Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
242 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 56 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 | #21663 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11539 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #7369 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #7356 | C2 | 1 | 2 | yes | maximal |
| 5 | #3135 | C4 | 1 | 1 | yes | characteristic |
| 6 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2931 | C4 | 1 | 1 | yes | characteristic |
| 8 | #2166 | — | 2 | 2 | no | — |
| 9 | #1995 | C4 | 1 | 1 | yes | characteristic |
| 10 | #1963 | C22 | 1 | 1 | yes | characteristic |
| 11 | #1937 | — | 2 | 2 | no | — |
| 12 | #1838 | C22 | 1 | 1 | yes | characteristic |
| 13 | #1765 | — | 2 | 2 | no | — |
| 14 | #1726 | C4 | 1 | 1 | yes | characteristic |
| 15 | #1328 | — | 2 | 2 | no | — |
| 16 | #1327 | C22 | 1 | 4 | yes | — |
| 17 | C2×#131 | — | 2 | 1 | no | — |
| 18 | #533 | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | #532 | C2×C4 | 1 | 1 | yes | characteristic |
| 20 | #491 | — | 2 | 1 | no | — |
| 21 | #490 | — | 2 | 1 | no | — |
| 22 | #489 | — | 2 | 2 | no | — |
| 23 | #446 | — | 4 | 1 | no | — |
| 24 | C8×Q8 | — | 4 | 1 | no | — |
| 25 | #436 | — | 4 | 1 | no | — |
| 26 | C8×D4 | — | 4 | 1 | no | — |
| 27 | #424 | D4 | 1 | 2 | yes | — |
| 28 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | #423 | C23 | 1 | 1 | yes | characteristic |
| 30 | C2×#104 | — | 2 | 2 | no | — |
| 31 | #416 | — | 2 | 2 | no | — |
| 32 | #408 | — | 2 | 1 | no | — |
| 33 | #405 | D4 | 1 | 2 | yes | — |
| 34 | C2×#97 | — | 2 | 1 | no | — |
| 35 | C2×#97 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | C2×C4×C8 | — | 2 | 1 | no | — |
| 37 | C2×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #376 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | #364 | — | 4 | 2 | no | — |
| 40 | #349 | — | 2 | 4 | no | — |
| 41 | #347 | — | 4 | 2 | no | — |
| 42 | #341 | — | 2 | 2 | no | — |
| 43 | #339 | — | 2 | 2 | no | — |
| 44 | C2×#41 | — | 2 | 1 | no | — |
| 45 | #131 | — | 4 | 4 | no | — |
| 46 | C2×M16 | — | 2 | 1 | no | — |
| 47 | C2×M16 | — | 4 | 1 | no | — |
| 48 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 49 | C22×C8 | — | 2 | 1 | no | — |
| 50 | C22×C8 | — | 4 | 1 | no | — |
| 51 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 52 | C22×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 53 | C4⋊Q8 | — | 2 | 1 | no | — |
| 54 | C4⋊Q8 | M16 | 1 | 1 | yes | characteristic |
| 55 | #127 | M16 | 1 | 1 | yes | characteristic |
| 56 | #125 | — | 2 | 1 | no | — |
| 57 | #124 | — | 2 | 1 | no | — |
| 58 | C22⋊Q8 | — | 4 | 1 | no | — |
| 59 | #121 | — | 4 | 1 | no | — |
| 60 | C4×Q8 | — | 4 | 1 | no | — |
| 61 | C4×D4 | — | 4 | 1 | no | — |
| 62 | #117 | C22⋊C4 | 1 | 2 | yes | — |
| 63 | C2×C42 | — | 2 | 1 | no | — |
| 64 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 65 | M32 | — | 4 | 4 | no | — |
| 66 | C4⋊C8 | — | 4 | 1 | no | — |
| 67 | C4⋊C8 | — | 4 | 2 | no | — |
| 68 | C4⋊C8 | — | 4 | 1 | no | — |
| 69 | #104 | — | 4 | 4 | no | — |
| 70 | #103 | — | 2 | 4 | no | — |
| 71 | #102 | — | 2 | 4 | no | — |
| 72 | C22⋊C8 | — | 4 | 2 | no | — |
| 73 | #97 | — | 2 | 2 | no | — |
| 74 | #97 | — | 2 | 1 | no | — |
| 75 | #97 | — | 4 | 2 | no | — |
| 76 | #97 | M16 | 1 | 2 | yes | — |
| 77 | C4×C8 | — | 2 | 2 | no | — |
| 78 | C4×C8 | — | 2 | 1 | no | — |
| 79 | C4×C8 | — | 2 | 1 | no | — |
| 80 | C4×C8 | — | 4 | 2 | no | — |
| 81 | #95 | — | 4 | 1 | no | — |
| 82 | #41 | — | 4 | 2 | no | — |
| 83 | #41 | — | 4 | 2 | no | — |
| 84 | C2×Q8 | — | 2 | 1 | no | — |
| 85 | C2×Q8 | — | 4 | 1 | no | — |
| 86 | C2×D4 | — | 2 | 1 | no | — |
| 87 | C2×D4 | — | 4 | 1 | no | — |
| 88 | C2×D4 | — | 4 | 1 | no | — |
| 89 | C22×C4 | — | 2 | 1 | no | — |
| 90 | C22×C4 | — | 4 | 1 | no | — |
| 91 | C22×C4 | — | 4 | 1 | no | — |
| 92 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 93 | M16 | — | 4 | 2 | no | — |
| 94 | M16 | — | 4 | 2 | no | — |
| 95 | M16 | — | 8 | 2 | no | — |
| 96 | C2×C8 | — | 2 | 2 | no | — |
| 97 | C2×C8 | — | 2 | 2 | no | — |
| 98 | C2×C8 | — | 2 | 1 | no | — |
| 99 | C2×C8 | — | 2 | 2 | no | — |
| 100 | C2×C8 | — | 4 | 1 | no | — |
| 101 | C2×C8 | — | 4 | 1 | no | — |
| 102 | C2×C8 | — | 4 | 1 | no | — |
| 103 | C2×C8 | — | 4 | 1 | no | — |
| 104 | C2×C8 | — | 8 | 2 | no | — |
| 105 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 106 | C2×C8 | #99 | 1 | 1 | yes | characteristic |
| 107 | C4⋊C4 | — | 2 | 2 | no | — |
| 108 | C4⋊C4 | — | 4 | 1 | no | — |
| 109 | C4⋊C4 | — | 4 | 2 | no | — |
| 110 | C4⋊C4 | — | 4 | 1 | no | — |
| 111 | C22⋊C4 | — | 2 | 2 | no | — |
| 112 | C22⋊C4 | — | 4 | 2 | no | — |
| 113 | C42 | — | 2 | 1 | no | — |
| 114 | C42 | — | 2 | 2 | no | — |
| 115 | C42 | — | 2 | 2 | no | — |
| 116 | C42 | — | 4 | 1 | no | — |
| 117 | C42 | C2×M16 | 1 | 1 | yes | characteristic |
| 118 | C42 | C2×M16 | 1 | 1 | yes | characteristic |
| 119 | C16 | — | 8 | 4 | no | — |
| 120 | C23 | — | 4 | 1 | no | — |
| 121 | C23 | #413 | 1 | 1 | yes | characteristic |
| 122 | Q8 | — | 4 | 2 | no | — |
| 123 | Q8 | — | 8 | 2 | no | — |
| 124 | D4 | — | 4 | 2 | no | — |
| 125 | D4 | — | 8 | 2 | no | — |
| 126 | D4 | — | 8 | 2 | no | — |
| 127 | C2×C4 | — | 2 | 1 | no | — |
| 128 | C2×C4 | — | 2 | 1 | no | — |
| 129 | C2×C4 | — | 2 | 1 | no | — |
| 130 | C2×C4 | — | 2 | 1 | no | — |
| 131 | C2×C4 | — | 4 | 2 | no | — |
| 132 | C2×C4 | — | 4 | 2 | no | — |
| 133 | C2×C4 | — | 4 | 1 | no | — |
| 134 | C2×C4 | — | 8 | 1 | no | — |
| 135 | C2×C4 | — | 8 | 2 | no | — |
| 136 | C2×C4 | — | 8 | 1 | no | — |
| 137 | C2×C4 | — | 8 | 1 | no | — |
| 138 | C2×C4 | #407 | 1 | 1 | yes | characteristic |
| 139 | C2×C4 | C2×#99 | 1 | 1 | yes | characteristic |
| 140 | C8 | — | 2 | 1 | no | — |
| 141 | C8 | — | 4 | 2 | no | — |
| 142 | C8 | — | 4 | 2 | no | — |
| 143 | C8 | — | 4 | 1 | no | — |
| 144 | C8 | — | 8 | 1 | no | — |
| 145 | C8 | #355 | 1 | 2 | yes | — |
| 146 | C22 | — | 2 | 1 | no | — |
| 147 | C22 | — | 4 | 1 | no | — |
| 148 | C22 | — | 8 | 2 | no | — |
| 149 | C22 | — | 8 | 1 | no | — |
| 150 | C22 | #1465 | 1 | 1 | yes | characteristic |
| 151 | C4 | — | 2 | 1 | no | — |
| 152 | C4 | — | 4 | 1 | no | — |
| 153 | C4 | — | 4 | 1 | no | — |
| 154 | C4 | — | 8 | 2 | no | — |
| 155 | C4 | — | 8 | 2 | no | — |
| 156 | C4 | — | 8 | 1 | no | — |
| 157 | C4 | #2126 | 1 | 1 | yes | characteristic |
| 158 | C4 | C2×#355 | 1 | 1 | yes | characteristic |
| 159 | C2 | — | 2 | 1 | no | — |
| 160 | C2 | — | 4 | 1 | no | — |
| 161 | C2 | — | 16 | 1 | no | — |
| 162 | C2 | #11530 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2126 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #423 | C23 |
| socle | C2 | #11530 |
| prime radical | #532 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2960 ◁ 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. #11539 | C2 | semidirect |
| 3. #7369 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21663 | 2 |
| #11539 | 2 |
| #7369 | 2 |
| #7356 | 2 |