Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
255 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 255 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #28734 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12807 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11485 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7410 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#499 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #3142 | — | 2 | 1 | no | — |
| 7 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 8 | #2126 | — | 2 | 1 | no | — |
| 9 | #2115 | — | 2 | 1 | no | — |
| 10 | #2079 | C4 | 1 | 2 | yes | complemented |
| 11 | #2079 | C22 | 1 | 2 | yes | — |
| 12 | #1998 | — | 2 | 1 | no | — |
| 13 | #1816 | — | 2 | 1 | no | — |
| 14 | #1816 | — | 2 | 1 | no | — |
| 15 | #1812 | — | 2 | 1 | no | — |
| 16 | C4×#108 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #1391 | C22 | 1 | 2 | yes | — |
| 18 | #1327 | — | 2 | 1 | no | — |
| 19 | #1321 | C22 | 1 | 2 | yes | — |
| 20 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 22 | #532 | D4 | 1 | 1 | yes | characteristic |
| 23 | C2×#125 | — | 2 | 1 | no | — |
| 24 | C2×M32 | — | 4 | 1 | no | — |
| 25 | #499 | — | 4 | 1 | no | — |
| 26 | #499 | — | 4 | 1 | no | — |
| 27 | #491 | — | 4 | 1 | no | — |
| 28 | #490 | — | 4 | 1 | no | — |
| 29 | #488 | — | 4 | 1 | no | — |
| 30 | #488 | — | 4 | 1 | no | — |
| 31 | #436 | — | 4 | 1 | no | — |
| 32 | C8×D4 | — | 4 | 1 | no | — |
| 33 | C2×#108 | — | 2 | 1 | no | — |
| 34 | #427 | — | 4 | 1 | no | — |
| 35 | C2×#107 | — | 2 | 1 | no | — |
| 36 | C2×#106 | C2×C4 | 1 | 2 | yes | — |
| 37 | #423 | D4 | 1 | 1 | yes | characteristic |
| 38 | #417 | — | 2 | 2 | no | — |
| 39 | #417 | — | 2 | 2 | no | — |
| 40 | #410 | — | 2 | 2 | no | — |
| 41 | #408 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | #408 | C23 | 1 | 1 | yes | characteristic |
| 43 | C4×M16 | — | 2 | 1 | no | — |
| 44 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | #354 | — | 4 | 1 | no | — |
| 46 | #353 | — | 4 | 1 | no | — |
| 47 | #349 | — | 2 | 2 | no | — |
| 48 | #349 | — | 4 | 1 | no | — |
| 49 | #343 | — | 2 | 2 | no | — |
| 50 | #343 | — | 2 | 2 | no | — |
| 51 | #341 | C2×C4 | 1 | 2 | yes | — |
| 52 | H(3,ℤ/4) | — | 4 | 1 | no | — |
| 53 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 55 | #131 | — | 4 | 1 | no | — |
| 56 | #131 | — | 4 | 1 | no | — |
| 57 | #131 | — | 4 | 1 | no | — |
| 58 | #131 | — | 4 | 1 | no | — |
| 59 | C2×M16 | — | 2 | 1 | no | — |
| 60 | C2×M16 | — | 2 | 1 | no | — |
| 61 | C2×M16 | — | 2 | 2 | no | — |
| 62 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C22×C8 | — | 2 | 1 | no | — |
| 65 | C22×C8 | — | 2 | 1 | no | — |
| 66 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C4⋊Q8 | SD16 | 1 | 1 | yes | characteristic |
| 69 | #127 | D8 | 1 | 1 | yes | characteristic |
| 70 | #125 | — | 4 | 2 | no | — |
| 71 | #125 | — | 4 | 1 | no | — |
| 72 | #125 | — | 4 | 1 | no | — |
| 73 | #124 | D8 | 1 | 1 | yes | characteristic |
| 74 | #124 | SD16 | 1 | 1 | yes | characteristic |
| 75 | #121 | — | 4 | 1 | no | — |
| 76 | C4×D4 | — | 4 | 1 | no | — |
| 77 | #117 | — | 4 | 2 | no | — |
| 78 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 79 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 80 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 81 | M32 | — | 4 | 2 | no | — |
| 82 | M32 | — | 8 | 1 | no | — |
| 83 | C2×C16 | — | 8 | 1 | no | — |
| 84 | #108 | — | 4 | 1 | no | — |
| 85 | #108 | — | 4 | 1 | no | — |
| 86 | #107 | — | 4 | 1 | no | — |
| 87 | #107 | — | 4 | 1 | no | — |
| 88 | #107 | — | 8 | 1 | no | — |
| 89 | #106 | — | 4 | 2 | no | — |
| 90 | #106 | — | 8 | 1 | no | — |
| 91 | C4⋊C8 | — | 4 | 1 | no | — |
| 92 | #103 | — | 4 | 2 | no | — |
| 93 | #103 | — | 4 | 2 | no | — |
| 94 | #102 | — | 4 | 2 | no | — |
| 95 | #102 | — | 4 | 2 | no | — |
| 96 | #99 | — | 4 | 2 | no | — |
| 97 | #99 | — | 4 | 2 | no | — |
| 98 | C22⋊C8 | — | 4 | 1 | no | — |
| 99 | C22⋊C8 | — | 4 | 1 | no | — |
| 100 | #97 | — | 4 | 1 | no | — |
| 101 | C4×C8 | — | 2 | 1 | no | — |
| 102 | C4×C8 | — | 2 | 1 | no | — |
| 103 | C4×C8 | — | 4 | 1 | no | — |
| 104 | #41 | — | 4 | 1 | no | — |
| 105 | #41 | — | 4 | 1 | no | — |
| 106 | #41 | — | 4 | 1 | no | — |
| 107 | #41 | — | 4 | 1 | no | — |
| 108 | C2×Q8 | #102 | 1 | 1 | yes | characteristic |
| 109 | C2×Q8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 110 | C2×D4 | — | 2 | 1 | no | — |
| 111 | C2×D4 | — | 2 | 1 | no | — |
| 112 | C2×D4 | — | 4 | 1 | no | — |
| 113 | C2×D4 | #102 | 1 | 1 | yes | characteristic |
| 114 | C2×D4 | C2×D8 | 1 | 1 | yes | characteristic |
| 115 | C22×C4 | — | 2 | 1 | no | — |
| 116 | C22×C4 | — | 2 | 1 | no | — |
| 117 | C22×C4 | — | 2 | 1 | no | — |
| 118 | C22×C4 | — | 4 | 2 | no | — |
| 119 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | M16 | — | 4 | 1 | no | — |
| 121 | M16 | — | 4 | 1 | no | — |
| 122 | M16 | — | 4 | 2 | no | — |
| 123 | M16 | — | 4 | 2 | no | — |
| 124 | M16 | — | 8 | 1 | no | — |
| 125 | M16 | — | 8 | 1 | no | — |
| 126 | C2×C8 | — | 2 | 1 | no | — |
| 127 | C2×C8 | — | 2 | 1 | no | — |
| 128 | C2×C8 | — | 2 | 1 | no | — |
| 129 | C2×C8 | — | 2 | 1 | no | — |
| 130 | C2×C8 | — | 2 | 1 | no | — |
| 131 | C2×C8 | — | 4 | 1 | no | — |
| 132 | C2×C8 | — | 4 | 1 | no | — |
| 133 | C2×C8 | — | 4 | 1 | no | — |
| 134 | C2×C8 | — | 4 | 1 | no | — |
| 135 | C2×C8 | — | 4 | 1 | no | — |
| 136 | C2×C8 | — | 4 | 2 | no | — |
| 137 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 138 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 139 | C4⋊C4 | — | 4 | 1 | no | — |
| 140 | C4⋊C4 | — | 4 | 2 | no | — |
| 141 | C4⋊C4 | — | 4 | 2 | no | — |
| 142 | C4⋊C4 | — | 8 | 2 | no | — |
| 143 | C4⋊C4 | — | 8 | 2 | no | — |
| 144 | C4⋊C4 | — | 8 | 2 | no | — |
| 145 | C22⋊C4 | — | 4 | 1 | no | — |
| 146 | C22⋊C4 | — | 4 | 1 | no | — |
| 147 | C22⋊C4 | — | 8 | 2 | no | — |
| 148 | C42 | — | 2 | 1 | no | — |
| 149 | C42 | — | 8 | 2 | no | — |
| 150 | C42 | #102 | 1 | 1 | yes | characteristic |
| 151 | C42 | #102 | 1 | 1 | yes | characteristic |
| 152 | C16 | — | 8 | 2 | no | — |
| 153 | C23 | — | 2 | 1 | no | — |
| 154 | C23 | — | 2 | 1 | no | — |
| 155 | C23 | #421 | 1 | 1 | yes | characteristic |
| 156 | Q8 | — | 4 | 1 | no | — |
| 157 | Q8 | — | 4 | 1 | no | — |
| 158 | D4 | — | 4 | 1 | no | — |
| 159 | D4 | — | 4 | 1 | no | — |
| 160 | D4 | — | 8 | 1 | no | — |
| 161 | D4 | — | 8 | 1 | no | — |
| 162 | D4 | — | 8 | 2 | no | — |
| 163 | C2×C4 | — | 2 | 1 | no | — |
| 164 | C2×C4 | — | 2 | 1 | no | — |
| 165 | C2×C4 | — | 2 | 1 | no | — |
| 166 | C2×C4 | — | 2 | 1 | no | — |
| 167 | C2×C4 | — | 2 | 1 | no | — |
| 168 | C2×C4 | — | 2 | 1 | no | — |
| 169 | C2×C4 | — | 4 | 1 | no | — |
| 170 | C2×C4 | — | 4 | 1 | no | — |
| 171 | C2×C4 | — | 4 | 1 | no | — |
| 172 | C2×C4 | — | 4 | 1 | no | — |
| 173 | C2×C4 | — | 8 | 1 | no | — |
| 174 | C2×C4 | — | 8 | 2 | no | — |
| 175 | C2×C4 | — | 16 | 2 | no | — |
| 176 | C2×C4 | — | 16 | 2 | no | — |
| 177 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 178 | C2×C4 | C2×#102 | 1 | 1 | yes | characteristic |
| 179 | C8 | — | 2 | 1 | no | — |
| 180 | C8 | — | 4 | 1 | no | — |
| 181 | C8 | — | 4 | 1 | no | — |
| 182 | C8 | — | 4 | 1 | no | — |
| 183 | C8 | — | 8 | 2 | no | — |
| 184 | C8 | #354 | 1 | 1 | yes | characteristic |
| 185 | C8 | #354 | 1 | 1 | yes | characteristic |
| 186 | C22 | — | 2 | 1 | no | — |
| 187 | C22 | — | 4 | 1 | no | — |
| 188 | C22 | — | 4 | 1 | no | — |
| 189 | C22 | — | 4 | 1 | no | — |
| 190 | C22 | — | 8 | 1 | no | — |
| 191 | C22 | — | 8 | 1 | no | — |
| 192 | C22 | #1519 | 1 | 1 | yes | characteristic |
| 193 | C4 | — | 2 | 1 | no | — |
| 194 | C4 | — | 4 | 1 | no | — |
| 195 | C4 | — | 4 | 1 | no | — |
| 196 | C4 | — | 4 | 1 | no | — |
| 197 | C4 | — | 4 | 1 | no | — |
| 198 | C4 | — | 16 | 2 | no | — |
| 199 | C4 | — | 32 | 2 | no | — |
| 200 | C4 | #2123 | 1 | 1 | yes | characteristic |
| 201 | C4 | C2×#354 | 1 | 1 | yes | characteristic |
| 202 | C2 | — | 2 | 1 | no | — |
| 203 | C2 | — | 4 | 1 | no | — |
| 204 | C2 | — | 8 | 1 | no | — |
| 205 | C2 | — | 8 | 1 | no | — |
| 206 | C2 | #11878 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11878 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | #408 | C23 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7410 | C2 | semidirect |
| 5. C2×#499 | C4 | semidirect |
| 10. #2079 | C4 | semidirect |
| 16. C4×#108 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28734 | 2 |
| #12807 | 2 |
| #11485 | 2 |
| #7410 | 2 |