Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
441 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 441 conjugacy classes of subgroups — too many to draw. Only its 124 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 | #33272 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #13515 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #13474 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #13448 | C2 | 1 | 3 | yes | maximal |
| 5 | #12899 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2644 | — | 2 | 2 | no | — |
| 8 | C2×#366 | C22 | 1 | 3 | yes | — |
| 9 | C2×C4⋊C16 | C22 | 1 | 3 | yes | — |
| 10 | C2×C4⋊C16 | C22 | 1 | 3 | yes | — |
| 11 | C2×#357 | C22 | 1 | 6 | yes | — |
| 12 | #1970 | C22 | 1 | 6 | yes | — |
| 13 | #1942 | C22 | 1 | 6 | yes | — |
| 14 | #1936 | — | 2 | 6 | no | — |
| 15 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1380 | C22 | 1 | 3 | yes | — |
| 17 | #1380 | C22 | 1 | 3 | yes | — |
| 18 | C22×D8 | C23 | 1 | 2 | yes | — |
| 19 | C2×D16 | — | 4 | 3 | no | — |
| 20 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 21 | C22×C16 | C23 | 1 | 3 | yes | — |
| 22 | C2×#107 | C23 | 1 | 6 | yes | — |
| 23 | C2×#102 | — | 2 | 6 | no | — |
| 24 | C2×#97 | D4 | 1 | 3 | yes | — |
| 25 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 26 | #397 | — | 4 | 6 | no | — |
| 27 | #396 | — | 4 | 2 | no | — |
| 28 | #390 | — | 2 | 6 | no | — |
| 29 | #389 | — | 2 | 6 | no | — |
| 30 | #376 | D4 | 1 | 1 | yes | characteristic |
| 31 | #366 | — | 4 | 3 | no | — |
| 32 | C4⋊C16 | — | 4 | 3 | no | — |
| 33 | C4⋊C16 | — | 4 | 3 | no | — |
| 34 | #357 | — | 4 | 6 | no | — |
| 35 | #340 | — | 2 | 6 | no | — |
| 36 | C22×D4 | — | 2 | 2 | no | — |
| 37 | C2×D8 | — | 2 | 6 | no | — |
| 38 | C2×D8 | — | 2 | 6 | no | — |
| 39 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 40 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 41 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 42 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 43 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 44 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 45 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 46 | D16 | — | 4 | 4 | no | — |
| 47 | C2×C16 | — | 2 | 6 | no | — |
| 48 | C2×C16 | — | 2 | 3 | no | — |
| 49 | C2×C16 | — | 2 | 3 | no | — |
| 50 | #107 | — | 2 | 6 | no | — |
| 51 | #107 | — | 2 | 6 | no | — |
| 52 | #102 | — | 4 | 12 | no | — |
| 53 | #97 | — | 4 | 3 | no | — |
| 54 | C4×C8 | — | 2 | 3 | no | — |
| 55 | C4×C8 | — | 2 | 3 | no | — |
| 56 | #95 | — | 4 | 6 | no | — |
| 57 | #95 | D8 | 1 | 1 | yes | characteristic |
| 58 | #95 | D8 | 1 | 3 | yes | — |
| 59 | C24 | — | 4 | 2 | no | — |
| 60 | C2×D4 | — | 4 | 6 | no | — |
| 61 | C2×D4 | — | 4 | 6 | no | — |
| 62 | C22×C4 | — | 4 | 6 | no | — |
| 63 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 64 | C22×C4 | C2×D8 | 1 | 3 | yes | — |
| 65 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 66 | D8 | — | 4 | 2 | no | — |
| 67 | D8 | — | 4 | 6 | no | — |
| 68 | C2×C8 | — | 2 | 3 | no | — |
| 69 | C2×C8 | — | 2 | 6 | no | — |
| 70 | C2×C8 | H(5,𝔽2) | 1 | 3 | yes | — |
| 71 | C2×C8 | 2−1+4 | 1 | 3 | yes | — |
| 72 | C4⋊C4 | — | 4 | 6 | no | — |
| 73 | C4⋊C4 | — | 4 | 6 | no | — |
| 74 | C4⋊C4 | — | 16 | 6 | no | — |
| 75 | C22⋊C4 | — | 16 | 6 | no | — |
| 76 | C22⋊C4 | — | 16 | 6 | no | — |
| 77 | C42 | — | 2 | 3 | no | — |
| 78 | C42 | — | 2 | 3 | no | — |
| 79 | C16 | — | 4 | 3 | no | — |
| 80 | C16 | — | 4 | 1 | no | — |
| 81 | C23 | — | 8 | 6 | no | — |
| 82 | C23 | — | 8 | 2 | no | — |
| 83 | C23 | — | 8 | 6 | no | — |
| 84 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 85 | D4 | — | 8 | 2 | no | — |
| 86 | D4 | — | 8 | 6 | no | — |
| 87 | C2×C4 | — | 2 | 3 | no | — |
| 88 | C2×C4 | — | 2 | 6 | no | — |
| 89 | C2×C4 | — | 2 | 3 | no | — |
| 90 | C2×C4 | — | 2 | 3 | no | — |
| 91 | C2×C4 | — | 2 | 3 | no | — |
| 92 | C2×C4 | — | 8 | 6 | no | — |
| 93 | C2×C4 | — | 8 | 6 | no | — |
| 94 | C2×C4 | — | 8 | 6 | no | — |
| 95 | C2×C4 | #537 | 1 | 3 | yes | — |
| 96 | C2×C4 | #575 | 1 | 3 | yes | — |
| 97 | C8 | — | 4 | 3 | no | — |
| 98 | C8 | #560 | 1 | 1 | yes | characteristic |
| 99 | C8 | #563 | 1 | 3 | yes | — |
| 100 | C22 | — | 16 | 6 | no | — |
| 101 | C22 | — | 16 | 6 | no | — |
| 102 | C22 | — | 16 | 2 | no | — |
| 103 | C22 | #3245 | 1 | 3 | yes | — |
| 104 | C22 | #3415 | 1 | 3 | yes | — |
| 105 | C22 | #3417 | 1 | 1 | yes | characteristic |
| 106 | C4 | — | 4 | 3 | no | — |
| 107 | C4 | — | 4 | 3 | no | — |
| 108 | C4 | — | 16 | 6 | no | — |
| 109 | C4 | #2823 | 1 | 1 | yes | characteristic |
| 110 | C4 | #3256 | 1 | 3 | yes | — |
| 111 | C2 | — | 32 | 2 | no | — |
| 112 | C2 | #32858 | 1 | 1 | yes | characteristic |
| 113 | C2 | #33825 | 1 | 3 | yes | — |
| 114 | C2 | #33828 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33272 | C2 | semidirect |
| 2. #13515 | C2 | semidirect |
| 3. #13474 | C2 | semidirect |
| 5. #12899 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33272 | 2 |
| #13515 | 2 |
| #13474 | 2 |
| #13448 | 2 |
| #12899 | 2 |