Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
503 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 136 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 | #19745 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #9428 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #9423 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #9417 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #8804 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #3184 | C22 | 1 | 3 | yes | — |
| 7 | #3151 | C22 | 1 | 3 | yes | — |
| 8 | #2980 | C22 | 1 | 3 | yes | — |
| 9 | #2380 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1721 | C4 | 1 | 3 | yes | complemented |
| 11 | #1715 | C4 | 1 | 3 | yes | complemented |
| 12 | #1712 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | #1705 | C4 | 1 | 1 | yes | characteristic, complemented |
| 14 | #1555 | C22 | 1 | 3 | yes | — |
| 15 | #1536 | C22 | 1 | 3 | yes | — |
| 16 | #1514 | C22 | 1 | 6 | yes | — |
| 17 | #1455 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1291 | C22 | 1 | 6 | yes | — |
| 19 | #1288 | C22 | 1 | 6 | yes | — |
| 20 | C2×AGL(1,ℤ/8) | — | 4 | 3 | no | — |
| 21 | #532 | C23 | 1 | 3 | yes | — |
| 22 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 23 | #498 | C2×C4 | 1 | 3 | yes | complemented |
| 24 | #496 | — | 4 | 3 | no | — |
| 25 | #493 | C2×C4 | 1 | 3 | yes | — |
| 26 | #493 | C2×C4 | 1 | 3 | yes | — |
| 27 | #493 | C2×C4 | 1 | 3 | yes | complemented |
| 28 | #492 | C2×C4 | 1 | 3 | yes | — |
| 29 | #492 | C2×C4 | 1 | 3 | yes | — |
| 30 | #486 | C2×C4 | 1 | 6 | yes | complemented |
| 31 | #469 | — | 4 | 3 | no | — |
| 32 | #442 | — | 4 | 3 | no | — |
| 33 | #435 | — | 4 | 6 | no | — |
| 34 | #432 | C23 | 1 | 6 | yes | — |
| 35 | #408 | — | 4 | 3 | no | — |
| 36 | C4×M16 | C23 | 1 | 3 | yes | — |
| 37 | #386 | — | 4 | 3 | no | — |
| 38 | #383 | C23 | 1 | 2 | yes | — |
| 39 | #331 | — | 4 | 3 | no | — |
| 40 | #329 | — | 4 | 3 | no | — |
| 41 | #327 | — | 4 | 6 | no | — |
| 42 | #322 | C2×C4 | 1 | 3 | yes | complemented |
| 43 | C82 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 44 | C2×#41 | — | 2 | 3 | no | — |
| 45 | C22×D4 | — | 2 | 3 | no | — |
| 46 | C23×C4 | — | 4 | 1 | no | — |
| 47 | AGL(1,ℤ/8) | — | 8 | 6 | no | — |
| 48 | AGL(1,ℤ/8) | — | 8 | 6 | no | — |
| 49 | C2×SD16 | — | 4 | 3 | no | — |
| 50 | C2×SD16 | — | 4 | 3 | no | — |
| 51 | C2×D8 | — | 4 | 3 | no | — |
| 52 | C2×D8 | — | 4 | 3 | no | — |
| 53 | C2×D8 | — | 4 | 3 | no | — |
| 54 | C2×M16 | — | 2 | 3 | no | — |
| 55 | C2×M16 | — | 4 | 3 | no | — |
| 56 | C22×C8 | — | 4 | 3 | no | — |
| 57 | C4⋊Q8 | C22×C4 | 1 | 3 | yes | — |
| 58 | #127 | — | 2 | 3 | no | — |
| 59 | #127 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | #127 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | #127 | C22×C4 | 1 | 3 | yes | — |
| 62 | #124 | — | 2 | 3 | no | — |
| 63 | #121 | — | 2 | 6 | no | — |
| 64 | C4×D4 | — | 2 | 6 | no | — |
| 65 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 66 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 67 | #106 | — | 4 | 3 | no | — |
| 68 | C4⋊C8 | — | 2 | 6 | no | — |
| 69 | #102 | — | 4 | 6 | no | — |
| 70 | C22⋊C8 | — | 2 | 6 | no | — |
| 71 | C22⋊C8 | — | 4 | 6 | no | — |
| 72 | #97 | — | 2 | 3 | no | — |
| 73 | #97 | — | 2 | 6 | no | — |
| 74 | C4×C8 | C22×C4 | 1 | 3 | yes | — |
| 75 | C4×C8 | C22×C4 | 1 | 3 | yes | — |
| 76 | #95 | — | 2 | 6 | no | — |
| 77 | C24 | — | 4 | 1 | no | — |
| 78 | #41 | — | 8 | 3 | no | — |
| 79 | #41 | — | 8 | 3 | no | — |
| 80 | C2×Q8 | — | 2 | 3 | no | — |
| 81 | C2×D4 | — | 2 | 6 | no | — |
| 82 | C2×D4 | — | 2 | 3 | no | — |
| 83 | C2×D4 | — | 2 | 3 | no | — |
| 84 | C2×D4 | — | 2 | 3 | no | — |
| 85 | C2×D4 | — | 4 | 3 | no | — |
| 86 | C2×D4 | — | 4 | 3 | no | — |
| 87 | C2×D4 | — | 8 | 3 | no | — |
| 88 | C2×D4 | — | 8 | 3 | no | — |
| 89 | C22×C4 | — | 4 | 3 | no | — |
| 90 | C22×C4 | — | 4 | 2 | no | — |
| 91 | C22×C4 | — | 16 | 3 | no | — |
| 92 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 93 | SD16 | — | 8 | 6 | no | — |
| 94 | SD16 | — | 8 | 6 | no | — |
| 95 | D8 | — | 8 | 6 | no | — |
| 96 | D8 | — | 8 | 6 | no | — |
| 97 | D8 | — | 8 | 6 | no | — |
| 98 | M16 | — | 4 | 6 | no | — |
| 99 | M16 | — | 16 | 3 | no | — |
| 100 | C2×C8 | — | 2 | 3 | no | — |
| 101 | C2×C8 | — | 2 | 3 | no | — |
| 102 | C2×C8 | — | 4 | 6 | no | — |
| 103 | C2×C8 | — | 16 | 3 | no | — |
| 104 | C4⋊C4 | — | 2 | 6 | no | — |
| 105 | C22⋊C4 | — | 4 | 6 | no | — |
| 106 | C22⋊C4 | — | 16 | 3 | no | — |
| 107 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 108 | C42 | H(5,𝔽2) | 1 | 3 | yes | — |
| 109 | C23 | — | 4 | 3 | no | — |
| 110 | C23 | — | 4 | 1 | no | — |
| 111 | C23 | — | 4 | 1 | no | — |
| 112 | C23 | — | 8 | 3 | no | — |
| 113 | C23 | — | 8 | 3 | no | — |
| 114 | C23 | #560 | 1 | 1 | yes | characteristic |
| 115 | Q8 | — | 8 | 3 | no | — |
| 116 | D4 | — | 8 | 3 | no | — |
| 117 | D4 | — | 8 | 3 | no | — |
| 118 | D4 | — | 8 | 3 | no | — |
| 119 | D4 | — | 8 | 6 | no | — |
| 120 | D4 | — | 16 | 3 | no | — |
| 121 | D4 | — | 16 | 3 | no | — |
| 122 | C2×C4 | — | 4 | 3 | no | — |
| 123 | C2×C4 | — | 4 | 3 | no | — |
| 124 | C2×C4 | — | 8 | 3 | no | — |
| 125 | C2×C4 | — | 8 | 3 | no | — |
| 126 | C2×C4 | — | 8 | 6 | no | — |
| 127 | C2×C4 | — | 16 | 4 | no | — |
| 128 | C2×C4 | #518 | 1 | 3 | yes | — |
| 129 | C2×C4 | #560 | 1 | 3 | yes | — |
| 130 | C8 | — | 2 | 6 | no | — |
| 131 | C8 | — | 4 | 3 | no | — |
| 132 | C8 | — | 8 | 6 | no | — |
| 133 | C22 | — | 2 | 3 | no | — |
| 134 | C22 | — | 8 | 3 | no | — |
| 135 | C22 | — | 8 | 3 | no | — |
| 136 | C22 | — | 8 | 3 | no | — |
| 137 | C22 | — | 8 | 6 | no | — |
| 138 | C22 | — | 16 | 1 | no | — |
| 139 | C22 | — | 16 | 3 | no | — |
| 140 | C22 | #2380 | 1 | 1 | yes | characteristic |
| 141 | C4 | — | 2 | 3 | no | — |
| 142 | C4 | — | 2 | 3 | no | — |
| 143 | C4 | — | 8 | 3 | no | — |
| 144 | C4 | — | 16 | 2 | no | — |
| 145 | C2 | — | 4 | 1 | no | — |
| 146 | C2 | — | 16 | 3 | no | — |
| 147 | C2 | — | 16 | 1 | no | — |
| 148 | C2 | — | 16 | 1 | no | — |
| 149 | C2 | #23361 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19745 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #9428 | C2 | semidirect |
| 3. #9423 | C2 | semidirect |
| 4. #9417 | C2 | semidirect |
| 5. #8804 | C2 | semidirect |
| 10. #1721 | C4 | semidirect |
| 11. #1715 | C4 | semidirect |
| 12. #1712 | C4 | semidirect |
| 13. #1705 | C4 | semidirect |
| 23. #498 | C2×C4 | semidirect |
| 27. #493 | C2×C4 | semidirect |
| 30. #486 | C2×C4 | semidirect |
| 42. #322 | C2×C4 | semidirect |
| 43. C82 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19745 | 2 |
| #9428 | 2 |
| #9423 | 2 |
| #9417 | 2 |
| #8804 | 2 |