Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
363 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #30237 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #13245 | C2 | 1 | 3 | yes | maximal |
| 3 | #13173 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #13158 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #12542 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×D16 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#368 | C22 | 1 | 3 | yes | — |
| 8 | C2×#361 | C22 | 1 | 6 | yes | — |
| 9 | #2110 | C22 | 1 | 3 | yes | — |
| 10 | #2108 | C22 | 1 | 3 | yes | — |
| 11 | #2081 | — | 2 | 6 | no | — |
| 12 | #2069 | — | 2 | 2 | no | — |
| 13 | #1978 | C22 | 1 | 6 | yes | — |
| 14 | #1953 | C22 | 1 | 6 | yes | — |
| 15 | #1747 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1388 | C22 | 1 | 6 | yes | — |
| 17 | C22×D8 | C23 | 1 | 2 | yes | — |
| 18 | C2×D16 | — | 2 | 3 | no | — |
| 19 | C2×D16 | — | 2 | 3 | no | — |
| 20 | C2×M32 | C23 | 1 | 3 | yes | — |
| 21 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 22 | #430 | — | 2 | 6 | no | — |
| 23 | C2×#108 | C23 | 1 | 6 | yes | — |
| 24 | #418 | — | 2 | 6 | no | — |
| 25 | C2×#100 | — | 2 | 6 | no | — |
| 26 | #405 | C23 | 1 | 3 | yes | — |
| 27 | #368 | — | 2 | 3 | no | — |
| 28 | #368 | — | 2 | 3 | no | — |
| 29 | #361 | — | 2 | 12 | no | — |
| 30 | #347 | — | 2 | 6 | no | — |
| 31 | #346 | — | 2 | 3 | no | — |
| 32 | C4×C16 | — | 2 | 3 | no | — |
| 33 | #344 | — | 2 | 6 | no | — |
| 34 | #338 | D4 | 1 | 1 | yes | characteristic |
| 35 | H(3,ℤ/4) | D4 | 1 | 3 | yes | — |
| 36 | C22×D4 | — | 2 | 2 | no | — |
| 37 | C2×D8 | — | 2 | 6 | no | — |
| 38 | C2×D8 | — | 2 | 6 | no | — |
| 39 | C2×M16 | — | 2 | 6 | no | — |
| 40 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 41 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 42 | #117 | C2×D4 | 1 | 3 | yes | — |
| 43 | D16 | — | 4 | 1 | no | — |
| 44 | D16 | — | 4 | 3 | no | — |
| 45 | M32 | — | 2 | 3 | no | — |
| 46 | M32 | #41 | 1 | 6 | yes | — |
| 47 | C2×C16 | — | 2 | 3 | no | — |
| 48 | C2×C16 | #41 | 1 | 3 | yes | — |
| 49 | C2×C16 | #41 | 1 | 3 | yes | — |
| 50 | #108 | — | 2 | 6 | no | — |
| 51 | #108 | — | 2 | 6 | no | — |
| 52 | #108 | — | 8 | 6 | no | — |
| 53 | #102 | — | 8 | 6 | no | — |
| 54 | #100 | — | 8 | 6 | no | — |
| 55 | #97 | — | 2 | 3 | no | — |
| 56 | C4×C8 | — | 2 | 3 | no | — |
| 57 | C24 | — | 4 | 2 | no | — |
| 58 | C2×D4 | — | 4 | 6 | no | — |
| 59 | C2×D4 | — | 4 | 6 | no | — |
| 60 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 61 | D8 | — | 4 | 2 | no | — |
| 62 | D8 | — | 4 | 6 | no | — |
| 63 | M16 | — | 2 | 3 | no | — |
| 64 | M16 | — | 2 | 3 | no | — |
| 65 | M16 | — | 4 | 6 | no | — |
| 66 | M16 | — | 4 | 6 | no | — |
| 67 | C2×C8 | — | 2 | 3 | no | — |
| 68 | C2×C8 | — | 4 | 6 | no | — |
| 69 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 70 | C2×C8 | C2×#41 | 1 | 3 | yes | — |
| 71 | C4⋊C4 | — | 2 | 3 | no | — |
| 72 | C22⋊C4 | — | 2 | 3 | no | — |
| 73 | C42 | — | 2 | 3 | no | — |
| 74 | C16 | — | 2 | 6 | no | — |
| 75 | C16 | #126 | 1 | 1 | yes | characteristic |
| 76 | C16 | #126 | 1 | 3 | yes | — |
| 77 | C23 | — | 8 | 6 | no | — |
| 78 | C23 | — | 32 | 2 | no | — |
| 79 | C23 | #578 | 1 | 1 | yes | characteristic |
| 80 | D4 | — | 8 | 2 | no | — |
| 81 | D4 | — | 8 | 6 | no | — |
| 82 | C2×C4 | — | 4 | 3 | no | — |
| 83 | C2×C4 | #532 | 1 | 3 | yes | — |
| 84 | C2×C4 | #576 | 1 | 3 | yes | — |
| 85 | C8 | — | 4 | 3 | no | — |
| 86 | C8 | — | 8 | 6 | no | — |
| 87 | C8 | C2×#126 | 1 | 1 | yes | characteristic |
| 88 | C8 | #529 | 1 | 3 | yes | — |
| 89 | C22 | — | 4 | 1 | no | — |
| 90 | C22 | — | 16 | 2 | no | — |
| 91 | C22 | — | 32 | 6 | no | — |
| 92 | C22 | #3175 | 1 | 3 | yes | — |
| 93 | C4 | — | 8 | 3 | no | — |
| 94 | C4 | #2860 | 1 | 1 | yes | characteristic |
| 95 | C4 | #3176 | 1 | 3 | yes | — |
| 96 | C2 | — | 2 | 3 | no | — |
| 97 | C2 | — | 32 | 2 | no | — |
| 98 | C2 | #33272 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #33272 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1747 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30237 | C2 | semidirect |
| 3. #13173 | C2 | semidirect |
| 4. #13158 | C2 | semidirect |
| 5. #12542 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30237 | 2 |
| #13245 | 2 |
| #13173 | 2 |
| #13158 | 2 |
| #12542 | 2 |