Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
330 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 68 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 | #21708 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11485 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #7374 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #7356 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #3150 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×#443 | — | 2 | 2 | no | — |
| 7 | #2931 | C4 | 1 | 1 | yes | characteristic |
| 8 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2115 | — | 2 | 2 | no | — |
| 10 | #1975 | — | 2 | 2 | no | — |
| 11 | #1963 | C4 | 1 | 1 | yes | characteristic |
| 12 | #1963 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1726 | C4 | 1 | 1 | yes | characteristic |
| 15 | #1327 | C22 | 1 | 4 | yes | — |
| 16 | #1321 | — | 2 | 2 | no | — |
| 17 | C2×#135 | — | 2 | 2 | no | — |
| 18 | C2×#131 | — | 2 | 1 | no | — |
| 19 | C2×#131 | D4 | 1 | 2 | yes | — |
| 20 | #532 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | #532 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C2×M32 | — | 4 | 2 | no | — |
| 23 | #497 | — | 2 | 1 | no | — |
| 24 | #497 | C8 | 1 | 2 | yes | — |
| 25 | #496 | — | 2 | 1 | no | — |
| 26 | #496 | C8 | 1 | 2 | yes | — |
| 27 | #443 | — | 4 | 4 | no | — |
| 28 | #443 | — | 4 | 4 | no | — |
| 29 | #436 | — | 4 | 1 | no | — |
| 30 | #436 | — | 4 | 1 | no | — |
| 31 | C8×D4 | — | 4 | 1 | no | — |
| 32 | C8×D4 | — | 4 | 1 | no | — |
| 33 | C2×#108 | — | 2 | 2 | no | — |
| 34 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | #423 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×#104 | — | 2 | 2 | no | — |
| 37 | C2×#104 | — | 2 | 2 | no | — |
| 38 | #408 | — | 2 | 1 | no | — |
| 39 | #408 | D4 | 1 | 2 | yes | — |
| 40 | C2×#97 | — | 2 | 1 | no | — |
| 41 | C2×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C4×C8 | — | 2 | 1 | no | — |
| 43 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | #376 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | #349 | — | 2 | 4 | no | — |
| 46 | #349 | — | 4 | 2 | no | — |
| 47 | C2×#41 | — | 2 | 1 | no | — |
| 48 | C2×#41 | C22⋊C4 | 1 | 2 | yes | — |
| 49 | #135 | — | 4 | 4 | no | — |
| 50 | #135 | — | 4 | 4 | no | — |
| 51 | C2×Q16 | — | 2 | 2 | no | — |
| 52 | C2×SD16 | — | 2 | 2 | no | — |
| 53 | C2×SD16 | — | 2 | 2 | no | — |
| 54 | C2×D8 | — | 2 | 2 | no | — |
| 55 | #131 | — | 4 | 2 | no | — |
| 56 | #131 | — | 4 | 2 | no | — |
| 57 | #131 | — | 4 | 2 | no | — |
| 58 | #131 | — | 4 | 2 | no | — |
| 59 | C2×M16 | — | 2 | 2 | no | — |
| 60 | C2×M16 | — | 2 | 1 | no | — |
| 61 | C2×M16 | — | 4 | 1 | no | — |
| 62 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 63 | C22×C8 | — | 2 | 1 | no | — |
| 64 | C22×C8 | — | 2 | 2 | no | — |
| 65 | C22×C8 | — | 4 | 1 | no | — |
| 66 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C4⋊Q8 | C2×C8 | 1 | 1 | yes | characteristic |
| 69 | C4⋊Q8 | M16 | 1 | 1 | yes | characteristic |
| 70 | #127 | C2×C8 | 1 | 1 | yes | characteristic |
| 71 | #127 | M16 | 1 | 1 | yes | characteristic |
| 72 | #124 | — | 2 | 1 | no | — |
| 73 | #124 | C2×C8 | 1 | 2 | yes | — |
| 74 | #121 | — | 4 | 1 | no | — |
| 75 | #121 | — | 4 | 1 | no | — |
| 76 | C4×D4 | — | 4 | 1 | no | — |
| 77 | C4×D4 | — | 4 | 1 | no | — |
| 78 | C2×C42 | — | 2 | 1 | no | — |
| 79 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 80 | M32 | — | 4 | 4 | no | — |
| 81 | M32 | — | 8 | 2 | no | — |
| 82 | C2×C16 | — | 8 | 2 | no | — |
| 83 | #108 | — | 4 | 4 | no | — |
| 84 | C4⋊C8 | — | 4 | 1 | no | — |
| 85 | C4⋊C8 | — | 4 | 1 | no | — |
| 86 | #104 | — | 4 | 4 | no | — |
| 87 | #104 | — | 4 | 4 | no | — |
| 88 | C22⋊C8 | — | 4 | 2 | no | — |
| 89 | C22⋊C8 | — | 4 | 2 | no | — |
| 90 | #97 | — | 2 | 1 | no | — |
| 91 | #97 | — | 4 | 2 | no | — |
| 92 | #97 | C2×C8 | 1 | 2 | yes | — |
| 93 | C4×C8 | — | 2 | 2 | no | — |
| 94 | C4×C8 | — | 2 | 2 | no | — |
| 95 | C4×C8 | — | 2 | 1 | no | — |
| 96 | C4×C8 | — | 2 | 1 | no | — |
| 97 | #95 | — | 4 | 1 | no | — |
| 98 | #41 | — | 4 | 2 | no | — |
| 99 | #41 | — | 4 | 2 | no | — |
| 100 | #41 | — | 4 | 2 | no | — |
| 101 | #41 | — | 4 | 2 | no | — |
| 102 | C2×Q8 | — | 2 | 1 | no | — |
| 103 | C2×Q8 | C22⋊C8 | 1 | 2 | yes | — |
| 104 | C2×D4 | — | 2 | 2 | no | — |
| 105 | C2×D4 | — | 2 | 1 | no | — |
| 106 | C2×D4 | — | 4 | 1 | no | — |
| 107 | C2×D4 | — | 4 | 1 | no | — |
| 108 | C2×D4 | — | 4 | 1 | no | — |
| 109 | C2×D4 | C22⋊C8 | 1 | 2 | yes | — |
| 110 | C22×C4 | — | 2 | 1 | no | — |
| 111 | C22×C4 | — | 2 | 2 | no | — |
| 112 | C22×C4 | — | 4 | 1 | no | — |
| 113 | C22×C4 | — | 4 | 1 | no | — |
| 114 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | Q16 | — | 4 | 4 | no | — |
| 116 | SD16 | — | 4 | 4 | no | — |
| 117 | SD16 | — | 4 | 4 | no | — |
| 118 | D8 | — | 4 | 4 | no | — |
| 119 | M16 | — | 4 | 2 | no | — |
| 120 | M16 | — | 4 | 2 | no | — |
| 121 | M16 | — | 8 | 2 | no | — |
| 122 | M16 | — | 8 | 2 | no | — |
| 123 | C2×C8 | — | 2 | 2 | no | — |
| 124 | C2×C8 | — | 2 | 2 | no | — |
| 125 | C2×C8 | — | 2 | 2 | no | — |
| 126 | C2×C8 | — | 2 | 1 | no | — |
| 127 | C2×C8 | — | 2 | 1 | no | — |
| 128 | C2×C8 | — | 2 | 1 | no | — |
| 129 | C2×C8 | — | 2 | 2 | no | — |
| 130 | C2×C8 | — | 4 | 2 | no | — |
| 131 | C2×C8 | — | 4 | 2 | no | — |
| 132 | C2×C8 | — | 4 | 1 | no | — |
| 133 | C2×C8 | — | 4 | 1 | no | — |
| 134 | C2×C8 | — | 8 | 1 | no | — |
| 135 | C2×C8 | — | 8 | 1 | no | — |
| 136 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 137 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 138 | C4⋊C4 | — | 4 | 1 | no | — |
| 139 | C4⋊C4 | — | 4 | 1 | no | — |
| 140 | C22⋊C4 | — | 4 | 2 | no | — |
| 141 | C22⋊C4 | — | 4 | 2 | no | — |
| 142 | C42 | — | 2 | 1 | no | — |
| 143 | C42 | — | 2 | 2 | no | — |
| 144 | C42 | — | 4 | 1 | no | — |
| 145 | C42 | C22×C8 | 1 | 1 | yes | characteristic |
| 146 | C42 | C2×M16 | 1 | 1 | yes | characteristic |
| 147 | C16 | — | 8 | 4 | no | — |
| 148 | C23 | — | 2 | 2 | no | — |
| 149 | C23 | — | 4 | 1 | no | — |
| 150 | C23 | #413 | 1 | 1 | yes | characteristic |
| 151 | Q8 | — | 4 | 2 | no | — |
| 152 | Q8 | — | 4 | 2 | no | — |
| 153 | D4 | — | 4 | 2 | no | — |
| 154 | D4 | — | 4 | 2 | no | — |
| 155 | D4 | — | 8 | 2 | no | — |
| 156 | D4 | — | 8 | 2 | no | — |
| 157 | D4 | — | 8 | 2 | no | — |
| 158 | D4 | — | 8 | 2 | no | — |
| 159 | C2×C4 | — | 2 | 1 | no | — |
| 160 | C2×C4 | — | 2 | 1 | no | — |
| 161 | C2×C4 | — | 2 | 1 | no | — |
| 162 | C2×C4 | — | 2 | 1 | no | — |
| 163 | C2×C4 | — | 2 | 2 | no | — |
| 164 | C2×C4 | — | 4 | 2 | no | — |
| 165 | C2×C4 | — | 4 | 2 | no | — |
| 166 | C2×C4 | — | 4 | 2 | no | — |
| 167 | C2×C4 | — | 4 | 1 | no | — |
| 168 | C2×C4 | — | 8 | 1 | no | — |
| 169 | C2×C4 | — | 8 | 2 | no | — |
| 170 | C2×C4 | — | 8 | 1 | no | — |
| 171 | C2×C4 | — | 8 | 1 | no | — |
| 172 | C2×C4 | C2×C22⋊C8 | 1 | 1 | yes | characteristic |
| 173 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 174 | C8 | — | 2 | 4 | no | — |
| 175 | C8 | — | 2 | 1 | no | — |
| 176 | C8 | — | 4 | 2 | no | — |
| 177 | C8 | — | 4 | 1 | no | — |
| 178 | C8 | — | 8 | 1 | no | — |
| 179 | C8 | #353 | 1 | 1 | yes | characteristic |
| 180 | C8 | #353 | 1 | 1 | yes | characteristic |
| 181 | C22 | — | 2 | 1 | no | — |
| 182 | C22 | — | 4 | 1 | no | — |
| 183 | C22 | — | 4 | 2 | no | — |
| 184 | C22 | — | 8 | 2 | no | — |
| 185 | C22 | — | 8 | 2 | no | — |
| 186 | C22 | — | 8 | 1 | no | — |
| 187 | C22 | #1459 | 1 | 1 | yes | characteristic |
| 188 | C4 | — | 2 | 1 | no | — |
| 189 | C4 | — | 4 | 1 | no | — |
| 190 | C4 | — | 4 | 1 | no | — |
| 191 | C4 | — | 4 | 2 | no | — |
| 192 | C4 | — | 8 | 2 | no | — |
| 193 | C4 | — | 8 | 1 | no | — |
| 194 | C4 | C2×#353 | 1 | 1 | yes | characteristic |
| 195 | C4 | #2132 | 1 | 1 | yes | characteristic |
| 196 | C2 | — | 2 | 1 | no | — |
| 197 | C2 | — | 4 | 1 | no | — |
| 198 | C2 | — | 8 | 2 | no | — |
| 199 | C2 | — | 16 | 1 | no | — |
| 200 | C2 | #11475 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | #11475 |
| prime radical | #3150 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11485 | C2 | semidirect |
| 3. #7374 | C2 | semidirect |
| 4. #7356 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21708 | 2 |
| #11485 | 2 |
| #7374 | 2 |
| #7356 | 2 |