Finite Group DatabasebetaFinite Group DatabasebetaOrder 12032.
452 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 39 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 | #20243308 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20241324 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C47×#1320 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23×D188 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×#13093940 | — | 2 | 1 | no | — |
| 6 | C2×#13093940 | — | 2 | 1 | no | — |
| 7 | #15689087 | C4 | 1 | 1 | yes | characteristic |
| 8 | #15688905 | — | 2 | 1 | no | — |
| 9 | C47×#407 | C22 | 1 | 1 | yes | characteristic |
| 10 | C47×#349 | — | 2 | 1 | no | — |
| 11 | C24×D47 | C8 | 1 | 2 | yes | — |
| 12 | C22×D188 | — | 2 | 1 | no | — |
| 13 | C22×D188 | — | 2 | 1 | no | — |
| 14 | C22×D188 | — | 4 | 2 | no | — |
| 15 | C22×D188 | — | 4 | 2 | no | — |
| 16 | C22×D188 | — | 4 | 1 | no | — |
| 17 | C22×D188 | — | 4 | 1 | no | — |
| 18 | C2×#10625329 | — | 2 | 1 | no | — |
| 19 | #13093964 | — | 4 | 1 | no | — |
| 20 | #13093940 | — | 4 | 2 | no | — |
| 21 | #13093940 | — | 4 | 2 | no | — |
| 22 | C22⋊(C47⋊C8) | — | 4 | 1 | no | — |
| 23 | C23×C4×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×M16×C47 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×M16×C47 | D4 | 1 | 1 | yes | characteristic |
| 26 | M32×C47 | — | 4 | 1 | no | — |
| 27 | C22⋊C8×C47 | — | 4 | 1 | no | — |
| 28 | C24×C47 | C2×C8 | 1 | 1 | yes | characteristic |
| 29 | C23×D47 | — | 2 | 2 | no | — |
| 30 | C23×D47 | — | 4 | 4 | no | — |
| 31 | C23×D47 | — | 4 | 2 | no | — |
| 32 | C23×D47 | — | 8 | 4 | no | — |
| 33 | C22×C4×C47 | — | 2 | 1 | no | — |
| 34 | C22×C4×C47 | — | 2 | 1 | no | — |
| 35 | C22×C4×C47 | — | 4 | 1 | no | — |
| 36 | C22×C4×C47 | — | 4 | 1 | no | — |
| 37 | C22×C4×C47 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 38 | C22×C4×C47 | M16 | 1 | 1 | yes | characteristic |
| 39 | C2×D188 | — | 2 | 2 | no | — |
| 40 | C2×D188 | — | 2 | 2 | no | — |
| 41 | C2×D188 | — | 4 | 1 | no | — |
| 42 | C2×D188 | — | 4 | 1 | no | — |
| 43 | C2×D188 | — | 4 | 2 | no | — |
| 44 | C2×D188 | — | 4 | 4 | no | — |
| 45 | C2×D188 | — | 4 | 8 | no | — |
| 46 | C2×D188 | — | 4 | 2 | no | — |
| 47 | C2×D188 | — | 4 | 2 | no | — |
| 48 | C2×D188 | — | 4 | 1 | no | — |
| 49 | C2×D188 | — | 4 | 1 | no | — |
| 50 | C2×D188 | — | 4 | 2 | no | — |
| 51 | C2×D188 | — | 4 | 2 | no | — |
| 52 | M16×C47 | — | 4 | 1 | no | — |
| 53 | M16×C47 | — | 4 | 1 | no | — |
| 54 | C2×C8×C47 | — | 2 | 1 | no | — |
| 55 | C2×C8×C47 | — | 2 | 1 | no | — |
| 56 | #10625329 | — | 4 | 1 | no | — |
| 57 | #10625329 | — | 4 | 1 | no | — |
| 58 | C2×C47⋊C8 | — | 4 | 1 | no | — |
| 59 | C16×C47 | — | 8 | 1 | no | — |
| 60 | C47⋊C16 | — | 8 | 1 | no | — |
| 61 | C23×C47 | — | 2 | 1 | no | — |
| 62 | C23×C47 | — | 4 | 1 | no | — |
| 63 | C23×C47 | — | 4 | 2 | no | — |
| 64 | C23×C47 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 65 | C22×D47 | — | 2 | 4 | no | — |
| 66 | C22×D47 | — | 4 | 2 | no | — |
| 67 | C22×D47 | — | 4 | 4 | no | — |
| 68 | C22×D47 | — | 4 | 2 | no | — |
| 69 | C22×D47 | — | 4 | 4 | no | — |
| 70 | C22×D47 | — | 8 | 4 | no | — |
| 71 | C22×D47 | — | 8 | 4 | no | — |
| 72 | C22×D47 | — | 8 | 8 | no | — |
| 73 | C22×D47 | — | 8 | 4 | no | — |
| 74 | C22×D47 | — | 8 | 8 | no | — |
| 75 | C2×C4×C47 | — | 2 | 1 | no | — |
| 76 | C2×C4×C47 | — | 2 | 1 | no | — |
| 77 | C2×C4×C47 | — | 2 | 1 | no | — |
| 78 | C2×C4×C47 | — | 4 | 2 | no | — |
| 79 | C2×C4×C47 | — | 4 | 1 | no | — |
| 80 | C2×C4×C47 | — | 4 | 1 | no | — |
| 81 | C2×C4×C47 | — | 4 | 1 | no | — |
| 82 | C2×C4×C47 | #99 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×C47 | #100 | 1 | 1 | yes | characteristic |
| 84 | D188 | — | 4 | 2 | no | — |
| 85 | D188 | — | 4 | 2 | no | — |
| 86 | D188 | — | 4 | 4 | no | — |
| 87 | D188 | — | 4 | 4 | no | — |
| 88 | D188 | — | 4 | 2 | no | — |
| 89 | D188 | — | 4 | 2 | no | — |
| 90 | C8×C47 | — | 4 | 1 | no | — |
| 91 | C8×C47 | — | 4 | 1 | no | — |
| 92 | C47⋊C8 | — | 8 | 1 | no | — |
| 93 | #7340 | — | 47 | 1 | no | maximal, Sylow |
| 94 | C22×C47 | — | 2 | 2 | no | — |
| 95 | C22×C47 | — | 2 | 1 | no | — |
| 96 | C22×C47 | — | 4 | 1 | no | — |
| 97 | C22×C47 | — | 4 | 1 | no | — |
| 98 | C22×C47 | — | 4 | 2 | no | — |
| 99 | C22×C47 | — | 4 | 2 | no | — |
| 100 | C22×C47 | — | 4 | 1 | no | — |
| 101 | C22×C47 | #323 | 1 | 1 | yes | characteristic |
| 102 | C2×D47 | — | 4 | 4 | no | — |
| 103 | C2×D47 | — | 8 | 4 | no | — |
| 104 | C2×D47 | — | 8 | 4 | no | — |
| 105 | C2×D47 | — | 8 | 8 | no | — |
| 106 | C2×D47 | — | 8 | 4 | no | — |
| 107 | C2×D47 | — | 8 | 8 | no | — |
| 108 | C4×C47 | — | 2 | 1 | no | — |
| 109 | C4×C47 | — | 4 | 1 | no | — |
| 110 | C4×C47 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 111 | C4×C47 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 112 | #1783 | — | 47 | 1 | no | — |
| 113 | #1320 | — | 47 | 1 | no | — |
| 114 | #1320 | D47 | 1 | 1 | yes | characteristic, complemented |
| 115 | C2×C47 | — | 2 | 1 | no | — |
| 116 | C2×C47 | — | 4 | 1 | no | — |
| 117 | C2×C47 | — | 4 | 2 | no | — |
| 118 | C2×C47 | #1316 | 1 | 1 | yes | characteristic |
| 119 | D47 | — | 8 | 4 | no | — |
| 120 | C23×D4 | — | 47 | 1 | no | — |
| 121 | C2×#100 | — | 94 | 1 | no | — |
| 122 | C2×#100 | — | 94 | 1 | no | — |
| 123 | #407 | — | 47 | 1 | no | — |
| 124 | #407 | C2×D47 | 1 | 1 | yes | characteristic |
| 125 | #349 | — | 2 | 1 | no | — |
| 126 | #349 | — | 94 | 1 | no | — |
| 127 | C47 | #7340 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 128 | C25 | — | 47 | 2 | no | — |
| 129 | C22×D4 | — | 94 | 1 | no | — |
| 130 | C22×D4 | — | 94 | 1 | no | — |
| 131 | C22×D4 | — | 188 | 1 | no | — |
| 132 | C22×D4 | — | 188 | 1 | no | — |
| 133 | C22×D4 | — | 188 | 2 | no | — |
| 134 | C22×D4 | — | 188 | 2 | no | — |
| 135 | C23×C4 | C4×D47 | 1 | 1 | yes | characteristic |
| 136 | C2×M16 | — | 94 | 1 | no | — |
| 137 | C2×M16 | D188 | 1 | 1 | yes | characteristic |
| 138 | C2×M16 | C22⋊D47 | 1 | 1 | yes | characteristic |
| 139 | M32 | — | 4 | 1 | no | — |
| 140 | M32 | — | 188 | 1 | no | — |
| 141 | #100 | — | 188 | 2 | no | — |
| 142 | #100 | — | 188 | 2 | no | — |
| 143 | C22⋊C8 | — | 4 | 1 | no | — |
| 144 | C22⋊C8 | — | 188 | 1 | no | — |
| 145 | C24 | — | 94 | 2 | no | — |
| 146 | C24 | — | 376 | 4 | no | — |
| 147 | C24 | — | 188 | 2 | no | — |
| 148 | C24 | — | 188 | 4 | no | — |
| 149 | C24 | C8×D47 | 1 | 1 | yes | characteristic |
| 150 | C2×D4 | — | 94 | 2 | no | — |
| 151 | C2×D4 | — | 94 | 2 | no | — |
| 152 | C2×D4 | — | 188 | 1 | no | — |
| 153 | C2×D4 | — | 188 | 1 | no | — |
| 154 | C2×D4 | — | 188 | 2 | no | — |
| 155 | C2×D4 | — | 188 | 4 | no | — |
| 156 | C2×D4 | — | 188 | 8 | no | — |
| 157 | C2×D4 | — | 188 | 2 | no | — |
| 158 | C2×D4 | — | 188 | 2 | no | — |
| 159 | C2×D4 | — | 188 | 2 | no | — |
| 160 | C2×D4 | — | 188 | 2 | no | — |
| 161 | C2×D4 | — | 188 | 1 | no | — |
| 162 | C2×D4 | — | 188 | 1 | no | — |
| 163 | C22×C4 | — | 2 | 1 | no | — |
| 164 | C22×C4 | — | 2 | 1 | no | — |
| 165 | C22×C4 | — | 4 | 1 | no | — |
| 166 | C22×C4 | — | 4 | 1 | no | — |
| 167 | C22×C4 | #10625324 | 1 | 1 | yes | characteristic |
| 168 | C22×C4 | (C2×D47)⋊C4 | 1 | 1 | yes | characteristic |
| 169 | M16 | — | 4 | 1 | no | — |
| 170 | M16 | — | 4 | 1 | no | — |
| 171 | M16 | — | 188 | 1 | no | — |
| 172 | M16 | — | 188 | 1 | no | — |
| 173 | C2×C8 | — | 2 | 1 | no | — |
| 174 | C2×C8 | — | 2 | 1 | no | — |
| 175 | C2×C8 | — | 188 | 1 | no | — |
| 176 | C16 | — | 8 | 1 | no | — |
| 177 | C16 | — | 376 | 1 | no | — |
| 178 | C23 | — | 2 | 1 | no | — |
| 179 | C23 | — | 4 | 1 | no | — |
| 180 | C23 | — | 4 | 2 | no | — |
| 181 | C23 | — | 94 | 4 | no | — |
| 182 | C23 | — | 376 | 4 | no | — |
| 183 | C23 | — | 376 | 4 | no | — |
| 184 | C23 | — | 376 | 8 | no | — |
| 185 | C23 | — | 376 | 4 | no | — |
| 186 | C23 | — | 376 | 8 | no | — |
| 187 | C23 | — | 188 | 2 | no | — |
| 188 | C23 | — | 188 | 4 | no | — |
| 189 | C23 | — | 188 | 4 | no | — |
| 190 | C23 | — | 188 | 2 | no | — |
| 191 | C23 | (C2×D47)⋊C8 | 1 | 1 | yes | characteristic |
| 192 | D4 | — | 188 | 2 | no | — |
| 193 | D4 | — | 188 | 2 | no | — |
| 194 | D4 | — | 188 | 4 | no | — |
| 195 | D4 | — | 188 | 4 | no | — |
| 196 | D4 | — | 188 | 2 | no | — |
| 197 | D4 | — | 188 | 2 | no | — |
| 198 | C2×C4 | — | 2 | 1 | no | — |
| 199 | C2×C4 | — | 2 | 1 | no | — |
| 200 | C2×C4 | — | 2 | 1 | no | — |
| 201 | C2×C4 | — | 4 | 2 | no | — |
| 202 | C2×C4 | — | 4 | 1 | no | — |
| 203 | C2×C4 | — | 4 | 1 | no | — |
| 204 | C2×C4 | — | 4 | 1 | no | — |
| 205 | C2×C4 | #13093938 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | #13093940 | 1 | 1 | yes | characteristic |
| 207 | C8 | — | 4 | 1 | no | — |
| 208 | C8 | — | 4 | 1 | no | — |
| 209 | C8 | — | 376 | 1 | no | — |
| 210 | C22 | — | 2 | 2 | no | — |
| 211 | C22 | — | 2 | 1 | no | — |
| 212 | C22 | — | 4 | 1 | no | — |
| 213 | C22 | — | 4 | 1 | no | — |
| 214 | C22 | — | 4 | 2 | no | — |
| 215 | C22 | — | 4 | 2 | no | — |
| 216 | C22 | — | 4 | 1 | no | — |
| 217 | C22 | — | 376 | 4 | no | — |
| 218 | C22 | — | 376 | 4 | no | — |
| 219 | C22 | — | 376 | 8 | no | — |
| 220 | C22 | — | 376 | 8 | no | — |
| 221 | C22 | — | 376 | 4 | no | — |
| 222 | C22 | — | 188 | 4 | no | — |
| 223 | C22 | #15688846 | 1 | 1 | yes | characteristic |
| 224 | C4 | — | 2 | 1 | no | — |
| 225 | C4 | — | 4 | 1 | no | — |
| 226 | C4 | #15688909 | 1 | 1 | yes | characteristic |
| 227 | C4 | #15688911 | 1 | 1 | yes | characteristic |
| 228 | C2 | — | 2 | 1 | no | — |
| 229 | C2 | — | 4 | 2 | no | — |
| 230 | C2 | — | 4 | 1 | no | — |
| 231 | C2 | — | 376 | 4 | no | — |
| 232 | C2 | #20241315 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20241324 | C2 | semidirect |
| 3. C47×#1320 | C2 | semidirect |
| 114. #1320 | D47 | semidirect |
| 127. C47 | #7340 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20243308 | 2 |
| #20241324 | 2 |
| C47×#1320 | 2 |
| #7340 | 47 |