← #13136483
Subgroups — #13136483
Order 1728.
516 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#88399 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×M16×C27 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#88363 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11720156 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | M16×D27 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #11720151 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10589847 | — | 3 | 1 | no | maximal |
| 8 | C22×C4×C27 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6171 | C4 | 1 | 2 | yes | — |
| 10 | #88399 | C4 | 1 | 2 | yes | — |
| 11 | #88399 | C4 | 1 | 2 | yes | — |
| 12 | #88399 | C22 | 1 | 2 | yes | — |
| 13 | #88399 | C22 | 1 | 2 | yes | — |
| 14 | C2×D108 | C4 | 1 | 1 | yes | characteristic |
| 15 | C2×C4×D27 | C22 | 1 | 2 | yes | — |
| 16 | C2×Dic54 | C4 | 1 | 1 | yes | characteristic |
| 17 | M16×C27 | C22 | 1 | 2 | yes | — |
| 18 | M16×C27 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×C8×C27 | C22 | 1 | 2 | yes | complemented |
| 20 | #88363 | C22 | 1 | 2 | yes | — |
| 21 | #88363 | C22 | 1 | 2 | yes | complemented |
| 22 | C2×C27⋊C8 | C22 | 1 | 2 | yes | complemented |
| 23 | #88359 | C22 | 1 | 8 | yes | complemented |
| 24 | C8×D27 | C22 | 1 | 8 | yes | complemented |
| 25 | C2×#3697 | — | 3 | 1 | no | — |
| 26 | C2×C9×M16 | D3 | 1 | 1 | yes | characteristic |
| 27 | C2×#3667 | — | 3 | 1 | no | — |
| 28 | #63469 | — | 3 | 4 | no | — |
| 29 | M16×D9 | — | 3 | 4 | no | — |
| 30 | #63464 | — | 3 | 4 | no | — |
| 31 | C23×C27 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C22×D27 | C2×C4 | 1 | 2 | yes | — |
| 33 | C2×C4×C27 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×C4×C27 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | C2×C4×C27 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 38 | C2×C4×C27 | C23 | 1 | 1 | yes | characteristic |
| 39 | #6171 | C2×C4 | 1 | 4 | yes | — |
| 40 | #6171 | C2×C4 | 1 | 4 | yes | — |
| 41 | C2×Dic27 | C2×C4 | 1 | 2 | yes | — |
| 42 | D108 | C2×C4 | 1 | 2 | yes | — |
| 43 | D108 | C2×C4 | 1 | 2 | yes | — |
| 44 | C4×D27 | C2×C4 | 1 | 4 | yes | — |
| 45 | C4×D27 | C23 | 1 | 4 | yes | — |
| 46 | Dic54 | C2×C4 | 1 | 2 | yes | — |
| 47 | Dic54 | C2×C4 | 1 | 2 | yes | — |
| 48 | C8×C27 | C23 | 1 | 4 | yes | — |
| 49 | C27⋊C8 | C23 | 1 | 4 | yes | — |
| 50 | #5745 | — | 9 | 1 | no | — |
| 51 | C22×C4×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 52 | C2×#611 | — | 3 | 2 | no | — |
| 53 | #3697 | — | 3 | 2 | no | — |
| 54 | #3697 | — | 3 | 2 | no | — |
| 55 | #3697 | — | 3 | 2 | no | — |
| 56 | #3697 | — | 3 | 2 | no | — |
| 57 | C2×D36 | — | 3 | 1 | no | — |
| 58 | C2×C4×D9 | — | 3 | 2 | no | — |
| 59 | C2×Dic18 | — | 3 | 1 | no | — |
| 60 | C9×M16 | C2×D3 | 1 | 2 | yes | — |
| 61 | C9×M16 | C2×D3 | 1 | 2 | yes | — |
| 62 | C2×C8×C9 | C2×D3 | 1 | 2 | yes | — |
| 63 | #3667 | — | 3 | 2 | no | — |
| 64 | #3667 | — | 3 | 2 | no | — |
| 65 | C2×C9⋊C8 | — | 3 | 2 | no | — |
| 66 | #3663 | — | 3 | 8 | no | — |
| 67 | C8×D9 | — | 3 | 8 | no | — |
| 68 | C22×C27 | — | 4 | 1 | no | — |
| 69 | C22×C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | C22×C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 71 | C22×C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 72 | C2×D27 | — | 4 | 2 | no | — |
| 73 | C2×D27 | C22×C4 | 1 | 4 | yes | — |
| 74 | C4×C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 75 | C4×C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 76 | C4×C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C4×C27 | C24 | 1 | 1 | yes | characteristic |
| 78 | Dic27 | C22×C4 | 1 | 4 | yes | — |
| 79 | C2×#235 | — | 9 | 1 | no | — |
| 80 | C2×C3×M16 | D9 | 1 | 1 | yes | characteristic |
| 81 | C2×#208 | — | 9 | 1 | no | — |
| 82 | #907 | — | 9 | 4 | no | — |
| 83 | D3×M16 | — | 9 | 4 | no | — |
| 84 | #901 | — | 9 | 4 | no | — |
| 85 | C23×C9 | C4×D3 | 1 | 1 | yes | characteristic |
| 86 | C22×D9 | — | 3 | 2 | no | — |
| 87 | C2×C4×C9 | C4×D3 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C9 | C4×D3 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C9 | C4×D3 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 93 | #611 | — | 3 | 4 | no | — |
| 94 | #611 | — | 3 | 4 | no | — |
| 95 | C2×Dic9 | — | 3 | 2 | no | — |
| 96 | D36 | — | 3 | 2 | no | — |
| 97 | D36 | — | 3 | 2 | no | — |
| 98 | C4×D9 | — | 3 | 4 | no | — |
| 99 | C4×D9 | — | 3 | 4 | no | — |
| 100 | Dic18 | — | 3 | 2 | no | — |
| 101 | Dic18 | — | 3 | 2 | no | — |
| 102 | C8×C9 | C22×D3 | 1 | 4 | yes | — |
| 103 | C9⋊C8 | — | 3 | 4 | no | — |
| 104 | #568 | — | 27 | 1 | no | Sylow |
| 105 | C2×C27 | — | 2 | 1 | no | — |
| 106 | C2×C27 | — | 2 | 1 | no | — |
| 107 | C2×C27 | — | 2 | 1 | no | — |
| 108 | C2×C27 | C23×C4 | 1 | 1 | yes | characteristic |
| 109 | D27 | — | 2 | 4 | no | — |
| 110 | C22×C3×C4 | C2×D9 | 1 | 1 | yes | characteristic |
| 111 | C2×#67 | — | 9 | 2 | no | — |
| 112 | #235 | — | 9 | 2 | no | — |
| 113 | #235 | — | 9 | 2 | no | — |
| 114 | #235 | — | 9 | 2 | no | — |
| 115 | #235 | — | 9 | 2 | no | — |
| 116 | C2×D12 | — | 9 | 1 | no | — |
| 117 | C2×C4×D3 | — | 9 | 2 | no | — |
| 118 | C2×Dic6 | — | 9 | 1 | no | — |
| 119 | C3×M16 | C2×D9 | 1 | 2 | yes | — |
| 120 | C3×M16 | C2×D9 | 1 | 2 | yes | — |
| 121 | C2×C3×C8 | C2×D9 | 1 | 2 | yes | — |
| 122 | #208 | — | 9 | 2 | no | — |
| 123 | #208 | — | 9 | 2 | no | — |
| 124 | C2×C3⋊C8 | — | 9 | 2 | no | — |
| 125 | ΓL(1,𝔽25) | — | 9 | 8 | no | — |
| 126 | D3×C8 | — | 9 | 8 | no | — |
| 127 | C22×C9 | — | 4 | 1 | no | — |
| 128 | C22×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 129 | C22×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 130 | C22×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 131 | C2×D9 | — | 3 | 4 | no | — |
| 132 | C2×D9 | — | 12 | 2 | no | — |
| 133 | C4×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 134 | C4×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 135 | C4×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 136 | C4×C9 | C23×D3 | 1 | 1 | yes | characteristic |
| 137 | Dic9 | — | 3 | 4 | no | — |
| 138 | C2×#41 | — | 27 | 1 | no | — |
| 139 | #131 | — | 27 | 4 | no | — |
| 140 | #131 | — | 27 | 4 | no | — |
| 141 | C2×M16 | — | 27 | 1 | no | — |
| 142 | C2×M16 | — | 27 | 4 | no | — |
| 143 | C2×M16 | D27 | 1 | 1 | yes | characteristic, complemented |
| 144 | C27 | #568 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | C23×C3 | C4×D9 | 1 | 1 | yes | characteristic |
| 146 | C22×D3 | — | 9 | 2 | no | — |
| 147 | C2×C3×C4 | C4×D9 | 1 | 1 | yes | characteristic |
| 148 | C2×C3×C4 | C4×D9 | 1 | 1 | yes | characteristic |
| 149 | C2×C3×C4 | C4×D9 | 1 | 1 | yes | characteristic |
| 150 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 151 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 152 | C2×C3×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 153 | #67 | — | 9 | 4 | no | — |
| 154 | #67 | — | 9 | 4 | no | — |
| 155 | C2×Dic3 | — | 9 | 2 | no | — |
| 156 | D12 | — | 9 | 2 | no | — |
| 157 | D12 | — | 9 | 2 | no | — |
| 158 | C4×D3 | — | 9 | 4 | no | — |
| 159 | C4×D3 | — | 9 | 4 | no | — |
| 160 | Dic6 | — | 9 | 2 | no | — |
| 161 | Dic6 | — | 9 | 2 | no | — |
| 162 | C3×C8 | C22×D9 | 1 | 4 | yes | — |
| 163 | C3⋊C8 | — | 9 | 4 | no | — |
| 164 | C2×C9 | — | 2 | 1 | no | — |
| 165 | C2×C9 | — | 2 | 1 | no | — |
| 166 | C2×C9 | — | 2 | 1 | no | — |
| 167 | C2×C9 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 168 | D9 | — | 6 | 4 | no | — |
| 169 | #41 | — | 27 | 2 | no | — |
| 170 | #41 | — | 27 | 2 | no | — |
| 171 | #41 | — | 27 | 2 | no | — |
| 172 | #41 | — | 27 | 2 | no | — |
| 173 | C2×Q8 | — | 27 | 1 | no | — |
| 174 | C2×D4 | — | 27 | 2 | no | — |
| 175 | C2×D4 | — | 27 | 1 | no | — |
| 176 | C22×C4 | — | 27 | 2 | no | — |
| 177 | C22×C4 | C2×D27 | 1 | 1 | yes | characteristic |
| 178 | M16 | — | 27 | 2 | no | — |
| 179 | M16 | — | 27 | 2 | no | — |
| 180 | M16 | — | 27 | 8 | no | — |
| 181 | M16 | C2×D27 | 1 | 2 | yes | — |
| 182 | M16 | C2×D27 | 1 | 2 | yes | complemented |
| 183 | C2×C8 | — | 27 | 2 | no | — |
| 184 | C2×C8 | — | 27 | 8 | no | — |
| 185 | C2×C8 | C2×D27 | 1 | 2 | yes | complemented |
| 186 | C22×C3 | — | 4 | 1 | no | — |
| 187 | C22×C3 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 188 | C22×C3 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 189 | C22×C3 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 190 | C2×D3 | — | 9 | 4 | no | — |
| 191 | C2×D3 | — | 36 | 2 | no | — |
| 192 | C3×C4 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 193 | C3×C4 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 194 | C3×C4 | C2×C4×D9 | 1 | 1 | yes | characteristic |
| 195 | C3×C4 | C23×D9 | 1 | 1 | yes | characteristic |
| 196 | Dic3 | — | 9 | 4 | no | — |
| 197 | C9 | #5745 | 1 | 1 | yes | characteristic |
| 198 | C23 | — | 27 | 2 | no | — |
| 199 | C23 | C4×D27 | 1 | 1 | yes | characteristic |
| 200 | Q8 | — | 27 | 2 | no | — |
| 201 | Q8 | — | 27 | 2 | no | — |
| 202 | D4 | — | 27 | 4 | no | — |
| 203 | D4 | — | 27 | 4 | no | — |
| 204 | D4 | — | 27 | 2 | no | — |
| 205 | D4 | — | 27 | 2 | no | — |
| 206 | C2×C4 | — | 27 | 2 | no | — |
| 207 | C2×C4 | — | 27 | 4 | no | — |
| 208 | C2×C4 | — | 27 | 4 | no | — |
| 209 | C2×C4 | C4×D27 | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | C4×D27 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | C4×D27 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | C22×D27 | 1 | 1 | yes | characteristic |
| 215 | C8 | — | 27 | 4 | no | — |
| 216 | C8 | C22×D27 | 1 | 4 | yes | — |
| 217 | C2×C3 | — | 2 | 1 | no | — |
| 218 | C2×C3 | — | 2 | 1 | no | — |
| 219 | C2×C3 | — | 2 | 1 | no | — |
| 220 | C2×C3 | C22×C4×D9 | 1 | 1 | yes | characteristic |
| 221 | D3 | — | 18 | 4 | no | — |
| 222 | C22 | — | 4 | 1 | no | — |
| 223 | C22 | — | 27 | 4 | no | — |
| 224 | C22 | — | 108 | 2 | no | — |
| 225 | C22 | C2×C4×D27 | 1 | 1 | yes | characteristic |
| 226 | C22 | C2×C4×D27 | 1 | 1 | yes | characteristic |
| 227 | C22 | C2×C4×D27 | 1 | 1 | yes | characteristic |
| 228 | C4 | — | 27 | 4 | no | — |
| 229 | C4 | C2×C4×D27 | 1 | 1 | yes | characteristic |
| 230 | C4 | C2×C4×D27 | 1 | 1 | yes | characteristic |
| 231 | C4 | C2×C4×D27 | 1 | 1 | yes | characteristic |
| 232 | C4 | C23×D27 | 1 | 1 | yes | characteristic |
| 233 | C3 | #10589847 | 1 | 1 | yes | characteristic |
| 234 | C2 | — | 2 | 1 | no | — |
| 235 | C2 | — | 2 | 1 | no | — |
| 236 | C2 | — | 2 | 1 | no | — |
| 237 | C2 | — | 54 | 4 | no | — |
| 238 | C2 | C22×C4×D27 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta