Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
347 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #33981 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33617 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#1417 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13554 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #7476 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#508 | C22 | 1 | 2 | yes | — |
| 7 | C2×#504 | C22 | 1 | 1 | yes | characteristic |
| 8 | #3067 | — | 2 | 2 | no | — |
| 9 | #2264 | — | 2 | 2 | no | — |
| 10 | #2263 | — | 2 | 2 | no | — |
| 11 | #2262 | — | 2 | 2 | no | — |
| 12 | C2×M64 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×C32 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2223 | — | 2 | 4 | no | — |
| 15 | #2222 | — | 2 | 2 | no | — |
| 16 | #2221 | — | 2 | 2 | no | — |
| 17 | C2×#368 | C22 | 1 | 2 | yes | — |
| 18 | #2148 | — | 2 | 2 | no | — |
| 19 | #1417 | — | 2 | 4 | no | — |
| 20 | C22⋊C32 | — | 2 | 2 | no | — |
| 21 | #575 | — | 4 | 2 | no | — |
| 22 | C2×#135 | — | 2 | 2 | no | — |
| 23 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 24 | #508 | — | 2 | 2 | no | — |
| 25 | #508 | — | 2 | 2 | no | — |
| 26 | #508 | — | 2 | 2 | no | — |
| 27 | #508 | — | 2 | 2 | no | — |
| 28 | C2×Q32 | D4 | 1 | 2 | yes | — |
| 29 | C2×SD32 | — | 2 | 2 | no | — |
| 30 | C2×D16 | D4 | 1 | 2 | yes | — |
| 31 | #504 | — | 2 | 2 | no | — |
| 32 | #504 | — | 2 | 2 | no | — |
| 33 | C2×M32 | — | 2 | 1 | no | — |
| 34 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 35 | C22×C16 | — | 2 | 1 | no | — |
| 36 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 37 | #473 | — | 2 | 4 | no | — |
| 38 | #472 | — | 2 | 4 | no | — |
| 39 | #471 | — | 4 | 4 | no | — |
| 40 | C2×#108 | — | 2 | 2 | no | — |
| 41 | #413 | — | 4 | 2 | no | — |
| 42 | Q64 | — | 2 | 2 | no | — |
| 43 | SD64 | — | 2 | 4 | no | — |
| 44 | D32 | — | 2 | 2 | no | — |
| 45 | M64 | — | 2 | 2 | no | — |
| 46 | C2×C32 | — | 2 | 1 | no | — |
| 47 | C2×C32 | — | 2 | 2 | no | — |
| 48 | C2×C32 | D4 | 1 | 2 | yes | — |
| 49 | #368 | — | 2 | 2 | no | — |
| 50 | #368 | D4 | 1 | 4 | yes | — |
| 51 | #362 | — | 2 | 4 | no | — |
| 52 | #361 | — | 2 | 4 | no | — |
| 53 | C2×#41 | — | 4 | 2 | no | — |
| 54 | C2×#41 | D8 | 1 | 1 | yes | characteristic |
| 55 | #137 | — | 4 | 4 | no | — |
| 56 | #137 | — | 8 | 2 | no | — |
| 57 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 58 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 59 | #135 | — | 4 | 2 | no | — |
| 60 | #135 | — | 4 | 2 | no | — |
| 61 | #135 | — | 4 | 2 | no | — |
| 62 | #135 | — | 4 | 2 | no | — |
| 63 | #135 | — | 8 | 4 | no | — |
| 64 | C2×Q16 | — | 2 | 2 | no | — |
| 65 | C2×SD16 | — | 4 | 2 | no | — |
| 66 | C2×D8 | — | 2 | 2 | no | — |
| 67 | #131 | — | 2 | 2 | no | — |
| 68 | #131 | — | 2 | 2 | no | — |
| 69 | C2×M16 | — | 2 | 1 | no | — |
| 70 | C2×M16 | — | 4 | 2 | no | — |
| 71 | C2×M16 | D8 | 1 | 1 | yes | characteristic |
| 72 | C22×C8 | — | 2 | 1 | no | — |
| 73 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | Q32 | — | 2 | 2 | no | — |
| 75 | Q32 | — | 2 | 2 | no | — |
| 76 | SD32 | — | 4 | 2 | no | — |
| 77 | SD32 | — | 4 | 2 | no | — |
| 78 | D16 | — | 2 | 2 | no | — |
| 79 | D16 | — | 2 | 2 | no | — |
| 80 | M32 | — | 2 | 2 | no | — |
| 81 | M32 | — | 4 | 2 | no | — |
| 82 | C2×C16 | — | 2 | 1 | no | — |
| 83 | C2×C16 | — | 4 | 1 | no | — |
| 84 | C2×C16 | — | 4 | 1 | no | — |
| 85 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 86 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 87 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 88 | C2×C16 | #41 | 1 | 1 | yes | characteristic |
| 89 | #108 | — | 2 | 4 | no | — |
| 90 | #108 | — | 4 | 2 | no | — |
| 91 | #101 | — | 4 | 4 | no | — |
| 92 | #100 | — | 4 | 4 | no | — |
| 93 | C32 | — | 2 | 2 | no | — |
| 94 | C32 | — | 4 | 1 | no | — |
| 95 | #41 | — | 2 | 2 | no | — |
| 96 | #41 | — | 2 | 2 | no | — |
| 97 | #41 | — | 8 | 2 | no | — |
| 98 | #41 | — | 8 | 2 | no | — |
| 99 | #41 | — | 8 | 2 | no | — |
| 100 | #41 | — | 8 | 2 | no | — |
| 101 | C2×Q8 | — | 4 | 2 | no | — |
| 102 | C2×Q8 | D16 | 1 | 1 | yes | characteristic |
| 103 | C2×D4 | — | 2 | 1 | no | — |
| 104 | C2×D4 | — | 4 | 2 | no | — |
| 105 | C2×D4 | — | 8 | 2 | no | — |
| 106 | C2×D4 | D16 | 1 | 1 | yes | characteristic |
| 107 | C22×C4 | — | 2 | 1 | no | — |
| 108 | C22×C4 | — | 8 | 2 | no | — |
| 109 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 110 | Q16 | — | 4 | 2 | no | — |
| 111 | Q16 | — | 4 | 2 | no | — |
| 112 | Q16 | — | 8 | 4 | no | — |
| 113 | SD16 | — | 8 | 4 | no | — |
| 114 | SD16 | — | 8 | 4 | no | — |
| 115 | SD16 | — | 8 | 2 | no | — |
| 116 | SD16 | — | 8 | 2 | no | — |
| 117 | D8 | — | 4 | 2 | no | — |
| 118 | D8 | — | 4 | 2 | no | — |
| 119 | D8 | — | 8 | 4 | no | — |
| 120 | M16 | — | 2 | 2 | no | — |
| 121 | M16 | — | 4 | 2 | no | — |
| 122 | M16 | — | 4 | 4 | no | — |
| 123 | M16 | — | 8 | 2 | no | — |
| 124 | C2×C8 | — | 2 | 1 | no | — |
| 125 | C2×C8 | — | 4 | 1 | no | — |
| 126 | C2×C8 | — | 4 | 1 | no | — |
| 127 | C2×C8 | — | 8 | 2 | no | — |
| 128 | C2×C8 | #120 | 1 | 2 | yes | — |
| 129 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 130 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 131 | C2×C8 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 132 | C16 | — | 4 | 1 | no | — |
| 133 | C16 | #120 | 1 | 1 | yes | characteristic |
| 134 | C16 | #121 | 1 | 2 | yes | — |
| 135 | C16 | #127 | 1 | 1 | yes | characteristic |
| 136 | C23 | — | 2 | 1 | no | — |
| 137 | C23 | — | 8 | 2 | no | — |
| 138 | C23 | #508 | 1 | 1 | yes | characteristic |
| 139 | Q8 | — | 2 | 2 | no | — |
| 140 | Q8 | — | 8 | 2 | no | — |
| 141 | Q8 | — | 8 | 2 | no | — |
| 142 | D4 | — | 2 | 2 | no | — |
| 143 | D4 | — | 4 | 2 | no | — |
| 144 | D4 | — | 8 | 2 | no | — |
| 145 | D4 | — | 8 | 2 | no | — |
| 146 | D4 | — | 16 | 2 | no | — |
| 147 | D4 | — | 16 | 2 | no | — |
| 148 | C2×C4 | — | 2 | 1 | no | — |
| 149 | C2×C4 | — | 4 | 1 | no | — |
| 150 | C2×C4 | — | 4 | 1 | no | — |
| 151 | C2×C4 | — | 8 | 2 | no | — |
| 152 | C2×C4 | — | 16 | 2 | no | — |
| 153 | C2×C4 | — | 16 | 2 | no | — |
| 154 | C2×C4 | #447 | 1 | 2 | yes | — |
| 155 | C2×C4 | #466 | 1 | 1 | yes | characteristic |
| 156 | C2×C4 | C2×D16 | 1 | 1 | yes | characteristic |
| 157 | C2×C4 | #509 | 1 | 2 | yes | — |
| 158 | C8 | — | 4 | 1 | no | — |
| 159 | C8 | — | 8 | 4 | no | — |
| 160 | C8 | #392 | 1 | 1 | yes | characteristic |
| 161 | C8 | #449 | 1 | 2 | yes | — |
| 162 | C8 | #469 | 1 | 1 | yes | characteristic |
| 163 | C22 | — | 4 | 1 | no | — |
| 164 | C22 | — | 4 | 2 | no | — |
| 165 | C22 | — | 4 | 1 | no | — |
| 166 | C22 | — | 16 | 2 | no | — |
| 167 | C22 | — | 32 | 2 | no | — |
| 168 | C22 | #2186 | 1 | 2 | yes | — |
| 169 | C22 | #2215 | 1 | 1 | yes | characteristic |
| 170 | C4 | — | 4 | 1 | no | — |
| 171 | C4 | — | 16 | 2 | no | — |
| 172 | C4 | #1999 | 1 | 1 | yes | characteristic |
| 173 | C4 | #2184 | 1 | 2 | yes | — |
| 174 | C4 | #2218 | 1 | 1 | yes | characteristic |
| 175 | C2 | — | 2 | 1 | no | — |
| 176 | C2 | — | 2 | 2 | no | — |
| 177 | C2 | — | 8 | 1 | no | — |
| 178 | C2 | — | 32 | 2 | no | — |
| 179 | C2 | #13304 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2218 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13304 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33981 | C2 | semidirect |
| 2. #33617 | C2 | semidirect |
| 3. C2×#1417 | C2 | semidirect |
| 4. #13554 | C2 | semidirect |
| 5. #7476 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33981 | 2 |
| #33617 | 2 |
| C2×#1417 | 2 |
| #13554 | 2 |
| #7476 | 2 |