Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
509 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 154 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 | #33749 | C2 | 1 | 4 | yes | maximal |
| 2 | #19604 | C2 | 1 | 2 | yes | maximal |
| 3 | #12535 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #12528 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #11402 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #3170 | — | 2 | 2 | no | — |
| 7 | #3170 | C22 | 1 | 4 | yes | — |
| 8 | #3160 | — | 2 | 2 | no | — |
| 9 | #2254 | C22 | 1 | 4 | yes | — |
| 10 | #2247 | C22 | 1 | 4 | yes | — |
| 11 | #2241 | C22 | 1 | 8 | yes | — |
| 12 | #2219 | — | 2 | 8 | no | — |
| 13 | #2176 | — | 2 | 8 | no | — |
| 14 | C4×M32 | C22 | 1 | 2 | yes | — |
| 15 | #1713 | C22 | 1 | 2 | yes | — |
| 16 | #1713 | C22 | 1 | 2 | yes | — |
| 17 | #1710 | — | 2 | 2 | no | — |
| 18 | #1707 | — | 2 | 2 | no | — |
| 19 | #1707 | C22 | 1 | 4 | yes | — |
| 20 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1312 | C22 | 1 | 4 | yes | — |
| 22 | #533 | — | 2 | 4 | no | — |
| 23 | C2×Q32 | — | 2 | 8 | no | — |
| 24 | C2×M32 | D4 | 1 | 4 | yes | — |
| 25 | #500 | — | 2 | 2 | no | — |
| 26 | #500 | — | 2 | 2 | no | — |
| 27 | #500 | — | 2 | 2 | no | — |
| 28 | #500 | C23 | 1 | 4 | yes | — |
| 29 | #499 | — | 4 | 4 | no | — |
| 30 | #498 | — | 2 | 4 | no | — |
| 31 | #494 | — | 2 | 2 | no | — |
| 32 | #494 | C23 | 1 | 4 | yes | — |
| 33 | #487 | — | 2 | 4 | no | — |
| 34 | #487 | — | 2 | 4 | no | — |
| 35 | #467 | — | 2 | 8 | no | — |
| 36 | #467 | — | 4 | 4 | no | — |
| 37 | C4×Q16 | — | 2 | 8 | no | — |
| 38 | C4×Q16 | — | 4 | 4 | no | — |
| 39 | #427 | — | 4 | 4 | no | — |
| 40 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 41 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | #367 | — | 2 | 8 | no | — |
| 43 | #358 | — | 2 | 16 | no | — |
| 44 | #346 | D4 | 1 | 4 | yes | — |
| 45 | C4×C16 | C23 | 1 | 4 | yes | — |
| 46 | C82 | D4 | 1 | 1 | yes | characteristic |
| 47 | C82 | D4 | 1 | 1 | yes | characteristic |
| 48 | C82 | C23 | 1 | 2 | yes | — |
| 49 | C2×Q16 | — | 2 | 8 | no | — |
| 50 | C2×Q16 | — | 4 | 4 | no | — |
| 51 | C22×C8 | D8 | 1 | 4 | yes | — |
| 52 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 53 | C4⋊Q8 | — | 2 | 4 | no | — |
| 54 | C4⋊Q8 | — | 2 | 4 | no | — |
| 55 | #125 | — | 4 | 4 | no | — |
| 56 | C22⋊Q8 | — | 4 | 4 | no | — |
| 57 | C22⋊Q8 | — | 4 | 4 | no | — |
| 58 | C4×Q8 | — | 4 | 4 | no | — |
| 59 | C4×Q8 | — | 4 | 4 | no | — |
| 60 | #117 | — | 4 | 4 | no | — |
| 61 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | Q32 | — | 4 | 16 | no | — |
| 63 | M32 | — | 4 | 4 | no | — |
| 64 | C2×C16 | C2×D4 | 1 | 8 | yes | — |
| 65 | #107 | — | 2 | 8 | no | — |
| 66 | #107 | — | 4 | 4 | no | — |
| 67 | #107 | — | 4 | 4 | no | — |
| 68 | #107 | — | 4 | 4 | no | — |
| 69 | #106 | — | 4 | 8 | no | — |
| 70 | #103 | — | 4 | 8 | no | — |
| 71 | #103 | — | 4 | 8 | no | — |
| 72 | C4×C8 | D8 | 1 | 4 | yes | — |
| 73 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 74 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 76 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 77 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 78 | C2×Q8 | — | 4 | 4 | no | — |
| 79 | C2×Q8 | — | 4 | 4 | no | — |
| 80 | C22×C4 | #127 | 1 | 1 | yes | characteristic |
| 81 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 82 | Q16 | — | 4 | 16 | no | — |
| 83 | Q16 | — | 8 | 8 | no | — |
| 84 | C2×C8 | — | 2 | 4 | no | — |
| 85 | C2×C8 | — | 2 | 8 | no | — |
| 86 | C2×C8 | #127 | 1 | 2 | yes | — |
| 87 | C2×C8 | C2×D8 | 1 | 4 | yes | — |
| 88 | C2×C8 | C2×D8 | 1 | 4 | yes | — |
| 89 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 90 | C4⋊C4 | — | 4 | 4 | no | — |
| 91 | C4⋊C4 | — | 4 | 4 | no | — |
| 92 | C4⋊C4 | — | 4 | 4 | no | — |
| 93 | C4⋊C4 | — | 4 | 4 | no | — |
| 94 | C4⋊C4 | — | 8 | 4 | no | — |
| 95 | C4⋊C4 | — | 8 | 4 | no | — |
| 96 | C22⋊C4 | — | 8 | 4 | no | — |
| 97 | C42 | — | 8 | 4 | no | — |
| 98 | C42 | #127 | 1 | 1 | yes | characteristic |
| 99 | C42 | C2×D8 | 1 | 2 | yes | — |
| 100 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 101 | C16 | — | 2 | 8 | no | — |
| 102 | C23 | #493 | 1 | 1 | yes | characteristic |
| 103 | Q8 | — | 8 | 8 | no | — |
| 104 | Q8 | — | 8 | 8 | no | — |
| 105 | C2×C4 | — | 2 | 2 | no | — |
| 106 | C2×C4 | — | 2 | 4 | no | — |
| 107 | C2×C4 | — | 8 | 4 | no | — |
| 108 | C2×C4 | — | 8 | 4 | no | — |
| 109 | C2×C4 | #493 | 1 | 2 | yes | — |
| 110 | C2×C4 | #493 | 1 | 1 | yes | characteristic |
| 111 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 112 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 113 | C8 | — | 2 | 2 | no | — |
| 114 | C8 | — | 2 | 4 | no | — |
| 115 | C8 | #510 | 1 | 8 | yes | — |
| 116 | C8 | #575 | 1 | 4 | yes | — |
| 117 | C22 | — | 2 | 1 | no | — |
| 118 | C22 | — | 2 | 2 | no | — |
| 119 | C22 | C2×#493 | 1 | 1 | yes | characteristic |
| 120 | C4 | — | 2 | 1 | no | — |
| 121 | C4 | — | 2 | 2 | no | — |
| 122 | C4 | — | 16 | 4 | no | — |
| 123 | C4 | — | 16 | 4 | no | — |
| 124 | C4 | #3150 | 1 | 2 | yes | — |
| 125 | C4 | C2×#510 | 1 | 4 | yes | — |
| 126 | C2 | — | 4 | 1 | no | — |
| 127 | C2 | #19616 | 1 | 1 | yes | characteristic |
| 128 | C2 | #33718 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C23 | #493 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12535 | C2 | semidirect |
| 4. #12528 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33749 | 2 |
| #19604 | 2 |
| #12535 | 2 |
| #12528 | 2 |
| #11402 | 2 |