Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
644 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 188 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 | #60715 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33428 | C2 | 1 | 2 | yes | maximal |
| 3 | #11916 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #11914 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #3509 | C4 | 1 | 2 | yes | — |
| 6 | #3185 | — | 2 | 2 | no | — |
| 7 | #3177 | C4 | 1 | 4 | yes | — |
| 8 | #3177 | C22 | 1 | 4 | yes | — |
| 9 | #3170 | — | 2 | 2 | no | — |
| 10 | #3160 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#494 | C4 | 1 | 1 | yes | characteristic |
| 12 | #3139 | C22 | 1 | 2 | yes | — |
| 13 | #3059 | — | 2 | 2 | no | — |
| 14 | #2950 | — | 2 | 2 | no | — |
| 15 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 16 | #2166 | — | 2 | 2 | no | — |
| 17 | #2166 | C22 | 1 | 4 | yes | — |
| 18 | #2165 | — | 2 | 2 | no | — |
| 19 | #2165 | C22 | 1 | 4 | yes | — |
| 20 | C2×#364 | — | 2 | 2 | no | — |
| 21 | #2115 | — | 2 | 2 | no | — |
| 22 | #2110 | C22 | 1 | 4 | yes | — |
| 23 | #1356 | C22 | 1 | 16 | yes | — |
| 24 | C22×Q16 | — | 2 | 2 | no | — |
| 25 | C2×#131 | D4 | 1 | 2 | yes | — |
| 26 | #557 | — | 4 | 2 | no | — |
| 27 | #556 | — | 4 | 2 | no | — |
| 28 | #552 | — | 4 | 4 | no | — |
| 29 | #541 | C2×C4 | 1 | 8 | yes | — |
| 30 | #536 | — | 2 | 2 | no | — |
| 31 | #533 | — | 2 | 4 | no | — |
| 32 | #533 | C2×C4 | 1 | 2 | yes | — |
| 33 | C2×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 34 | #519 | — | 2 | 2 | no | — |
| 35 | C4×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×M32 | — | 2 | 4 | no | — |
| 37 | #500 | — | 2 | 1 | no | — |
| 38 | #499 | C2×C4 | 1 | 4 | yes | — |
| 39 | #498 | — | 2 | 1 | no | — |
| 40 | #494 | — | 2 | 1 | no | — |
| 41 | #494 | — | 2 | 1 | no | — |
| 42 | #494 | C2×C4 | 1 | 4 | yes | — |
| 43 | #491 | C2×C4 | 1 | 4 | yes | — |
| 44 | #491 | C23 | 1 | 4 | yes | — |
| 45 | #487 | — | 2 | 4 | no | — |
| 46 | #470 | — | 2 | 4 | no | — |
| 47 | #470 | — | 2 | 4 | no | — |
| 48 | #467 | — | 2 | 4 | no | — |
| 49 | #467 | — | 2 | 4 | no | — |
| 50 | #441 | — | 2 | 4 | no | — |
| 51 | #441 | — | 2 | 4 | no | — |
| 52 | C4×Q16 | — | 2 | 4 | no | — |
| 53 | C4×Q16 | — | 2 | 4 | no | — |
| 54 | #427 | — | 2 | 2 | no | — |
| 55 | #419 | — | 2 | 4 | no | — |
| 56 | #405 | C2×C4 | 1 | 2 | yes | — |
| 57 | #405 | D4 | 1 | 2 | yes | — |
| 58 | #405 | C23 | 1 | 2 | yes | — |
| 59 | C4×M16 | D4 | 1 | 2 | yes | — |
| 60 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 63 | #364 | — | 2 | 4 | no | — |
| 64 | #364 | — | 2 | 4 | no | — |
| 65 | #349 | — | 2 | 4 | no | — |
| 66 | #349 | — | 2 | 4 | no | — |
| 67 | #347 | — | 2 | 4 | no | — |
| 68 | #347 | C23 | 1 | 8 | yes | — |
| 69 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C22×Q8 | — | 2 | 2 | no | — |
| 71 | C2×Q16 | — | 2 | 4 | no | — |
| 72 | C2×Q16 | — | 2 | 4 | no | — |
| 73 | C2×Q16 | — | 4 | 4 | no | — |
| 74 | C2×Q16 | — | 4 | 4 | no | — |
| 75 | #131 | — | 4 | 2 | no | — |
| 76 | #131 | — | 4 | 2 | no | — |
| 77 | C2×M16 | C22⋊C4 | 1 | 2 | yes | — |
| 78 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 79 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 80 | C22×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 81 | C22×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 82 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 83 | C4⋊Q8 | — | 2 | 2 | no | — |
| 84 | C4⋊Q8 | — | 2 | 2 | no | — |
| 85 | C4⋊Q8 | — | 4 | 2 | no | — |
| 86 | C4⋊Q8 | — | 4 | 2 | no | — |
| 87 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 88 | #126 | — | 4 | 8 | no | — |
| 89 | #125 | — | 4 | 4 | no | — |
| 90 | #125 | — | 4 | 8 | no | — |
| 91 | #125 | C22×C4 | 1 | 4 | yes | — |
| 92 | #124 | — | 4 | 2 | no | — |
| 93 | #123 | — | 4 | 4 | no | — |
| 94 | C22⋊Q8 | — | 2 | 8 | no | — |
| 95 | C22⋊Q8 | — | 4 | 4 | no | — |
| 96 | C22⋊Q8 | — | 4 | 2 | no | — |
| 97 | C4×Q8 | — | 2 | 1 | no | — |
| 98 | C4×Q8 | — | 2 | 8 | no | — |
| 99 | C4×Q8 | — | 4 | 4 | no | — |
| 100 | C4×Q8 | — | 4 | 2 | no | — |
| 101 | C4×Q8 | — | 4 | 4 | no | — |
| 102 | C4×D4 | — | 2 | 2 | no | — |
| 103 | C4×D4 | — | 2 | 1 | no | — |
| 104 | #117 | — | 2 | 2 | no | — |
| 105 | #117 | — | 4 | 4 | no | — |
| 106 | #117 | — | 4 | 4 | no | — |
| 107 | #117 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 108 | #117 | C22×C4 | 1 | 2 | yes | — |
| 109 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 110 | C2×C42 | C22⋊C4 | 1 | 2 | yes | — |
| 111 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 112 | M32 | — | 2 | 8 | no | — |
| 113 | M32 | — | 4 | 4 | no | — |
| 114 | C2×C16 | — | 4 | 4 | no | — |
| 115 | #107 | — | 2 | 4 | no | — |
| 116 | #106 | — | 2 | 4 | no | — |
| 117 | #103 | — | 2 | 8 | no | — |
| 118 | #103 | — | 2 | 8 | no | — |
| 119 | #97 | C22⋊C4 | 1 | 2 | yes | — |
| 120 | #97 | C22×C4 | 1 | 1 | yes | characteristic |
| 121 | #97 | C2×D4 | 1 | 2 | yes | — |
| 122 | #97 | C24 | 1 | 1 | yes | characteristic |
| 123 | C4×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 124 | C4×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 125 | C4×C8 | C22⋊C4 | 1 | 2 | yes | — |
| 126 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 127 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 128 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 129 | #41 | — | 4 | 2 | no | — |
| 130 | C2×Q8 | — | 2 | 4 | no | — |
| 131 | C2×Q8 | — | 4 | 2 | no | — |
| 132 | C2×Q8 | — | 4 | 8 | no | — |
| 133 | C2×Q8 | — | 4 | 2 | no | — |
| 134 | C2×Q8 | #100 | 1 | 1 | yes | characteristic |
| 135 | C2×D4 | #100 | 1 | 2 | yes | — |
| 136 | C2×D4 | #100 | 1 | 1 | yes | characteristic |
| 137 | C22×C4 | — | 2 | 2 | no | — |
| 138 | C22×C4 | — | 2 | 1 | no | — |
| 139 | C22×C4 | — | 4 | 2 | no | — |
| 140 | C22×C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 141 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 142 | Q16 | — | 4 | 8 | no | — |
| 143 | Q16 | — | 4 | 4 | no | — |
| 144 | Q16 | — | 4 | 4 | no | — |
| 145 | M16 | — | 4 | 2 | no | — |
| 146 | M16 | — | 4 | 2 | no | — |
| 147 | M16 | — | 4 | 2 | no | — |
| 148 | C2×C8 | — | 2 | 2 | no | — |
| 149 | C2×C8 | — | 2 | 2 | no | — |
| 150 | C2×C8 | — | 2 | 2 | no | — |
| 151 | C2×C8 | — | 2 | 2 | no | — |
| 152 | C2×C8 | — | 2 | 2 | no | — |
| 153 | C2×C8 | — | 2 | 2 | no | — |
| 154 | C2×C8 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 155 | C2×C8 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 156 | C2×C8 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 157 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 158 | C4⋊C4 | — | 2 | 1 | no | — |
| 159 | C4⋊C4 | — | 2 | 2 | no | — |
| 160 | C4⋊C4 | — | 2 | 4 | no | — |
| 161 | C4⋊C4 | — | 4 | 4 | no | — |
| 162 | C4⋊C4 | — | 4 | 4 | no | — |
| 163 | C4⋊C4 | — | 4 | 4 | no | — |
| 164 | C4⋊C4 | — | 4 | 2 | no | — |
| 165 | C4⋊C4 | — | 4 | 8 | no | — |
| 166 | C4⋊C4 | — | 4 | 8 | no | — |
| 167 | C22⋊C4 | — | 2 | 1 | no | — |
| 168 | C22⋊C4 | — | 2 | 2 | no | — |
| 169 | C22⋊C4 | — | 4 | 2 | no | — |
| 170 | C22⋊C4 | — | 4 | 8 | no | — |
| 171 | C42 | — | 2 | 2 | no | — |
| 172 | C42 | — | 2 | 1 | no | — |
| 173 | C42 | — | 4 | 8 | no | — |
| 174 | C42 | — | 4 | 2 | no | — |
| 175 | C42 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 176 | C42 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 177 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 178 | C16 | — | 4 | 8 | no | — |
| 179 | C23 | C2×#100 | 1 | 2 | yes | — |
| 180 | C23 | C2×#100 | 1 | 1 | yes | characteristic |
| 181 | Q8 | — | 4 | 1 | no | — |
| 182 | Q8 | — | 4 | 8 | no | — |
| 183 | Q8 | — | 8 | 4 | no | — |
| 184 | Q8 | — | 8 | 4 | no | — |
| 185 | D4 | — | 4 | 2 | no | — |
| 186 | D4 | — | 4 | 1 | no | — |
| 187 | C2×C4 | — | 2 | 1 | no | — |
| 188 | C2×C4 | — | 2 | 2 | no | — |
| 189 | C2×C4 | — | 2 | 2 | no | — |
| 190 | C2×C4 | — | 2 | 2 | no | — |
| 191 | C2×C4 | — | 2 | 2 | no | — |
| 192 | C2×C4 | — | 2 | 1 | no | — |
| 193 | C2×C4 | — | 4 | 4 | no | — |
| 194 | C2×C4 | — | 4 | 2 | no | — |
| 195 | C2×C4 | — | 4 | 4 | no | — |
| 196 | C2×C4 | — | 4 | 4 | no | — |
| 197 | C2×C4 | — | 8 | 4 | no | — |
| 198 | C2×C4 | C2×#100 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | C2×#100 | 1 | 2 | yes | — |
| 200 | C2×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 201 | C8 | — | 2 | 2 | no | — |
| 202 | C8 | — | 2 | 2 | no | — |
| 203 | C8 | — | 2 | 2 | no | — |
| 204 | C8 | #421 | 1 | 2 | yes | — |
| 205 | C8 | #421 | 1 | 2 | yes | — |
| 206 | C22 | — | 2 | 1 | no | — |
| 207 | C22 | — | 2 | 2 | no | — |
| 208 | C22 | — | 4 | 2 | no | — |
| 209 | C22 | — | 4 | 1 | no | — |
| 210 | C22 | C22×#100 | 1 | 1 | yes | characteristic |
| 211 | C4 | — | 2 | 2 | no | — |
| 212 | C4 | — | 2 | 2 | no | — |
| 213 | C4 | — | 2 | 1 | no | — |
| 214 | C4 | — | 4 | 2 | no | — |
| 215 | C4 | — | 4 | 1 | no | — |
| 216 | C4 | — | 8 | 4 | no | — |
| 217 | C4 | — | 8 | 4 | no | — |
| 218 | C4 | C2×#421 | 1 | 2 | yes | — |
| 219 | C2 | — | 2 | 1 | no | — |
| 220 | C2 | — | 4 | 1 | no | — |
| 221 | C2 | — | 4 | 2 | no | — |
| 222 | C2 | #17675 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17675 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #11916 | C2 | semidirect |
| 4. #11914 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60715 | 2 |
| #33428 | 2 |
| #11916 | 2 |
| #11914 | 2 |