Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
498 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 498 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 | #11722410 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C9⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9×#983 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D9×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #11722235 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11722234 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #11722232 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #10596446 | — | 3 | 1 | no | maximal |
| 9 | #10589880 | — | 3 | 1 | no | maximal |
| 10 | C2×C9×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #88668 | — | 4 | 1 | no | maximal |
| 12 | D9×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | #88615 | C22 | 1 | 1 | yes | characteristic, complemented |
| 14 | C9⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C9⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | C9×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C9×#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #1007⋊C3 | — | 3 | 1 | no | — |
| 19 | C2×C3⋊GL(2,𝔽3) | — | 3 | 1 | no | — |
| 20 | C3×#983 | D3 | 1 | 1 | yes | characteristic |
| 21 | D3×GL(2,𝔽3) | — | 3 | 1 | no | — |
| 22 | #64202 | — | 3 | 1 | no | — |
| 23 | #64199 | — | 3 | 1 | no | — |
| 24 | #64198 | — | 3 | 1 | no | — |
| 25 | #63715 | — | 3 | 1 | no | — |
| 26 | #63713 | D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C2×#3689 | — | 6 | 1 | no | — |
| 28 | C9×#137 | — | 3 | 1 | no | — |
| 29 | #63513 | — | 3 | 1 | no | — |
| 30 | C2×#3675 | — | 3 | 1 | no | — |
| 31 | #63481 | — | 3 | 1 | no | — |
| 32 | #63481 | — | 3 | 1 | no | — |
| 33 | #63480 | — | 3 | 1 | no | — |
| 34 | #63480 | — | 3 | 1 | no | — |
| 35 | #63476 | — | 3 | 1 | no | — |
| 36 | #63476 | — | 3 | 1 | no | — |
| 37 | SD16×D9 | — | 3 | 1 | no | — |
| 38 | SD16×D9 | — | 3 | 1 | no | — |
| 39 | #63470 | — | 3 | 1 | no | — |
| 40 | #63469 | — | 3 | 1 | no | — |
| 41 | C22×C3⋊D9 | — | 4 | 1 | no | — |
| 42 | C2×D3×D9 | — | 4 | 1 | no | — |
| 43 | C9×#67 | — | 4 | 1 | no | — |
| 44 | C3×#611 | — | 4 | 1 | no | — |
| 45 | C9×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 46 | #6195 | — | 4 | 1 | no | — |
| 47 | #6192 | — | 4 | 1 | no | — |
| 48 | (C3⋊D9)⋊C4 | — | 4 | 1 | no | — |
| 49 | #5926 | — | 9 | 1 | no | — |
| 50 | #5778 | — | 9 | 1 | no | — |
| 51 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 52 | #3811 | — | 12 | 1 | no | — |
| 53 | D3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 54 | C3⋊(#41⋊C3) | — | 3 | 1 | no | — |
| 55 | C3⋊GL(2,𝔽3) | — | 3 | 1 | no | — |
| 56 | C3⋊GL(2,𝔽3) | — | 3 | 1 | no | — |
| 57 | C3×GL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | C3×#226 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | C9×#41 | — | 3 | 1 | no | — |
| 60 | C2×Q8×C9 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 61 | #3701 | — | 3 | 1 | no | — |
| 62 | #3701 | — | 3 | 1 | no | — |
| 63 | #3701 | — | 3 | 1 | no | — |
| 64 | #3701 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 65 | Q8×D9 | — | 3 | 1 | no | — |
| 66 | Q8×D9 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 67 | D4×D9 | — | 3 | 1 | no | — |
| 68 | D4×D9 | — | 3 | 1 | no | — |
| 69 | D4×D9 | — | 6 | 1 | no | — |
| 70 | D4×D9 | — | 6 | 1 | no | — |
| 71 | #3697 | — | 3 | 1 | no | — |
| 72 | #3697 | — | 3 | 1 | no | — |
| 73 | #3697 | — | 6 | 1 | no | — |
| 74 | C2×D36 | — | 3 | 1 | no | — |
| 75 | C2×D36 | — | 6 | 1 | no | — |
| 76 | C2×Q8⋊C9 | — | 2 | 1 | no | — |
| 77 | #3689 | — | 6 | 1 | no | — |
| 78 | #3689 | — | 6 | 1 | no | — |
| 79 | C9×Q16 | — | 3 | 1 | no | — |
| 80 | C9×Q16 | — | 3 | 1 | no | — |
| 81 | C9×SD16 | — | 3 | 1 | no | — |
| 82 | C9×SD16 | — | 3 | 1 | no | — |
| 83 | C9×M16 | — | 3 | 1 | no | — |
| 84 | #3675 | — | 3 | 1 | no | — |
| 85 | #3675 | — | 3 | 1 | no | — |
| 86 | #3675 | — | 3 | 1 | no | — |
| 87 | #3675 | — | 3 | 1 | no | — |
| 88 | #3675 | — | 3 | 1 | no | — |
| 89 | #3674 | — | 3 | 1 | no | — |
| 90 | #3673 | — | 3 | 1 | no | — |
| 91 | #3672 | — | 3 | 1 | no | — |
| 92 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 93 | D72 | — | 3 | 1 | no | — |
| 94 | D72 | — | 3 | 1 | no | — |
| 95 | #3664 | — | 3 | 1 | no | — |
| 96 | #3664 | — | 3 | 1 | no | — |
| 97 | #3663 | — | 3 | 1 | no | — |
| 98 | #3663 | — | 3 | 1 | no | — |
| 99 | C8×D9 | — | 3 | 1 | no | — |
| 100 | C8×D9 | — | 3 | 1 | no | — |
| 101 | C22×C3×C9 | — | 4 | 1 | no | — |
| 102 | C2×C3⋊D9 | — | 4 | 1 | no | — |
| 103 | C2×C3⋊D9 | — | 4 | 1 | no | — |
| 104 | C2×C3⋊D9 | — | 8 | 2 | no | — |
| 105 | C2×D3×C9 | — | 4 | 1 | no | — |
| 106 | C2×C3×D9 | — | 4 | 1 | no | — |
| 107 | D3×D9 | — | 8 | 2 | no | — |
| 108 | C9×Dic3 | — | 4 | 1 | no | — |
| 109 | C3×Dic9 | — | 4 | 1 | no | — |
| 110 | #1009 | — | 9 | 1 | no | — |
| 111 | #1007 | — | 3 | 1 | no | — |
| 112 | 2−1+4⋊C3 | — | 9 | 1 | no | — |
| 113 | #986 | — | 9 | 1 | no | — |
| 114 | #985 | — | 9 | 1 | no | — |
| 115 | #983 | — | 9 | 1 | no | — |
| 116 | #983 | D9 | 1 | 1 | yes | characteristic, complemented |
| 117 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 118 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 119 | C2×GL(2,𝔽3) | — | 18 | 1 | no | — |
| 120 | C3×#137 | — | 3 | 1 | no | — |
| 121 | #950 | — | 9 | 1 | no | — |
| 122 | C2×#215 | — | 9 | 1 | no | — |
| 123 | #919 | — | 9 | 1 | no | — |
| 124 | #919 | — | 9 | 1 | no | — |
| 125 | #918 | — | 9 | 1 | no | — |
| 126 | #918 | — | 9 | 1 | no | — |
| 127 | #914 | — | 9 | 1 | no | — |
| 128 | #914 | — | 9 | 1 | no | — |
| 129 | D3×SD16 | — | 9 | 1 | no | — |
| 130 | D3×SD16 | — | 9 | 1 | no | — |
| 131 | #908 | — | 9 | 1 | no | — |
| 132 | #907 | — | 9 | 1 | no | — |
| 133 | C22×C3⋊D3 | — | 12 | 1 | no | — |
| 134 | C2×D32 | — | 12 | 1 | no | — |
| 135 | C3×#67 | — | 4 | 1 | no | — |
| 136 | C3×#67 | — | 12 | 1 | no | — |
| 137 | C3×SL(2,𝔽3) | C22×D3 | 1 | 1 | yes | characteristic |
| 138 | #626 | — | 12 | 1 | no | — |
| 139 | #626 | — | 12 | 1 | no | — |
| 140 | #624 | — | 12 | 1 | no | — |
| 141 | C22×D9 | — | 6 | 1 | no | — |
| 142 | C22×D9 | — | 6 | 1 | no | — |
| 143 | C22×D9 | — | 6 | 1 | no | — |
| 144 | C22×D9 | — | 24 | 1 | no | — |
| 145 | Q8×C9 | — | 3 | 1 | no | — |
| 146 | Q8×C9 | — | 3 | 1 | no | — |
| 147 | Q8×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 148 | D4×C9 | — | 3 | 1 | no | — |
| 149 | D4×C9 | — | 6 | 1 | no | — |
| 150 | C2×C4×C9 | — | 3 | 1 | no | — |
| 151 | C2×C4×C9 | — | 6 | 1 | no | — |
| 152 | #611 | — | 3 | 1 | no | — |
| 153 | #611 | — | 6 | 1 | no | — |
| 154 | #611 | — | 6 | 1 | no | — |
| 155 | #611 | S4 | 1 | 1 | yes | characteristic |
| 156 | D36 | — | 3 | 1 | no | — |
| 157 | D36 | — | 3 | 1 | no | — |
| 158 | D36 | — | 3 | 1 | no | — |
| 159 | D36 | — | 3 | 1 | no | — |
| 160 | D36 | — | 6 | 1 | no | — |
| 161 | D36 | — | 6 | 1 | no | — |
| 162 | D36 | — | 6 | 1 | no | — |
| 163 | C4×D9 | — | 3 | 1 | no | — |
| 164 | C4×D9 | — | 3 | 1 | no | — |
| 165 | C4×D9 | — | 6 | 1 | no | — |
| 166 | C4×D9 | — | 6 | 1 | no | — |
| 167 | C4×D9 | — | 6 | 1 | no | — |
| 168 | Dic18 | — | 3 | 1 | no | — |
| 169 | Dic18 | — | 3 | 1 | no | — |
| 170 | Q8⋊C9 | — | 2 | 1 | no | — |
| 171 | C8×C9 | — | 3 | 1 | no | — |
| 172 | C8×C9 | — | 3 | 1 | no | — |
| 173 | C9⋊C8 | — | 3 | 1 | no | — |
| 174 | C9⋊C8 | — | 3 | 1 | no | — |
| 175 | #577 | — | 27 | 1 | no | Sylow |
| 176 | C2×C3×C9 | — | 4 | 1 | no | — |
| 177 | C2×C3×C9 | — | 8 | 1 | no | — |
| 178 | C3⋊D9 | — | 4 | 2 | no | — |
| 179 | C3⋊D9 | — | 8 | 1 | no | — |
| 180 | D3×C9 | — | 8 | 1 | no | — |
| 181 | C3×D9 | — | 8 | 1 | no | — |
| 182 | C3×#41 | — | 3 | 1 | no | — |
| 183 | C2×C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 184 | #239 | — | 3 | 1 | no | — |
| 185 | #239 | — | 9 | 1 | no | — |
| 186 | #239 | — | 9 | 1 | no | — |
| 187 | #239 | — | 9 | 1 | no | — |
| 188 | D3×Q8 | — | 3 | 1 | no | — |
| 189 | D3×Q8 | — | 9 | 1 | no | — |
| 190 | D3×D4 | — | 9 | 1 | no | — |
| 191 | D3×D4 | — | 9 | 1 | no | — |
| 192 | D3×D4 | — | 18 | 1 | no | — |
| 193 | D3×D4 | — | 18 | 1 | no | — |
| 194 | D3×D4 | — | 36 | 1 | no | — |
| 195 | #235 | — | 9 | 1 | no | — |
| 196 | #235 | — | 9 | 1 | no | — |
| 197 | #235 | — | 18 | 1 | no | — |
| 198 | C2×D12 | — | 9 | 1 | no | — |
| 199 | C2×D12 | — | 18 | 1 | no | — |
| 200 | #41⋊C3 | — | 9 | 1 | no | — |
| 201 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 202 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 203 | C2×SL(2,𝔽3) | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 204 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 205 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 206 | GL(2,𝔽3) | — | 18 | 1 | no | — |
| 207 | GL(2,𝔽3) | — | 18 | 1 | no | — |
| 208 | GL(2,𝔽3) | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 209 | #226 | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 210 | C3×Q16 | — | 3 | 1 | no | — |
| 211 | C3×Q16 | — | 3 | 1 | no | — |
| 212 | C3×SD16 | — | 3 | 1 | no | — |
| 213 | C3×SD16 | — | 3 | 1 | no | — |
| 214 | C3×M16 | — | 3 | 1 | no | — |
| 215 | #216 | — | 9 | 1 | no | — |
| 216 | #215 | — | 9 | 1 | no | — |
| 217 | #215 | — | 9 | 1 | no | — |
| 218 | #215 | — | 9 | 1 | no | — |
| 219 | #215 | — | 9 | 1 | no | — |
| 220 | #215 | — | 9 | 1 | no | — |
| 221 | #214 | — | 9 | 1 | no | — |
| 222 | #213 | — | 9 | 1 | no | — |
| 223 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 224 | D24 | — | 9 | 1 | no | — |
| 225 | D24 | — | 9 | 1 | no | — |
| 226 | #204 | — | 9 | 1 | no | — |
| 227 | #204 | — | 9 | 1 | no | — |
| 228 | ΓL(1,𝔽25) | — | 9 | 1 | no | — |
| 229 | ΓL(1,𝔽25) | — | 9 | 1 | no | — |
| 230 | D3×C8 | — | 9 | 1 | no | — |
| 231 | D3×C8 | — | 9 | 1 | no | — |
| 232 | C22×C32 | — | 4 | 1 | no | — |
| 233 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 234 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 235 | C2×C3⋊D3 | — | 24 | 2 | no | — |
| 236 | C2×C3×D3 | — | 4 | 1 | no | — |
| 237 | C2×C3×D3 | — | 12 | 1 | no | — |
| 238 | D32 | — | 24 | 2 | no | — |
| 239 | C3×Dic3 | — | 4 | 1 | no | — |
| 240 | C3×Dic3 | — | 12 | 1 | no | — |
| 241 | C22×C9 | — | 6 | 1 | no | — |
| 242 | C22×C9 | — | 8 | 1 | no | — |
| 243 | C22×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 244 | C2×D9 | — | 3 | 1 | no | — |
| 245 | C2×D9 | — | 6 | 1 | no | — |
| 246 | C2×D9 | — | 6 | 1 | no | — |
| 247 | C2×D9 | — | 24 | 1 | no | — |
| 248 | C2×D9 | — | 24 | 1 | no | — |
| 249 | C2×D9 | — | 24 | 1 | no | — |
| 250 | C2×D9 | — | 24 | 1 | no | — |
| 251 | C2×D9 | — | 24 | 1 | no | — |
| 252 | C2×D9 | — | 24 | 4 | no | — |
| 253 | C2×D9 | C2×S4 | 1 | 1 | yes | characteristic |
| 254 | C4×C9 | — | 3 | 1 | no | — |
| 255 | C4×C9 | — | 3 | 1 | no | — |
| 256 | C4×C9 | — | 6 | 1 | no | — |
| 257 | Dic9 | — | 3 | 1 | no | — |
| 258 | Dic9 | C2×S4 | 1 | 1 | yes | characteristic |
| 259 | 2−1+4 | — | 9 | 1 | no | — |
| 260 | H(5,𝔽2) | — | 27 | 1 | no | — |
| 261 | #137 | — | 3 | 1 | no | — |
| 262 | #137 | — | 27 | 1 | no | — |
| 263 | #137 | — | 27 | 1 | no | — |
| 264 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 265 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 266 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 267 | #135 | — | 27 | 1 | no | — |
| 268 | #135 | — | 27 | 1 | no | — |
| 269 | #135 | — | 27 | 1 | no | — |
| 270 | C2×SD16 | — | 27 | 1 | no | — |
| 271 | C2×SD16 | — | 27 | 1 | no | — |
| 272 | C2×SD16 | — | 27 | 1 | no | — |
| 273 | #131 | — | 27 | 1 | no | — |
| 274 | C3×C9 | — | 4 | 1 | no | Sylow |
| 275 | C22×D3 | — | 18 | 1 | no | — |
| 276 | C22×D3 | — | 18 | 1 | no | — |
| 277 | C22×D3 | — | 18 | 1 | no | — |
| 278 | C22×D3 | — | 36 | 1 | no | — |
| 279 | C22×D3 | — | 36 | 1 | no | — |
| 280 | C22×D3 | — | 72 | 1 | no | — |
| 281 | C3×Q8 | — | 3 | 1 | no | — |
| 282 | C3×Q8 | — | 3 | 1 | no | — |
| 283 | C3×Q8 | C2×D32 | 1 | 1 | yes | characteristic |
| 284 | C3×D4 | — | 3 | 1 | no | — |
| 285 | C3×D4 | — | 6 | 1 | no | — |
| 286 | C3×D4 | — | 36 | 1 | no | — |
| 287 | C2×C3×C4 | — | 3 | 1 | no | — |
| 288 | C2×C3×C4 | — | 6 | 1 | no | — |
| 289 | #67 | — | 3 | 1 | no | — |
| 290 | #67 | — | 4 | 1 | no | — |
| 291 | #67 | — | 9 | 1 | no | — |
| 292 | #67 | — | 18 | 1 | no | — |
| 293 | #67 | — | 18 | 1 | no | — |
| 294 | #67 | — | 36 | 1 | no | — |
| 295 | D12 | — | 9 | 1 | no | — |
| 296 | D12 | — | 9 | 1 | no | — |
| 297 | D12 | — | 9 | 1 | no | — |
| 298 | D12 | — | 9 | 1 | no | — |
| 299 | D12 | — | 18 | 1 | no | — |
| 300 | D12 | — | 18 | 1 | no | — |
| 301 | D12 | — | 18 | 1 | no | — |
| 302 | D12 | — | 36 | 1 | no | — |
| 303 | C4×D3 | — | 9 | 1 | no | — |
| 304 | C4×D3 | — | 9 | 1 | no | — |
| 305 | C4×D3 | — | 18 | 1 | no | — |
| 306 | C4×D3 | — | 18 | 1 | no | — |
| 307 | C4×D3 | — | 18 | 1 | no | — |
| 308 | C4×D3 | — | 36 | 1 | no | — |
| 309 | Dic6 | — | 9 | 1 | no | — |
| 310 | Dic6 | — | 9 | 1 | no | — |
| 311 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 312 | SL(2,𝔽3) | C22×D9 | 1 | 1 | yes | characteristic |
| 313 | C3×C8 | — | 3 | 1 | no | — |
| 314 | C3×C8 | — | 3 | 1 | no | — |
| 315 | C3⋊C8 | — | 9 | 1 | no | — |
| 316 | C3⋊C8 | — | 9 | 1 | no | — |
| 317 | C2×C32 | — | 4 | 1 | no | — |
| 318 | C2×C32 | — | 8 | 1 | no | — |
| 319 | C3⋊D3 | — | 12 | 2 | no | — |
| 320 | C3⋊D3 | — | 24 | 1 | no | — |
| 321 | C3×D3 | — | 8 | 1 | no | — |
| 322 | C3×D3 | — | 24 | 1 | no | — |
| 323 | C2×C9 | — | 2 | 1 | no | — |
| 324 | C2×C9 | — | 8 | 2 | no | — |
| 325 | C2×C9 | — | 8 | 1 | no | — |
| 326 | C2×C9 | — | 12 | 1 | no | — |
| 327 | C2×C9 | C22×S4 | 1 | 1 | yes | characteristic |
| 328 | D9 | — | 2 | 1 | no | — |
| 329 | D9 | — | 6 | 1 | no | — |
| 330 | D9 | — | 12 | 1 | no | — |
| 331 | D9 | — | 12 | 1 | no | — |
| 332 | D9 | — | 24 | 2 | no | — |
| 333 | D9 | — | 24 | 2 | no | — |
| 334 | #41 | — | 3 | 1 | no | — |
| 335 | #41 | — | 9 | 1 | no | — |
| 336 | #41 | — | 27 | 1 | no | — |
| 337 | #41 | — | 27 | 1 | no | — |
| 338 | #41 | — | 27 | 1 | no | — |
| 339 | #41 | — | 27 | 1 | no | — |
| 340 | #41 | — | 27 | 1 | no | — |
| 341 | #41 | — | 54 | 1 | no | — |
| 342 | C2×Q8 | — | 9 | 1 | no | — |
| 343 | C2×Q8 | — | 27 | 1 | no | — |
| 344 | C2×Q8 | D3×D9 | 1 | 1 | yes | characteristic, complemented |
| 345 | C2×D4 | — | 27 | 1 | no | — |
| 346 | C2×D4 | — | 27 | 1 | no | — |
| 347 | C2×D4 | — | 27 | 1 | no | — |
| 348 | C2×D4 | — | 54 | 1 | no | — |
| 349 | C2×D4 | — | 54 | 1 | no | — |
| 350 | C2×D4 | — | 54 | 1 | no | — |
| 351 | Q16 | — | 3 | 1 | no | — |
| 352 | Q16 | — | 3 | 1 | no | — |
| 353 | Q16 | — | 27 | 1 | no | — |
| 354 | SD16 | — | 3 | 1 | no | — |
| 355 | SD16 | — | 3 | 1 | no | — |
| 356 | SD16 | — | 27 | 1 | no | — |
| 357 | SD16 | — | 27 | 1 | no | — |
| 358 | SD16 | — | 27 | 1 | no | — |
| 359 | SD16 | — | 27 | 1 | no | — |
| 360 | SD16 | — | 27 | 1 | no | — |
| 361 | SD16 | — | 27 | 1 | no | — |
| 362 | SD16 | — | 27 | 1 | no | — |
| 363 | SD16 | — | 27 | 1 | no | — |
| 364 | D8 | — | 27 | 1 | no | — |
| 365 | D8 | — | 27 | 1 | no | — |
| 366 | D8 | — | 27 | 1 | no | — |
| 367 | M16 | — | 3 | 1 | no | — |
| 368 | M16 | — | 27 | 1 | no | — |
| 369 | M16 | — | 27 | 1 | no | — |
| 370 | C2×C8 | — | 27 | 1 | no | — |
| 371 | C2×C8 | — | 27 | 1 | no | — |
| 372 | C2×C8 | — | 27 | 1 | no | — |
| 373 | C22×C3 | — | 4 | 1 | no | — |
| 374 | C22×C3 | — | 6 | 1 | no | — |
| 375 | C22×C3 | — | 8 | 1 | no | — |
| 376 | C22×C3 | — | 36 | 1 | no | — |
| 377 | C22×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 378 | C2×D3 | — | 3 | 1 | no | — |
| 379 | C2×D3 | — | 4 | 1 | no | — |
| 380 | C2×D3 | — | 9 | 1 | no | — |
| 381 | C2×D3 | — | 18 | 1 | no | — |
| 382 | C2×D3 | — | 18 | 1 | no | — |
| 383 | C2×D3 | — | 36 | 1 | no | — |
| 384 | C2×D3 | — | 36 | 1 | no | — |
| 385 | C2×D3 | — | 72 | 1 | no | — |
| 386 | C2×D3 | — | 72 | 1 | no | — |
| 387 | C2×D3 | — | 72 | 1 | no | — |
| 388 | C2×D3 | — | 72 | 2 | no | — |
| 389 | C2×D3 | — | 72 | 1 | no | — |
| 390 | C2×D3 | — | 72 | 1 | no | — |
| 391 | C2×D3 | — | 72 | 2 | no | — |
| 392 | C2×D3 | — | 72 | 4 | no | — |
| 393 | C3×C4 | — | 3 | 1 | no | — |
| 394 | C3×C4 | — | 3 | 1 | no | — |
| 395 | C3×C4 | — | 6 | 1 | no | — |
| 396 | C3×C4 | — | 36 | 1 | no | — |
| 397 | Dic3 | — | 3 | 1 | no | — |
| 398 | Dic3 | — | 4 | 1 | no | — |
| 399 | Dic3 | — | 9 | 1 | no | — |
| 400 | C32 | — | 4 | 1 | no | — |
| 401 | C9 | — | 8 | 1 | no | — |
| 402 | C9 | #5926 | 1 | 1 | yes | characteristic, complemented |
| 403 | C23 | — | 54 | 1 | no | — |
| 404 | C23 | — | 54 | 1 | no | — |
| 405 | C23 | — | 54 | 1 | no | — |
| 406 | Q8 | — | 3 | 1 | no | — |
| 407 | Q8 | — | 3 | 1 | no | — |
| 408 | Q8 | — | 27 | 1 | no | — |
| 409 | Q8 | — | 27 | 1 | no | — |
| 410 | Q8 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 411 | D4 | — | 3 | 1 | no | — |
| 412 | D4 | — | 6 | 1 | no | — |
| 413 | D4 | — | 9 | 1 | no | — |
| 414 | D4 | — | 27 | 1 | no | — |
| 415 | D4 | — | 27 | 1 | no | — |
| 416 | D4 | — | 27 | 1 | no | — |
| 417 | D4 | — | 27 | 1 | no | — |
| 418 | D4 | — | 27 | 1 | no | — |
| 419 | D4 | — | 54 | 1 | no | — |
| 420 | D4 | — | 54 | 1 | no | — |
| 421 | D4 | — | 54 | 1 | no | — |
| 422 | D4 | — | 54 | 1 | no | — |
| 423 | D4 | — | 54 | 1 | no | — |
| 424 | C2×C4 | — | 3 | 1 | no | — |
| 425 | C2×C4 | — | 6 | 1 | no | — |
| 426 | C2×C4 | — | 27 | 1 | no | — |
| 427 | C2×C4 | — | 27 | 1 | no | — |
| 428 | C2×C4 | — | 54 | 1 | no | — |
| 429 | C2×C4 | — | 54 | 1 | no | — |
| 430 | C2×C4 | — | 54 | 1 | no | — |
| 431 | C8 | — | 3 | 1 | no | — |
| 432 | C8 | — | 3 | 1 | no | — |
| 433 | C8 | — | 27 | 1 | no | — |
| 434 | C8 | — | 27 | 1 | no | — |
| 435 | C2×C3 | — | 2 | 1 | no | — |
| 436 | C2×C3 | — | 4 | 1 | no | — |
| 437 | C2×C3 | — | 8 | 1 | no | — |
| 438 | C2×C3 | — | 8 | 2 | no | — |
| 439 | C2×C3 | — | 8 | 1 | no | — |
| 440 | C2×C3 | — | 12 | 1 | no | — |
| 441 | C2×C3 | — | 72 | 1 | no | — |
| 442 | C2×C3 | C2×D3×S4 | 1 | 1 | yes | characteristic |
| 443 | D3 | — | 6 | 1 | no | — |
| 444 | D3 | — | 8 | 1 | no | — |
| 445 | D3 | — | 18 | 1 | no | — |
| 446 | D3 | — | 36 | 2 | no | — |
| 447 | D3 | — | 36 | 1 | no | — |
| 448 | D3 | — | 36 | 1 | no | — |
| 449 | D3 | — | 72 | 1 | no | — |
| 450 | D3 | — | 72 | 2 | no | — |
| 451 | D3 | — | 72 | 2 | no | — |
| 452 | C22 | — | 6 | 1 | no | — |
| 453 | C22 | — | 9 | 1 | no | — |
| 454 | C22 | — | 27 | 1 | no | — |
| 455 | C22 | — | 54 | 1 | no | — |
| 456 | C22 | — | 54 | 1 | no | — |
| 457 | C22 | — | 216 | 1 | no | — |
| 458 | C22 | — | 216 | 1 | no | — |
| 459 | C22 | — | 216 | 1 | no | — |
| 460 | C22 | D9×S4 | 1 | 1 | yes | characteristic |
| 461 | C4 | — | 3 | 1 | no | — |
| 462 | C4 | — | 3 | 1 | no | — |
| 463 | C4 | — | 6 | 1 | no | — |
| 464 | C4 | — | 9 | 1 | no | — |
| 465 | C4 | — | 27 | 1 | no | — |
| 466 | C3 | — | 4 | 1 | no | — |
| 467 | C3 | — | 8 | 1 | no | — |
| 468 | C3 | #10596446 | 1 | 1 | yes | characteristic |
| 469 | C2 | — | 2 | 1 | no | — |
| 470 | C2 | — | 12 | 1 | no | — |
| 471 | C2 | — | 18 | 1 | no | — |
| 472 | C2 | — | 54 | 1 | no | — |
| 473 | C2 | — | 108 | 1 | no | — |
| 474 | C2 | — | 108 | 1 | no | — |
| 475 | C2 | C2×D9×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D9×S4 |
| commutator, cosocle, hypocenter | C9×SL(2,𝔽3) | C23 |
| Frattini, socle | C2×C3 | C2×D3×S4 |
| Fitting | C2×Q8×C9 | C2×D3 |
| hypercenter | C22 | D9×S4 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722410 | C2 | semidirect |
| 2. C2×C9⋊GL(2,𝔽3) | C2 | semidirect |
| 3. C9×#983 | C2 | semidirect |
| 4. D9×GL(2,𝔽3) | C2 | semidirect |
| 5. #11722235 | C2 | semidirect |
| 6. #11722234 | C2 | semidirect |
| 7. #11722232 | C2 | semidirect |
| 10. C2×C9×SL(2,𝔽3) | C22 | semidirect |
| 12. D9×SL(2,𝔽3) | C22 | semidirect |
| 13. #88615 | C22 | semidirect |
| 14. C9⋊GL(2,𝔽3) | C22 | semidirect |
| 15. C9⋊GL(2,𝔽3) | C22 | semidirect |
| 16. C9×GL(2,𝔽3) | C22 | semidirect |
| 17. C9×#226 | C22 | semidirect |
| 26. #63713 | D3 | semidirect |
| 60. C2×Q8×C9 | C2×D3 | semidirect |
| 64. #3701 | C2×D3 | semidirect |
| 66. Q8×D9 | C2×D3 | semidirect |
| 116. #983 | D9 | semidirect |
| 203. C2×SL(2,𝔽3) | C2×D9 | semidirect |
| 208. GL(2,𝔽3) | C2×D9 | semidirect |
| 209. #226 | C2×D9 | semidirect |
| 344. C2×Q8 | D3×D9 | semidirect |
| 402. C9 | #5926 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722410 | 2 |
| C2×C9⋊GL(2,𝔽3) | 2 |
| C9×#983 | 2 |
| D9×GL(2,𝔽3) | 2 |
| #11722235 | 2 |
| #11722234 | 2 |
| #11722232 | 2 |
| #10596446 | 3 |
| #10589880 | 3 |
| #88668 | 4 |