |
| 85 | D28 | — | 4 | 2 | no | — |
| 86 | D28 | — | 4 | 2 | no | — |
| 87 | D28 | — | 8 | 1 | no | — |
| 88 | D28 | — | 8 | 1 | no | — |
| 89 | C7×C8 | — | 4 | 1 | no | — |
| 90 | C7×C8 | — | 4 | 1 | no | — |
| 91 | C7⋊C8 | — | 4 | 2 | no | — |
| 92 | C2×#41 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 93 | C22×D4 | — | 14 | 1 | no | — |
| 94 | AGL(1,ℤ/8) | — | 28 | 2 | no | — |
| 95 | AGL(1,ℤ/8) | — | 28 | 2 | no | — |
| 96 | C2×SD16 | — | 28 | 1 | no | — |
| 97 | C2×D8 | — | 28 | 1 | no | — |
| 98 | C2×D8 | — | 28 | 2 | no | — |
| 99 | C2×D8 | — | 28 | 1 | no | — |
| 100 | C2×M16 | — | 14 | 1 | no | — |
| 101 | C2×M16 | D28 | 1 | 1 | yes | characteristic |
| 102 | C22×C8 | C4×D7 | 1 | 1 | yes | characteristic |
| 103 | #117 | — | 4 | 1 | no | — |
| 104 | M32 | — | 28 | 1 | no | — |
| 105 | #100 | — | 28 | 2 | no | — |
| 106 | #99 | — | 2 | 2 | no | — |
| 107 | #99 | — | 4 | 1 | no | — |
| 108 | C22⋊C8 | — | 4 | 1 | no | — |
| 109 | C22×C7 | — | 2 | 1 | no | — |
| 110 | C22×C7 | — | 4 | 1 | no | — |
| 111 | C22×C7 | — | 4 | 1 | no | — |
| 112 | C22×C7 | — | 8 | 1 | no | — |
| 113 | C22×C7 | #327 | 1 | 1 | yes | characteristic |
| 114 | C2×D7 | — | 8 | 2 | no | — |
| 115 | C2×D7 | — | 16 | 2 | no | — |
| 116 | C2×D7 | — | 16 | 2 | no | — |
| 117 | C2×D7 | — | 16 | 2 | no | — |
| 118 | C4×C7 | — | 2 | 1 | no | — |
| 119 | C4×C7 | — | 4 | 1 | no | — |
| 120 | C4×C7 | — | 16 | 2 | no | — |
| 121 | C4×C7 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 122 | C4×C7 | #353 | 1 | 1 | yes | characteristic |
| 123 | C24 | — | 28 | 1 | no | — |
| 124 | #41 | — | 4 | 1 | no | — |
| 125 | #41 | — | 4 | 1 | no | — |
| 126 | C2×Q8 | #1143 | 1 | 1 | yes | characteristic |
| 127 | C2×D4 | — | 2 | 1 | no | — |
| 128 | C2×D4 | — | 14 | 2 | no | — |
| 129 | C2×D4 | — | 28 | 1 | no | — |
| 130 | C2×D4 | — | 28 | 1 | no | — |
| 131 | C2×D4 | — | 28 | 1 | no | — |
| 132 | C2×D4 | — | 28 | 1 | no | — |
| 133 | C2×D4 | — | 28 | 1 | no | — |
| 134 | C2×D4 | #1141 | 1 | 1 | yes | characteristic |
| 135 | C22×C4 | — | 2 | 1 | no | — |
| 136 | C22×C4 | (C2×D7)⋊C4 | 1 | 1 | yes | characteristic |
| 137 | SD16 | — | 28 | 4 | no | — |
| 138 | D8 | — | 28 | 2 | no | — |
| 139 | D8 | — | 28 | 2 | no | — |
| 140 | D8 | — | 28 | 4 | no | — |
| 141 | M16 | — | 4 | 1 | no | — |
| 142 | M16 | — | 14 | 2 | no | — |
| 143 | M16 | — | 28 | 1 | no | — |
| 144 | C2×C8 | — | 2 | 1 | no | — |
| 145 | C2×C8 | — | 2 | 1 | no | — |
| 146 | C2×C8 | — | 4 | 1 | no | — |
| 147 | C2×C8 | — | 28 | 1 | no | — |
| 148 | C4⋊C4 | — | 8 | 1 | no | — |
| 149 | C22⋊C4 | — | 4 | 2 | no | — |
| 150 | C42 | — | 8 | 1 | no | — |
| 151 | C16 | — | 56 | 1 | no | — |
| 152 | C2×C7 | — | 2 | 1 | no | — |
| 153 | C2×C7 | — | 4 | 1 | no | — |
| 154 | C2×C7 | — | 8 | 1 | no | — |
| 155 | C2×C7 | #1343 | 1 | 1 | yes | characteristic |
| 156 | D7 | — | 16 | 2 | no | — |
| 157 | C23 | — | 2 | 1 | no | — |
| 158 | C23 | — | 28 | 2 | no | — |
| 159 | C23 | — | 56 | 1 | no | — |
| 160 | C23 | — | 56 | 1 | no | — |
| 161 | C23 | — | 112 | 2 | no | — |
| 162 | C23 | #6379 | 1 | 1 | yes | characteristic |
| 163 | Q8 | — | 4 | 1 | no | — |
| 164 | D4 | — | 4 | 1 | no | — |
| 165 | D4 | — | 8 | 1 | no | — |
| 166 | D4 | — | 28 | 2 | no | — |
| 167 | D4 | — | 28 | 2 | no | — |
| 168 | D4 | — | 56 | 1 | no | — |
| 169 | D4 | — | 56 | 1 | no | — |
| 170 | C2×C4 | — | 2 | 1 | no | — |
| 171 | C2×C4 | — | 2 | 1 | no | — |
| 172 | C2×C4 | — | 2 | 1 | no | — |
| 173 | C2×C4 | — | 4 | 1 | no | — |
| 174 | C2×C4 | — | 4 | 1 | no | — |
| 175 | C2×C4 | — | 8 | 2 | no | — |
| 176 | C2×C4 | #6380 | 1 | 1 | yes | characteristic |
| 177 | C2×C4 | #6383 | 1 | 1 | yes | characteristic |
| 178 | C8 | — | 4 | 1 | no | — |
| 179 | C8 | — | 4 | 1 | no | — |
| 180 | C8 | — | 28 | 2 | no | — |
| 181 | C7 | #7398 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 182 | C22 | — | 2 | 1 | no | — |
| 183 | C22 | — | 4 | 1 | no | — |
| 184 | C22 | — | 4 | 1 | no | — |
| 185 | C22 | — | 8 | 1 | no | — |
| 186 | C22 | — | 56 | 2 | no | — |
| 187 | C22 | — | 112 | 2 | no | — |
| 188 | C22 | — | 112 | 2 | no | — |
| 189 | C22 | — | 112 | 2 | no | — |
| 190 | C22 | #89246 | 1 | 1 | yes | characteristic |
| 191 | C4 | — | 2 | 1 | no | — |
| 192 | C4 | — | 4 | 1 | no | — |
| 193 | C4 | — | 16 | 2 | no | — |
| 194 | C4 | #89266 | 1 | 1 | yes | characteristic |
| 195 | C4 | #89269 | 1 | 1 | yes | characteristic |
| 196 | C2 | — | 2 | 1 | no | — |
| 197 | C2 | — | 4 | 1 | no | — |
| 198 | C2 | — | 8 | 1 | no | — |
| 199 | C2 | — | 112 | 2 | no | — |
| 200 | C2 | #11725295 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#1391 | C2 | semidirect |
| 3. #11725496 | C2 | semidirect |
| 5. C22×D56 | C4 | semidirect |
| 10. #89347 | C4 | semidirect |
| 30. #1391 | D7 | semidirect |
| 181. C7 | #7398 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11727760 | 2 |
| C7×#1391 | 2 |
| #11725496 | 2 |
| #7398 | 7 |