Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
481 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 112 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 | #24810 | C2 | 1 | 2 | yes | maximal |
| 2 | #13520 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13441 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13390 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13315 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13198 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #13157 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #13006 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12878 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C22×Q32 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3069 | — | 2 | 2 | no | — |
| 12 | C2×#473 | C22 | 1 | 2 | yes | — |
| 13 | C2×#473 | C22 | 1 | 2 | yes | — |
| 14 | #2955 | C22 | 1 | 1 | yes | characteristic |
| 15 | #2916 | C22 | 1 | 2 | yes | — |
| 16 | #2161 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2151 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#362 | C22 | 1 | 2 | yes | — |
| 19 | C2×#362 | C22 | 1 | 2 | yes | — |
| 20 | C2×#362 | C22 | 1 | 2 | yes | — |
| 21 | #2117 | C22 | 1 | 2 | yes | — |
| 22 | #2113 | C22 | 1 | 2 | yes | — |
| 23 | #2110 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2070 | — | 2 | 2 | no | — |
| 25 | #2049 | — | 2 | 2 | no | — |
| 26 | #1979 | C22 | 1 | 2 | yes | — |
| 27 | #1953 | C22 | 1 | 2 | yes | — |
| 28 | #1902 | — | 2 | 2 | no | — |
| 29 | #1813 | C22 | 1 | 2 | yes | — |
| 30 | #1388 | C22 | 1 | 2 | yes | — |
| 31 | #1388 | C22 | 1 | 2 | yes | — |
| 32 | #1385 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1381 | C22 | 1 | 2 | yes | — |
| 34 | #1381 | C22 | 1 | 1 | yes | characteristic |
| 35 | #575 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×#137 | — | 2 | 2 | no | — |
| 37 | C2×#137 | — | 2 | 2 | no | — |
| 38 | C22×Q16 | C23 | 1 | 2 | yes | — |
| 39 | #568 | — | 2 | 2 | no | — |
| 40 | C2×#131 | C23 | 1 | 2 | yes | — |
| 41 | C2×Q32 | — | 4 | 2 | no | — |
| 42 | C2×Q32 | — | 4 | 1 | no | — |
| 43 | C2×M32 | C23 | 1 | 2 | yes | — |
| 44 | C2×M32 | C23 | 1 | 1 | yes | characteristic |
| 45 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 46 | #473 | — | 2 | 4 | no | — |
| 47 | #473 | — | 4 | 2 | no | — |
| 48 | #473 | — | 4 | 2 | no | — |
| 49 | #473 | — | 4 | 2 | no | — |
| 50 | #473 | — | 4 | 4 | no | — |
| 51 | #471 | — | 4 | 4 | no | — |
| 52 | #444 | — | 4 | 1 | no | — |
| 53 | #443 | — | 4 | 1 | no | — |
| 54 | #430 | — | 2 | 4 | no | — |
| 55 | #430 | — | 2 | 2 | no | — |
| 56 | #430 | — | 2 | 2 | no | — |
| 57 | #430 | D4 | 1 | 1 | yes | characteristic |
| 58 | C2×#108 | — | 4 | 2 | no | — |
| 59 | C2×#108 | C23 | 1 | 2 | yes | — |
| 60 | C2×#108 | C23 | 1 | 2 | yes | — |
| 61 | C2×#108 | C23 | 1 | 2 | yes | — |
| 62 | #421 | D4 | 1 | 2 | yes | — |
| 63 | #419 | — | 2 | 2 | no | — |
| 64 | #413 | — | 2 | 2 | no | — |
| 65 | C2×#101 | — | 2 | 2 | no | — |
| 66 | C2×#101 | — | 2 | 2 | no | — |
| 67 | C2×#101 | — | 2 | 2 | no | — |
| 68 | #405 | C23 | 1 | 1 | yes | characteristic |
| 69 | #368 | — | 4 | 1 | no | — |
| 70 | C4⋊C16 | — | 4 | 1 | no | — |
| 71 | #362 | — | 2 | 4 | no | — |
| 72 | #362 | — | 4 | 2 | no | — |
| 73 | #362 | — | 4 | 2 | no | — |
| 74 | #350 | — | 4 | 2 | no | — |
| 75 | C22⋊C16 | — | 4 | 2 | no | — |
| 76 | #347 | — | 4 | 1 | no | — |
| 77 | #344 | — | 2 | 2 | no | — |
| 78 | #338 | — | 2 | 2 | no | — |
| 79 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 80 | C22×Q8 | — | 2 | 2 | no | — |
| 81 | #137 | — | 2 | 2 | no | — |
| 82 | #137 | — | 4 | 4 | no | — |
| 83 | #137 | — | 4 | 4 | no | — |
| 84 | #137 | — | 4 | 4 | no | — |
| 85 | #137 | — | 8 | 2 | no | — |
| 86 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 87 | #135 | — | 4 | 1 | no | — |
| 88 | C2×Q16 | — | 2 | 2 | no | — |
| 89 | C2×Q16 | — | 2 | 2 | no | — |
| 90 | C2×Q16 | — | 2 | 2 | no | — |
| 91 | C2×Q16 | — | 2 | 2 | no | — |
| 92 | C2×Q16 | — | 2 | 2 | no | — |
| 93 | C2×Q16 | — | 2 | 2 | no | — |
| 94 | C2×Q16 | — | 4 | 2 | no | — |
| 95 | C2×Q16 | — | 4 | 2 | no | — |
| 96 | C2×SD16 | — | 4 | 2 | no | — |
| 97 | C2×SD16 | — | 4 | 2 | no | — |
| 98 | #131 | — | 2 | 2 | no | — |
| 99 | #131 | — | 2 | 2 | no | — |
| 100 | #131 | — | 2 | 2 | no | — |
| 101 | #131 | — | 2 | 2 | no | — |
| 102 | #131 | — | 4 | 4 | no | — |
| 103 | #131 | — | 8 | 2 | no | — |
| 104 | C2×M16 | — | 2 | 2 | no | — |
| 105 | C2×M16 | — | 2 | 2 | no | — |
| 106 | C2×M16 | — | 2 | 2 | no | — |
| 107 | C2×M16 | — | 2 | 2 | no | — |
| 108 | C2×M16 | — | 4 | 4 | no | — |
| 109 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 110 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C22×C8 | — | 2 | 2 | no | — |
| 112 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 113 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | Q32 | — | 4 | 4 | no | — |
| 115 | M32 | — | 2 | 2 | no | — |
| 116 | M32 | — | 2 | 2 | no | — |
| 117 | M32 | — | 2 | 2 | no | — |
| 118 | C2×C16 | — | 2 | 2 | no | — |
| 119 | C2×C16 | — | 2 | 2 | no | — |
| 120 | C2×C16 | — | 2 | 1 | no | — |
| 121 | C2×C16 | — | 2 | 1 | no | — |
| 122 | #108 | — | 2 | 2 | no | — |
| 123 | #108 | — | 2 | 2 | no | — |
| 124 | #108 | — | 2 | 2 | no | — |
| 125 | #108 | — | 2 | 2 | no | — |
| 126 | #108 | — | 2 | 2 | no | — |
| 127 | #108 | — | 4 | 1 | no | — |
| 128 | #108 | — | 4 | 4 | no | — |
| 129 | #108 | — | 4 | 4 | no | — |
| 130 | #108 | #41 | 1 | 4 | yes | — |
| 131 | #104 | — | 4 | 2 | no | — |
| 132 | #103 | — | 8 | 2 | no | — |
| 133 | #101 | — | 4 | 4 | no | — |
| 134 | #101 | — | 4 | 4 | no | — |
| 135 | #101 | — | 4 | 4 | no | — |
| 136 | #101 | — | 4 | 2 | no | — |
| 137 | #100 | — | 4 | 2 | no | — |
| 138 | #97 | — | 2 | 1 | no | — |
| 139 | C4×C8 | — | 2 | 1 | no | — |
| 140 | #41 | — | 2 | 2 | no | — |
| 141 | #41 | — | 2 | 2 | no | — |
| 142 | #41 | — | 2 | 2 | no | — |
| 143 | #41 | — | 2 | 2 | no | — |
| 144 | C2×Q8 | — | 4 | 2 | no | — |
| 145 | C2×Q8 | — | 4 | 2 | no | — |
| 146 | C2×Q8 | — | 4 | 2 | no | — |
| 147 | C2×Q8 | — | 4 | 2 | no | — |
| 148 | C2×Q8 | — | 4 | 2 | no | — |
| 149 | C2×Q8 | — | 4 | 2 | no | — |
| 150 | C2×Q8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 151 | C2×D4 | — | 2 | 2 | no | — |
| 152 | C2×D4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 153 | C22×C4 | — | 2 | 2 | no | — |
| 154 | C22×C4 | — | 4 | 2 | no | — |
| 155 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 156 | Q16 | — | 4 | 1 | no | — |
| 157 | Q16 | — | 4 | 2 | no | — |
| 158 | Q16 | — | 4 | 2 | no | — |
| 159 | Q16 | — | 4 | 2 | no | — |
| 160 | Q16 | — | 4 | 2 | no | — |
| 161 | Q16 | — | 8 | 4 | no | — |
| 162 | Q16 | — | 8 | 4 | no | — |
| 163 | SD16 | — | 4 | 2 | no | — |
| 164 | SD16 | — | 8 | 4 | no | — |
| 165 | SD16 | — | 8 | 4 | no | — |
| 166 | D8 | — | 4 | 1 | no | — |
| 167 | M16 | — | 2 | 2 | no | — |
| 168 | M16 | — | 2 | 2 | no | — |
| 169 | M16 | — | 2 | 4 | no | — |
| 170 | M16 | — | 2 | 2 | no | — |
| 171 | M16 | — | 4 | 2 | no | — |
| 172 | M16 | — | 4 | 2 | no | — |
| 173 | M16 | — | 4 | 2 | no | — |
| 174 | M16 | — | 4 | 2 | no | — |
| 175 | M16 | — | 4 | 2 | no | — |
| 176 | M16 | — | 4 | 2 | no | — |
| 177 | M16 | — | 4 | 2 | no | — |
| 178 | M16 | — | 8 | 4 | no | — |
| 179 | C2×C8 | — | 2 | 1 | no | — |
| 180 | C2×C8 | — | 2 | 2 | no | — |
| 181 | C2×C8 | — | 4 | 2 | no | — |
| 182 | C2×C8 | — | 4 | 2 | no | — |
| 183 | C2×C8 | — | 4 | 2 | no | — |
| 184 | C2×C8 | — | 4 | 2 | no | — |
| 185 | C2×C8 | — | 4 | 2 | no | — |
| 186 | C2×C8 | — | 8 | 4 | no | — |
| 187 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 188 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 189 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 190 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 191 | C4⋊C4 | — | 2 | 1 | no | — |
| 192 | C22⋊C4 | — | 2 | 1 | no | — |
| 193 | C42 | — | 2 | 1 | no | — |
| 194 | C16 | — | 4 | 2 | no | — |
| 195 | C16 | — | 4 | 1 | no | — |
| 196 | C16 | — | 4 | 1 | no | — |
| 197 | C23 | — | 2 | 2 | no | — |
| 198 | C23 | #578 | 1 | 1 | yes | characteristic |
| 199 | Q8 | — | 2 | 2 | no | — |
| 200 | Q8 | — | 2 | 2 | no | — |
| 201 | Q8 | — | 8 | 2 | no | — |
| 202 | Q8 | — | 8 | 2 | no | — |
| 203 | Q8 | — | 8 | 2 | no | — |
| 204 | Q8 | — | 8 | 2 | no | — |
| 205 | D4 | — | 2 | 2 | no | — |
| 206 | D4 | — | 2 | 2 | no | — |
| 207 | D4 | — | 4 | 2 | no | — |
| 208 | D4 | — | 4 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 2 | no | — |
| 210 | C2×C4 | — | 4 | 2 | no | — |
| 211 | C2×C4 | — | 4 | 2 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 8 | 2 | no | — |
| 214 | C2×C4 | — | 8 | 2 | no | — |
| 215 | C2×C4 | — | 8 | 2 | no | — |
| 216 | C2×C4 | #534 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | #546 | 1 | 2 | yes | — |
| 218 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 219 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 220 | C8 | — | 4 | 1 | no | — |
| 221 | C8 | — | 4 | 2 | no | — |
| 222 | C8 | — | 4 | 4 | no | — |
| 223 | C8 | — | 8 | 2 | no | — |
| 224 | C8 | — | 8 | 2 | no | — |
| 225 | C8 | #540 | 1 | 2 | yes | — |
| 226 | C8 | #555 | 1 | 1 | yes | characteristic |
| 227 | C8 | #560 | 1 | 1 | yes | characteristic |
| 228 | C22 | — | 4 | 1 | no | — |
| 229 | C22 | — | 4 | 2 | no | — |
| 230 | C22 | — | 8 | 2 | no | — |
| 231 | C22 | #3203 | 1 | 1 | yes | characteristic |
| 232 | C22 | #3308 | 1 | 2 | yes | — |
| 233 | C4 | — | 4 | 2 | no | — |
| 234 | C4 | — | 8 | 1 | no | — |
| 235 | C4 | — | 16 | 2 | no | — |
| 236 | C4 | #2699 | 1 | 1 | yes | characteristic |
| 237 | C4 | #3192 | 1 | 1 | yes | characteristic |
| 238 | C4 | #3318 | 1 | 2 | yes | — |
| 239 | C2 | — | 2 | 1 | no | — |
| 240 | C2 | — | 2 | 2 | no | — |
| 241 | C2 | — | 8 | 2 | no | — |
| 242 | C2 | #31736 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31736 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #575 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2955 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13520 | C2 | semidirect |
| 3. #13441 | C2 | semidirect |
| 4. #13390 | C2 | semidirect |
| 5. #13315 | C2 | semidirect |
| 7. #13157 | C2 | semidirect |
| 8. #13006 | C2 | semidirect |
| 9. #12878 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #24810 | 2 |
| #13520 | 2 |
| #13441 | 2 |
| #13390 | 2 |
| #13315 | 2 |
| #13198 | 2 |
| #13157 | 2 |
| #13006 | 2 |
| #12878 | 2 |