Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
214 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 48 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 | #33827 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #12902 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11990 | C2 | 1 | 2 | yes | maximal |
| 4 | #11771 | C2 | 1 | 2 | yes | maximal |
| 5 | #7239 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #3177 | C22 | 1 | 2 | yes | — |
| 7 | #2229 | — | 2 | 2 | no | — |
| 8 | #2219 | — | 2 | 1 | no | — |
| 9 | #2216 | — | 2 | 1 | no | — |
| 10 | #2209 | — | 2 | 2 | no | — |
| 11 | #2166 | C22 | 1 | 2 | yes | — |
| 12 | #2153 | — | 2 | 1 | no | — |
| 13 | #2153 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2151 | — | 2 | 1 | no | — |
| 15 | #2151 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1747 | C22 | 1 | 1 | yes | characteristic |
| 17 | #1366 | — | 2 | 2 | no | — |
| 18 | #1358 | — | 2 | 2 | no | — |
| 19 | #1342 | — | 2 | 2 | no | — |
| 20 | #1340 | — | 2 | 2 | no | — |
| 21 | #541 | — | 2 | 2 | no | — |
| 22 | C2×Q32 | — | 2 | 1 | no | — |
| 23 | C2×Q32 | — | 2 | 1 | no | — |
| 24 | C2×M32 | — | 2 | 1 | no | — |
| 25 | C2×M32 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×C16 | D4 | 1 | 1 | yes | characteristic |
| 27 | #499 | D4 | 1 | 2 | yes | — |
| 28 | #494 | D4 | 1 | 2 | yes | — |
| 29 | #491 | — | 2 | 2 | no | — |
| 30 | #470 | — | 2 | 2 | no | — |
| 31 | #467 | — | 2 | 2 | no | — |
| 32 | #441 | — | 2 | 2 | no | — |
| 33 | C4×Q16 | — | 2 | 2 | no | — |
| 34 | #405 | — | 2 | 1 | no | — |
| 35 | #405 | C23 | 1 | 1 | yes | characteristic |
| 36 | #367 | — | 2 | 1 | no | — |
| 37 | #366 | — | 2 | 1 | no | — |
| 38 | #364 | — | 2 | 2 | no | — |
| 39 | #364 | — | 2 | 2 | no | — |
| 40 | #364 | — | 4 | 2 | no | — |
| 41 | C4⋊C16 | — | 2 | 2 | no | — |
| 42 | C4⋊C16 | — | 4 | 2 | no | — |
| 43 | #358 | — | 2 | 2 | no | — |
| 44 | #358 | — | 2 | 2 | no | — |
| 45 | #349 | — | 2 | 2 | no | — |
| 46 | #347 | — | 2 | 2 | no | — |
| 47 | #338 | D4 | 1 | 1 | yes | characteristic |
| 48 | H(3,ℤ/4) | — | 2 | 1 | no | — |
| 49 | H(3,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 50 | C2×Q16 | — | 2 | 2 | no | — |
| 51 | C2×Q16 | — | 2 | 2 | no | — |
| 52 | C2×M16 | — | 2 | 1 | no | — |
| 53 | C2×M16 | C2×D4 | 1 | 1 | yes | characteristic |
| 54 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 55 | C4⋊Q8 | — | 2 | 2 | no | — |
| 56 | #126 | — | 8 | 2 | no | — |
| 57 | #125 | — | 2 | 2 | no | — |
| 58 | #125 | — | 8 | 2 | no | — |
| 59 | C22⋊Q8 | — | 4 | 2 | no | — |
| 60 | C4×Q8 | — | 4 | 2 | no | — |
| 61 | #117 | — | 2 | 1 | no | — |
| 62 | #117 | #41 | 1 | 1 | yes | characteristic |
| 63 | Q32 | — | 8 | 1 | no | — |
| 64 | Q32 | — | 8 | 1 | no | — |
| 65 | M32 | — | 4 | 1 | no | — |
| 66 | M32 | — | 4 | 1 | no | — |
| 67 | M32 | — | 4 | 1 | no | — |
| 68 | M32 | — | 4 | 2 | no | — |
| 69 | C2×C16 | — | 2 | 1 | no | — |
| 70 | C2×C16 | — | 2 | 1 | no | — |
| 71 | C2×C16 | — | 4 | 1 | no | — |
| 72 | C2×C16 | — | 4 | 1 | no | — |
| 73 | #107 | — | 2 | 2 | no | — |
| 74 | #106 | — | 2 | 2 | no | — |
| 75 | #103 | — | 4 | 2 | no | — |
| 76 | #103 | — | 4 | 2 | no | — |
| 77 | #97 | — | 2 | 2 | no | — |
| 78 | #97 | #41 | 1 | 2 | yes | — |
| 79 | C4×C8 | — | 2 | 2 | no | — |
| 80 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 81 | C2×Q8 | — | 4 | 2 | no | — |
| 82 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 83 | Q16 | — | 4 | 2 | no | — |
| 84 | Q16 | — | 4 | 2 | no | — |
| 85 | Q16 | — | 8 | 2 | no | — |
| 86 | M16 | — | 4 | 1 | no | — |
| 87 | M16 | — | 4 | 1 | no | — |
| 88 | M16 | — | 4 | 1 | no | — |
| 89 | C2×C8 | — | 2 | 1 | no | — |
| 90 | C2×C8 | — | 2 | 1 | no | — |
| 91 | C2×C8 | — | 2 | 2 | no | — |
| 92 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 93 | C2×C8 | #121 | 1 | 2 | yes | — |
| 94 | C2×C8 | #124 | 1 | 1 | yes | characteristic |
| 95 | C4⋊C4 | — | 2 | 1 | no | — |
| 96 | C4⋊C4 | — | 4 | 1 | no | — |
| 97 | C4⋊C4 | — | 4 | 2 | no | — |
| 98 | C4⋊C4 | — | 8 | 2 | no | — |
| 99 | C4⋊C4 | — | 8 | 2 | no | — |
| 100 | C4⋊C4 | — | 8 | 2 | no | — |
| 101 | C4⋊C4 | — | 8 | 2 | no | — |
| 102 | C22⋊C4 | — | 2 | 1 | no | — |
| 103 | C22⋊C4 | — | 4 | 1 | no | — |
| 104 | C22⋊C4 | — | 8 | 2 | no | — |
| 105 | C42 | — | 2 | 2 | no | — |
| 106 | C42 | — | 8 | 2 | no | — |
| 107 | C42 | #123 | 1 | 2 | yes | — |
| 108 | C16 | — | 4 | 1 | no | — |
| 109 | C16 | — | 4 | 1 | no | — |
| 110 | C16 | — | 8 | 2 | no | — |
| 111 | C16 | — | 8 | 1 | no | — |
| 112 | C23 | #472 | 1 | 1 | yes | characteristic |
| 113 | Q8 | — | 8 | 2 | no | — |
| 114 | Q8 | — | 8 | 2 | no | — |
| 115 | C2×C4 | — | 2 | 1 | no | — |
| 116 | C2×C4 | — | 2 | 1 | no | — |
| 117 | C2×C4 | — | 2 | 2 | no | — |
| 118 | C2×C4 | — | 4 | 2 | no | — |
| 119 | C2×C4 | — | 8 | 2 | no | — |
| 120 | C2×C4 | — | 8 | 2 | no | — |
| 121 | C2×C4 | #394 | 1 | 1 | yes | characteristic |
| 122 | C2×C4 | #472 | 1 | 1 | yes | characteristic |
| 123 | C8 | — | 2 | 2 | no | — |
| 124 | C8 | — | 2 | 1 | no | — |
| 125 | C8 | — | 4 | 2 | no | — |
| 126 | C8 | #453 | 1 | 1 | yes | characteristic |
| 127 | C8 | #456 | 1 | 1 | yes | characteristic |
| 128 | C22 | — | 2 | 1 | no | — |
| 129 | C22 | — | 4 | 1 | no | — |
| 130 | C22 | #2045 | 1 | 1 | yes | characteristic |
| 131 | C4 | — | 2 | 1 | no | — |
| 132 | C4 | — | 4 | 2 | no | — |
| 133 | C4 | — | 8 | 2 | no | — |
| 134 | C4 | — | 16 | 2 | no | — |
| 135 | C4 | — | 16 | 2 | no | — |
| 136 | C4 | #2039 | 1 | 1 | yes | characteristic |
| 137 | C4 | #2053 | 1 | 1 | yes | characteristic |
| 138 | C2 | — | 2 | 1 | no | — |
| 139 | C2 | — | 4 | 1 | no | — |
| 140 | C2 | #10512 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10512 |
| commutator, Frattini, cosocle | #405 | C23 |
| prime radical | C23 | #472 |
| LCS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33827 | 2 |
| #12902 | 2 |
| #11990 | 2 |
| #11771 | 2 |
| #7239 | 2 |