Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
822 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 822 conjugacy classes of subgroups — too many to draw. Only its 47 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 | C2×C9×C22⋊A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C9⋊#1020 | C2 | 1 | 2 | yes | maximal, direct product factor |
| 3 | C2×C3⋊#1020 | — | 3 | 1 | no | maximal |
| 4 | C2×#64188 | — | 3 | 2 | no | maximal |
| 5 | C2×#63515 | — | 3 | 1 | no | maximal |
| 6 | C9×C22⋊A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×C9⋊S4 | — | 4 | 3 | no | maximal |
| 8 | C2×C3×C22⋊A4 | D3 | 1 | 1 | yes | characteristic |
| 9 | C3⋊#1020 | — | 3 | 2 | no | — |
| 10 | C25×C9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C22⋊(C22⋊C9) | D3 | 1 | 2 | yes | complemented |
| 12 | #64188 | — | 3 | 4 | no | — |
| 13 | C22×#611 | — | 3 | 3 | no | — |
| 14 | #63515 | — | 3 | 6 | no | — |
| 15 | #63515 | — | 3 | 2 | no | — |
| 16 | C2×#3676 | — | 3 | 3 | no | — |
| 17 | C2×C9×A4 | — | 4 | 3 | no | — |
| 18 | C2×C9×A4 | — | 8 | 1 | no | — |
| 19 | C9⋊S4 | — | 4 | 6 | no | — |
| 20 | C2×#1020 | — | 9 | 3 | no | — |
| 21 | C2×#953 | — | 9 | 1 | no | — |
| 22 | C3×C22⋊A4 | C2×D3 | 1 | 1 | yes | characteristic |
| 23 | C2×C3⋊S4 | — | 12 | 3 | no | — |
| 24 | C24×C9 | — | 3 | 3 | no | — |
| 25 | C24×C9 | — | 3 | 3 | no | — |
| 26 | C24×C9 | — | 6 | 1 | no | — |
| 27 | C24×C9 | — | 6 | 1 | no | — |
| 28 | C24×C9 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 29 | C23×D9 | — | 3 | 1 | no | — |
| 30 | C22⋊(C22⋊C9) | C2×D3 | 1 | 2 | yes | complemented |
| 31 | C2×(C22⋊C9)⋊C2 | — | 12 | 6 | no | — |
| 32 | C2×#611 | — | 3 | 6 | no | — |
| 33 | C2×#611 | — | 3 | 6 | no | — |
| 34 | C2×#611 | — | 6 | 3 | no | — |
| 35 | C2×#611 | — | 6 | 3 | no | — |
| 36 | C2×#611 | — | 6 | 3 | no | — |
| 37 | C2×#611 | — | 6 | 3 | no | — |
| 38 | C22×Dic9 | — | 3 | 3 | no | — |
| 39 | #3676 | — | 3 | 6 | no | — |
| 40 | #3676 | — | 3 | 6 | no | — |
| 41 | C2×C3⋊D9 | — | 16 | 1 | no | — |
| 42 | C9×A4 | — | 4 | 3 | no | — |
| 43 | C9×A4 | — | 8 | 1 | no | — |
| 44 | C25×C3 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 45 | C2×C22⋊A4 | D9 | 1 | 3 | yes | complemented |
| 46 | #1020 | — | 9 | 6 | no | — |
| 47 | C22×#67 | — | 9 | 3 | no | — |
| 48 | #953 | — | 9 | 6 | no | — |
| 49 | #953 | — | 9 | 2 | no | — |
| 50 | C2×#217 | — | 9 | 3 | no | — |
| 51 | C2×C3×A4 | — | 4 | 3 | no | — |
| 52 | C2×C3×A4 | — | 8 | 1 | no | — |
| 53 | C3⋊S4 | — | 12 | 6 | no | — |
| 54 | C23×C9 | — | 2 | 1 | no | — |
| 55 | C23×C9 | — | 3 | 1 | no | — |
| 56 | C23×C9 | — | 3 | 3 | no | — |
| 57 | C23×C9 | — | 3 | 3 | no | — |
| 58 | C23×C9 | — | 3 | 3 | no | — |
| 59 | C23×C9 | — | 3 | 3 | no | — |
| 60 | C23×C9 | — | 3 | 3 | no | — |
| 61 | C23×C9 | — | 6 | 1 | no | — |
| 62 | C23×C9 | — | 6 | 1 | no | — |
| 63 | C23×C9 | — | 6 | 6 | no | — |
| 64 | C23×C9 | — | 6 | 3 | no | — |
| 65 | C23×C9 | — | 6 | 3 | no | — |
| 66 | C23×C9 | — | 6 | 3 | no | — |
| 67 | C23×C9 | S4 | 1 | 3 | yes | complemented |
| 68 | C22×D9 | — | 3 | 2 | no | — |
| 69 | C22×D9 | — | 6 | 3 | no | — |
| 70 | C22×D9 | — | 6 | 3 | no | — |
| 71 | C2×C22⋊C9 | — | 4 | 6 | no | — |
| 72 | C2×C22⋊C9 | — | 8 | 2 | no | — |
| 73 | (C22⋊C9)⋊C2 | — | 12 | 12 | no | — |
| 74 | #611 | — | 6 | 6 | no | — |
| 75 | #611 | — | 6 | 6 | no | — |
| 76 | #611 | — | 6 | 6 | no | — |
| 77 | #611 | — | 6 | 6 | no | — |
| 78 | C2×Dic9 | — | 3 | 6 | no | — |
| 79 | C2×Dic9 | — | 6 | 3 | no | — |
| 80 | C2×Dic9 | — | 6 | 3 | no | — |
| 81 | C2×#120 | — | 27 | 1 | no | Sylow |
| 82 | C2×C3×C9 | — | 16 | 1 | no | — |
| 83 | C3⋊D9 | — | 16 | 2 | no | — |
| 84 | C24×C3 | — | 3 | 3 | no | — |
| 85 | C24×C3 | — | 3 | 3 | no | — |
| 86 | C24×C3 | — | 6 | 1 | no | — |
| 87 | C24×C3 | — | 6 | 1 | no | — |
| 88 | C24×C3 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 89 | C23×D3 | — | 9 | 1 | no | — |
| 90 | C22⋊A4 | C2×D9 | 1 | 3 | yes | complemented |
| 91 | C2×S4 | — | 36 | 9 | no | — |
| 92 | C2×#67 | — | 9 | 6 | no | — |
| 93 | C2×#67 | — | 9 | 6 | no | — |
| 94 | C2×#67 | — | 18 | 3 | no | — |
| 95 | C2×#67 | — | 18 | 3 | no | — |
| 96 | C2×#67 | — | 18 | 3 | no | — |
| 97 | C2×#67 | — | 18 | 3 | no | — |
| 98 | C22×Dic3 | — | 9 | 3 | no | — |
| 99 | #217 | — | 9 | 6 | no | — |
| 100 | #217 | — | 9 | 6 | no | — |
| 101 | C2×C3⋊D3 | — | 48 | 1 | no | — |
| 102 | C3×A4 | — | 4 | 3 | no | — |
| 103 | C3×A4 | — | 8 | 1 | no | — |
| 104 | C22×C9 | — | 2 | 1 | no | — |
| 105 | C22×C9 | — | 3 | 3 | no | — |
| 106 | C22×C9 | — | 3 | 1 | no | — |
| 107 | C22×C9 | — | 3 | 3 | no | — |
| 108 | C22×C9 | — | 3 | 3 | no | — |
| 109 | C22×C9 | — | 3 | 3 | no | — |
| 110 | C22×C9 | — | 3 | 3 | no | — |
| 111 | C22×C9 | — | 6 | 3 | no | — |
| 112 | C22×C9 | — | 6 | 6 | no | — |
| 113 | C22×C9 | — | 6 | 3 | no | — |
| 114 | C22×C9 | — | 6 | 3 | no | — |
| 115 | C22×C9 | — | 6 | 1 | no | — |
| 116 | C22×C9 | — | 6 | 1 | no | — |
| 117 | C22×C9 | C2×S4 | 1 | 3 | yes | complemented |
| 118 | C2×D9 | — | 6 | 6 | no | — |
| 119 | C2×D9 | — | 12 | 3 | no | — |
| 120 | C2×D9 | — | 12 | 1 | no | — |
| 121 | C2×D9 | — | 48 | 2 | no | — |
| 122 | C22⋊C9 | — | 4 | 6 | no | — |
| 123 | C22⋊C9 | — | 8 | 2 | no | — |
| 124 | Dic9 | — | 6 | 6 | no | — |
| 125 | C25 | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 126 | C22×D4 | — | 27 | 3 | no | — |
| 127 | #120 | — | 27 | 6 | no | — |
| 128 | #120 | — | 27 | 2 | no | — |
| 129 | C2×C22⋊C4 | — | 27 | 3 | no | — |
| 130 | C3×C9 | — | 16 | 1 | no | Sylow |
| 131 | C23×C3 | — | 2 | 1 | no | — |
| 132 | C23×C3 | — | 3 | 1 | no | — |
| 133 | C23×C3 | — | 3 | 3 | no | — |
| 134 | C23×C3 | — | 3 | 3 | no | — |
| 135 | C23×C3 | — | 3 | 3 | no | — |
| 136 | C23×C3 | — | 3 | 3 | no | — |
| 137 | C23×C3 | — | 3 | 3 | no | — |
| 138 | C23×C3 | — | 6 | 1 | no | — |
| 139 | C23×C3 | — | 6 | 1 | no | — |
| 140 | C23×C3 | — | 6 | 6 | no | — |
| 141 | C23×C3 | — | 6 | 3 | no | — |
| 142 | C23×C3 | — | 6 | 3 | no | — |
| 143 | C23×C3 | — | 6 | 3 | no | — |
| 144 | C23×C3 | C3⋊S4 | 1 | 3 | yes | — |
| 145 | C22×D3 | — | 9 | 2 | no | — |
| 146 | C22×D3 | — | 18 | 3 | no | — |
| 147 | C22×D3 | — | 18 | 3 | no | — |
| 148 | C2×A4 | — | 4 | 9 | no | — |
| 149 | C2×A4 | — | 8 | 3 | no | — |
| 150 | S4 | — | 36 | 18 | no | — |
| 151 | #67 | — | 18 | 6 | no | — |
| 152 | #67 | — | 18 | 6 | no | — |
| 153 | #67 | — | 18 | 6 | no | — |
| 154 | #67 | — | 18 | 6 | no | — |
| 155 | C2×Dic3 | — | 9 | 6 | no | — |
| 156 | C2×Dic3 | — | 18 | 3 | no | — |
| 157 | C2×Dic3 | — | 18 | 3 | no | — |
| 158 | C2×C32 | — | 16 | 1 | no | — |
| 159 | C3⋊D3 | — | 48 | 2 | no | — |
| 160 | C2×C9 | — | 3 | 3 | no | — |
| 161 | C2×C9 | — | 3 | 3 | no | — |
| 162 | C2×C9 | — | 6 | 1 | no | — |
| 163 | C2×C9 | — | 6 | 1 | no | — |
| 164 | C2×C9 | — | 16 | 2 | no | — |
| 165 | C2×C9 | #1020 | 1 | 1 | yes | characteristic, complemented |
| 166 | D9 | — | 12 | 2 | no | — |
| 167 | D9 | — | 48 | 4 | no | — |
| 168 | C24 | — | 3 | 3 | no | — |
| 169 | C24 | — | 3 | 3 | no | — |
| 170 | C24 | — | 6 | 1 | no | — |
| 171 | C24 | — | 6 | 1 | no | — |
| 172 | C24 | — | 27 | 1 | no | — |
| 173 | C24 | C2×C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 174 | C2×D4 | — | 27 | 6 | no | — |
| 175 | C2×D4 | — | 27 | 6 | no | — |
| 176 | C2×D4 | — | 54 | 3 | no | — |
| 177 | C2×D4 | — | 54 | 3 | no | — |
| 178 | C2×D4 | — | 54 | 3 | no | — |
| 179 | C2×D4 | — | 54 | 3 | no | — |
| 180 | C22×C4 | — | 27 | 3 | no | — |
| 181 | C22⋊C4 | — | 27 | 6 | no | — |
| 182 | C22⋊C4 | — | 27 | 6 | no | — |
| 183 | C22×C3 | — | 2 | 1 | no | — |
| 184 | C22×C3 | — | 3 | 3 | no | — |
| 185 | C22×C3 | — | 3 | 1 | no | — |
| 186 | C22×C3 | — | 3 | 3 | no | — |
| 187 | C22×C3 | — | 3 | 3 | no | — |
| 188 | C22×C3 | — | 3 | 3 | no | — |
| 189 | C22×C3 | — | 3 | 3 | no | — |
| 190 | C22×C3 | — | 6 | 3 | no | — |
| 191 | C22×C3 | — | 6 | 6 | no | — |
| 192 | C22×C3 | — | 6 | 3 | no | — |
| 193 | C22×C3 | — | 6 | 3 | no | — |
| 194 | C22×C3 | — | 6 | 1 | no | — |
| 195 | C22×C3 | — | 6 | 1 | no | — |
| 196 | C22×C3 | C2×C3⋊S4 | 1 | 3 | yes | — |
| 197 | C2×D3 | — | 18 | 6 | no | — |
| 198 | C2×D3 | — | 36 | 3 | no | — |
| 199 | C2×D3 | — | 36 | 1 | no | — |
| 200 | C2×D3 | — | 144 | 3 | no | — |
| 201 | A4 | — | 4 | 9 | no | — |
| 202 | A4 | — | 8 | 3 | no | — |
| 203 | Dic3 | — | 18 | 6 | no | — |
| 204 | C32 | — | 16 | 1 | no | — |
| 205 | C9 | — | 16 | 2 | no | — |
| 206 | C9 | C2×#1020 | 1 | 1 | yes | characteristic, complemented |
| 207 | C23 | — | 2 | 1 | no | — |
| 208 | C23 | — | 3 | 1 | no | — |
| 209 | C23 | — | 3 | 3 | no | — |
| 210 | C23 | — | 3 | 3 | no | — |
| 211 | C23 | — | 3 | 3 | no | — |
| 212 | C23 | — | 3 | 3 | no | — |
| 213 | C23 | — | 3 | 3 | no | — |
| 214 | C23 | — | 6 | 1 | no | — |
| 215 | C23 | — | 6 | 1 | no | — |
| 216 | C23 | — | 6 | 6 | no | — |
| 217 | C23 | — | 6 | 3 | no | — |
| 218 | C23 | — | 6 | 3 | no | — |
| 219 | C23 | — | 6 | 3 | no | — |
| 220 | C23 | — | 27 | 2 | no | — |
| 221 | C23 | — | 54 | 3 | no | — |
| 222 | C23 | — | 54 | 3 | no | — |
| 223 | C23 | C9⋊S4 | 1 | 3 | yes | complemented |
| 224 | D4 | — | 54 | 6 | no | — |
| 225 | D4 | — | 54 | 6 | no | — |
| 226 | D4 | — | 54 | 6 | no | — |
| 227 | D4 | — | 54 | 6 | no | — |
| 228 | C2×C4 | — | 27 | 6 | no | — |
| 229 | C2×C4 | — | 54 | 3 | no | — |
| 230 | C2×C4 | — | 54 | 3 | no | — |
| 231 | C2×C3 | — | 3 | 3 | no | — |
| 232 | C2×C3 | — | 3 | 3 | no | — |
| 233 | C2×C3 | — | 6 | 1 | no | — |
| 234 | C2×C3 | — | 6 | 1 | no | — |
| 235 | C2×C3 | — | 16 | 3 | no | — |
| 236 | C2×C3 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 237 | D3 | — | 36 | 2 | no | — |
| 238 | D3 | — | 144 | 6 | no | — |
| 239 | C22 | — | 2 | 1 | no | — |
| 240 | C22 | — | 3 | 3 | no | — |
| 241 | C22 | — | 3 | 1 | no | — |
| 242 | C22 | — | 3 | 3 | no | — |
| 243 | C22 | — | 3 | 3 | no | — |
| 244 | C22 | — | 3 | 3 | no | — |
| 245 | C22 | — | 3 | 3 | no | — |
| 246 | C22 | — | 6 | 3 | no | — |
| 247 | C22 | — | 6 | 6 | no | — |
| 248 | C22 | — | 6 | 3 | no | — |
| 249 | C22 | — | 6 | 3 | no | — |
| 250 | C22 | — | 6 | 1 | no | — |
| 251 | C22 | — | 6 | 1 | no | — |
| 252 | C22 | — | 54 | 6 | no | — |
| 253 | C22 | — | 108 | 3 | no | — |
| 254 | C22 | — | 108 | 1 | no | — |
| 255 | C22 | C2×C9⋊S4 | 1 | 3 | yes | complemented |
| 256 | C4 | — | 54 | 6 | no | — |
| 257 | C3 | — | 16 | 3 | no | — |
| 258 | C3 | C2×C3⋊#1020 | 1 | 1 | yes | characteristic |
| 259 | C2 | — | 3 | 3 | no | — |
| 260 | C2 | — | 3 | 3 | no | — |
| 261 | C2 | — | 6 | 1 | no | — |
| 262 | C2 | — | 6 | 1 | no | — |
| 263 | C2 | — | 108 | 2 | no | — |
| 264 | C2 | C9⋊#1020 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊#1020 |
| commutator, cosocle, hypocenter | C9×C22⋊A4 | C22 |
| Frattini | C3 | C2×C3⋊#1020 |
| Fitting | C25×C9 | D3 |
| socle | C25×C3 | C3⋊D3 |
| LCS | G ▷ C9×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C9×C22⋊A4 | C2 | semidirect |
| 2. C9⋊#1020 | C2 | direct (N × H) |
| 6. C9×C22⋊A4 | C22 | semidirect |
| 10. C25×C9 | D3 | semidirect |
| 11. C2×C22⋊(C22⋊C9) | D3 | semidirect |
| 28. C24×C9 | C2×D3 | semidirect |
| 30. C22⋊(C22⋊C9) | C2×D3 | semidirect |
| 45. C2×C22⋊A4 | D9 | semidirect |
| 67. C23×C9 | S4 | semidirect |
| 90. C22⋊A4 | C2×D9 | semidirect |
| 117. C22×C9 | C2×S4 | semidirect |
| 125. C25 | C3⋊D9 | semidirect |
| 165. C2×C9 | #1020 | semidirect |
| 173. C24 | C2×C3⋊D9 | semidirect |
| 206. C9 | C2×#1020 | semidirect |
| 223. C23 | C9⋊S4 | semidirect |
| 255. C22 | C2×C9⋊S4 | semidirect |
| 264. C2 | C9⋊#1020 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×C22⋊A4 | 2 |
| C9⋊#1020 | 2 |
| C2×C3⋊#1020 | 3 |
| C2×#64188 | 3 |
| C2×#63515 | 3 |
| C2×C9⋊S4 | 4 |