Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
839 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 839 conjugacy classes of subgroups — too many to draw. Only its 160 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 | #23474 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23470 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11395 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11277 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #11130 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #11119 | C2 | 1 | 3 | yes | maximal |
| 7 | #11013 | C2 | 1 | 3 | yes | maximal |
| 8 | C22×#137 | C22 | 1 | 1 | yes | characteristic |
| 9 | #3458 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2911 | C22 | 1 | 3 | yes | — |
| 11 | C2×#421 | C22 | 1 | 3 | yes | — |
| 12 | C2×#419 | C22 | 1 | 3 | yes | — |
| 13 | C2×#410 | C22 | 1 | 3 | yes | — |
| 14 | #2310 | C22 | 1 | 3 | yes | — |
| 15 | #2032 | — | 2 | 4 | no | — |
| 16 | #2030 | — | 2 | 4 | no | — |
| 17 | C2×#344 | C22 | 1 | 3 | yes | — |
| 18 | C2×#344 | C22 | 1 | 3 | yes | — |
| 19 | #1740 | C22 | 1 | 3 | yes | — |
| 20 | #1740 | C22 | 1 | 3 | yes | — |
| 21 | #1733 | C22 | 1 | 3 | yes | — |
| 22 | C2×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 23 | C22×#41 | C23 | 1 | 1 | yes | characteristic |
| 24 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 25 | C2×#137 | — | 4 | 3 | no | — |
| 26 | C2×#137 | — | 4 | 3 | no | — |
| 27 | C22×Q16 | Q8 | 1 | 2 | yes | complemented |
| 28 | C22×SD16 | Q8 | 1 | 2 | yes | complemented |
| 29 | C22×M16 | C23 | 1 | 3 | yes | — |
| 30 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 31 | #544 | — | 4 | 3 | no | — |
| 32 | #543 | D4 | 1 | 4 | yes | — |
| 33 | #533 | — | 4 | 3 | no | — |
| 34 | C2×#117 | C23 | 1 | 3 | yes | — |
| 35 | C2×#117 | C23 | 1 | 3 | yes | — |
| 36 | C2×#117 | C23 | 1 | 3 | yes | — |
| 37 | #428 | — | 4 | 3 | no | — |
| 38 | #428 | — | 4 | 3 | no | — |
| 39 | #427 | — | 4 | 3 | no | — |
| 40 | #421 | — | 2 | 6 | no | — |
| 41 | #421 | — | 4 | 3 | no | — |
| 42 | C2×#104 | — | 4 | 3 | no | — |
| 43 | #419 | — | 2 | 6 | no | — |
| 44 | #419 | — | 4 | 3 | no | — |
| 45 | C2×#103 | — | 4 | 3 | no | — |
| 46 | #410 | — | 4 | 3 | no | — |
| 47 | #410 | — | 4 | 3 | no | — |
| 48 | C2×#99 | — | 2 | 6 | no | — |
| 49 | #391 | — | 4 | 3 | no | — |
| 50 | #391 | — | 4 | 3 | no | — |
| 51 | #381 | — | 2 | 6 | no | — |
| 52 | #344 | — | 4 | 3 | no | — |
| 53 | #344 | — | 4 | 3 | no | — |
| 54 | #342 | — | 2 | 6 | no | — |
| 55 | #342 | — | 2 | 6 | no | — |
| 56 | #339 | — | 2 | 6 | no | — |
| 57 | H(3,ℤ/4) | — | 4 | 3 | no | — |
| 58 | C2×#41 | — | 2 | 3 | no | — |
| 59 | C2×#41 | — | 2 | 3 | no | — |
| 60 | C2×#41 | — | 2 | 3 | no | — |
| 61 | C2×#41 | — | 2 | 3 | no | — |
| 62 | C22×Q8 | — | 2 | 1 | no | — |
| 63 | C22×Q8 | — | 2 | 3 | no | — |
| 64 | C22×Q8 | — | 2 | 3 | no | — |
| 65 | C22×Q8 | — | 2 | 3 | no | — |
| 66 | C22×Q8 | — | 2 | 3 | no | — |
| 67 | C22×Q8 | C2×Q8 | 1 | 2 | yes | — |
| 68 | C22×Q8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 69 | C22×D4 | — | 2 | 1 | no | — |
| 70 | C22×D4 | C2×Q8 | 1 | 1 | yes | characteristic |
| 71 | C23×C4 | — | 2 | 1 | no | — |
| 72 | C23×C4 | — | 2 | 1 | no | — |
| 73 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 74 | #137 | — | 4 | 12 | no | — |
| 75 | #137 | — | 4 | 4 | no | — |
| 76 | C2×Q16 | — | 4 | 6 | no | — |
| 77 | C2×SD16 | — | 4 | 6 | no | — |
| 78 | C2×M16 | — | 2 | 3 | no | — |
| 79 | C2×M16 | — | 2 | 3 | no | — |
| 80 | C2×M16 | — | 2 | 3 | no | — |
| 81 | C2×M16 | — | 2 | 3 | no | — |
| 82 | C2×M16 | — | 2 | 3 | no | — |
| 83 | C2×M16 | — | 2 | 3 | no | — |
| 84 | C2×M16 | — | 2 | 3 | no | — |
| 85 | C2×M16 | #41 | 1 | 6 | yes | — |
| 86 | C22×C8 | — | 2 | 3 | no | — |
| 87 | C22×C8 | C2×Q8 | 1 | 2 | yes | — |
| 88 | C4⋊Q8 | — | 4 | 6 | no | — |
| 89 | #125 | — | 4 | 6 | no | — |
| 90 | C22⋊Q8 | — | 4 | 12 | no | — |
| 91 | C4×Q8 | — | 4 | 4 | no | — |
| 92 | #117 | — | 2 | 3 | no | — |
| 93 | #117 | — | 2 | 3 | no | — |
| 94 | #117 | — | 2 | 3 | no | — |
| 95 | #117 | — | 2 | 3 | no | — |
| 96 | #117 | — | 2 | 3 | no | — |
| 97 | #117 | — | 2 | 3 | no | — |
| 98 | #117 | — | 2 | 3 | no | — |
| 99 | #117 | — | 2 | 3 | no | — |
| 100 | #117 | — | 2 | 3 | no | — |
| 101 | #117 | — | 2 | 3 | no | — |
| 102 | #117 | — | 2 | 3 | no | — |
| 103 | #117 | — | 2 | 3 | no | — |
| 104 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 105 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 106 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 107 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 108 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 109 | C2×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 110 | C2×C42 | — | 2 | 3 | no | — |
| 111 | C2×C42 | — | 2 | 3 | no | — |
| 112 | C2×C42 | — | 2 | 3 | no | — |
| 113 | #107 | — | 4 | 6 | no | — |
| 114 | #106 | — | 4 | 6 | no | — |
| 115 | #104 | — | 4 | 12 | no | — |
| 116 | #103 | — | 4 | 12 | no | — |
| 117 | #99 | — | 4 | 12 | no | — |
| 118 | #95 | — | 4 | 12 | no | — |
| 119 | C24 | — | 2 | 1 | no | — |
| 120 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 121 | #41 | — | 4 | 3 | no | — |
| 122 | #41 | — | 4 | 3 | no | — |
| 123 | #41 | — | 4 | 3 | no | — |
| 124 | #41 | — | 4 | 3 | no | — |
| 125 | #41 | — | 4 | 3 | no | — |
| 126 | #41 | — | 4 | 1 | no | — |
| 127 | C2×Q8 | — | 2 | 6 | no | — |
| 128 | C2×Q8 | — | 2 | 3 | no | — |
| 129 | C2×Q8 | — | 2 | 6 | no | — |
| 130 | C2×Q8 | — | 2 | 3 | no | — |
| 131 | C2×Q8 | — | 4 | 4 | no | — |
| 132 | C2×Q8 | — | 4 | 3 | no | — |
| 133 | C2×Q8 | — | 4 | 3 | no | — |
| 134 | C2×Q8 | — | 4 | 3 | no | — |
| 135 | C2×Q8 | — | 4 | 3 | no | — |
| 136 | C2×Q8 | — | 4 | 3 | no | — |
| 137 | C2×Q8 | — | 4 | 3 | no | — |
| 138 | C2×Q8 | — | 4 | 3 | no | — |
| 139 | C2×Q8 | — | 4 | 1 | no | — |
| 140 | C2×D4 | — | 2 | 3 | no | — |
| 141 | C2×D4 | — | 2 | 3 | no | — |
| 142 | C2×D4 | — | 4 | 3 | no | — |
| 143 | C2×D4 | — | 4 | 3 | no | — |
| 144 | C22×C4 | — | 2 | 1 | no | — |
| 145 | C22×C4 | — | 2 | 2 | no | — |
| 146 | C22×C4 | — | 2 | 6 | no | — |
| 147 | C22×C4 | — | 4 | 3 | no | — |
| 148 | C22×C4 | — | 4 | 3 | no | — |
| 149 | C22×C4 | — | 4 | 3 | no | — |
| 150 | C22×C4 | — | 4 | 3 | no | — |
| 151 | C22×C4 | — | 4 | 3 | no | — |
| 152 | C22×C4 | — | 4 | 3 | no | — |
| 153 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 154 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 155 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 156 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 157 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 158 | C22×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 159 | Q16 | — | 4 | 8 | no | — |
| 160 | SD16 | — | 4 | 8 | no | — |
| 161 | M16 | — | 2 | 6 | no | — |
| 162 | M16 | — | 2 | 6 | no | — |
| 163 | M16 | — | 4 | 3 | no | — |
| 164 | M16 | — | 4 | 3 | no | — |
| 165 | M16 | — | 4 | 1 | no | — |
| 166 | M16 | — | 4 | 3 | no | — |
| 167 | C2×C8 | — | 2 | 6 | no | — |
| 168 | C2×C8 | — | 2 | 6 | no | — |
| 169 | C2×C8 | — | 4 | 3 | no | — |
| 170 | C2×C8 | — | 4 | 3 | no | — |
| 171 | C4⋊C4 | — | 4 | 12 | no | — |
| 172 | C4⋊C4 | — | 4 | 3 | no | — |
| 173 | C4⋊C4 | — | 4 | 3 | no | — |
| 174 | C4⋊C4 | — | 4 | 3 | no | — |
| 175 | C4⋊C4 | — | 4 | 3 | no | — |
| 176 | C4⋊C4 | — | 4 | 3 | no | — |
| 177 | C4⋊C4 | — | 4 | 3 | no | — |
| 178 | C22⋊C4 | — | 2 | 6 | no | — |
| 179 | C22⋊C4 | — | 2 | 6 | no | — |
| 180 | C22⋊C4 | — | 4 | 3 | no | — |
| 181 | C22⋊C4 | — | 4 | 3 | no | — |
| 182 | C22⋊C4 | — | 4 | 3 | no | — |
| 183 | C22⋊C4 | — | 4 | 3 | no | — |
| 184 | C42 | — | 4 | 3 | no | — |
| 185 | C42 | — | 4 | 3 | no | — |
| 186 | C42 | — | 4 | 3 | no | — |
| 187 | C42 | — | 4 | 3 | no | — |
| 188 | C42 | — | 4 | 3 | no | — |
| 189 | C42 | — | 4 | 3 | no | — |
| 190 | C23 | — | 4 | 3 | no | — |
| 191 | C23 | — | 8 | 1 | no | — |
| 192 | C23 | — | 16 | 1 | no | — |
| 193 | C23 | #534 | 1 | 3 | yes | — |
| 194 | C23 | D4×Q8 | 1 | 1 | yes | characteristic |
| 195 | C23 | #550 | 1 | 3 | yes | — |
| 196 | Q8 | — | 4 | 2 | no | — |
| 197 | Q8 | — | 4 | 1 | no | — |
| 198 | Q8 | — | 4 | 6 | no | — |
| 199 | Q8 | — | 4 | 3 | no | — |
| 200 | Q8 | — | 8 | 3 | no | — |
| 201 | Q8 | — | 8 | 1 | no | — |
| 202 | Q8 | — | 16 | 4 | no | — |
| 203 | D4 | — | 4 | 1 | no | — |
| 204 | D4 | — | 4 | 3 | no | — |
| 205 | D4 | — | 8 | 3 | no | — |
| 206 | D4 | — | 8 | 1 | no | — |
| 207 | C2×C4 | — | 4 | 3 | no | — |
| 208 | C2×C4 | — | 4 | 6 | no | — |
| 209 | C2×C4 | — | 4 | 6 | no | — |
| 210 | C2×C4 | — | 8 | 3 | no | — |
| 211 | C2×C4 | — | 8 | 1 | no | — |
| 212 | C2×C4 | — | 8 | 3 | no | — |
| 213 | C2×C4 | — | 8 | 1 | no | — |
| 214 | C2×C4 | — | 8 | 6 | no | — |
| 215 | C2×C4 | — | 8 | 6 | no | — |
| 216 | C2×C4 | — | 8 | 3 | no | — |
| 217 | C2×C4 | — | 8 | 3 | no | — |
| 218 | C2×C4 | — | 16 | 3 | no | — |
| 219 | C2×C4 | #538 | 1 | 3 | yes | — |
| 220 | C2×C4 | #538 | 1 | 3 | yes | — |
| 221 | C2×C4 | #539 | 1 | 3 | yes | — |
| 222 | C2×C4 | #543 | 1 | 3 | yes | — |
| 223 | C2×C4 | #554 | 1 | 3 | yes | — |
| 224 | C2×C4 | #555 | 1 | 3 | yes | — |
| 225 | C2×C4 | #560 | 1 | 3 | yes | — |
| 226 | C2×C4 | H(3,𝔽4) | 1 | 3 | yes | — |
| 227 | C2×C4 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #562 | 1 | 3 | yes | — |
| 229 | C8 | — | 4 | 6 | no | — |
| 230 | C8 | — | 4 | 2 | no | — |
| 231 | C22 | — | 4 | 3 | no | — |
| 232 | C22 | — | 4 | 3 | no | — |
| 233 | C22 | — | 4 | 1 | no | — |
| 234 | C22 | — | 8 | 1 | no | — |
| 235 | C22 | — | 16 | 3 | no | — |
| 236 | C22 | #2732 | 1 | 3 | yes | — |
| 237 | C22 | #2733 | 1 | 3 | yes | — |
| 238 | C22 | #2812 | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 8 | 6 | no | — |
| 240 | C4 | — | 8 | 1 | no | — |
| 241 | C4 | — | 8 | 2 | no | — |
| 242 | C4 | — | 16 | 3 | no | — |
| 243 | C4 | — | 16 | 3 | no | — |
| 244 | C4 | #2784 | 1 | 3 | yes | — |
| 245 | C4 | #2785 | 1 | 1 | yes | characteristic |
| 246 | C4 | #2824 | 1 | 3 | yes | — |
| 247 | C4 | #2840 | 1 | 1 | yes | characteristic |
| 248 | C2 | — | 2 | 1 | no | — |
| 249 | C2 | — | 2 | 3 | no | — |
| 250 | C2 | — | 2 | 3 | no | — |
| 251 | C2 | — | 16 | 1 | no | — |
| 252 | C2 | #17216 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17216 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23470 | C2 | semidirect |
| 3. #11395 | C2 | semidirect |
| 4. #11277 | C2 | semidirect |
| 5. #11130 | C2 | semidirect |
| 27. C22×Q16 | Q8 | semidirect |
| 28. C22×SD16 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23474 | 2 |
| #23470 | 2 |
| #11395 | 2 |
| #11277 | 2 |
| #11130 | 2 |
| #11119 | 2 |
| #11013 | 2 |