Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
120 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #2017 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1392 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #1315 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #575 | — | 2 | 1 | no | — |
| 5 | #575 | C4 | 1 | 1 | yes | characteristic |
| 6 | #413 | — | 2 | 1 | no | — |
| 7 | #408 | — | 2 | 1 | no | — |
| 8 | #408 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22⋊C16 | — | 2 | 1 | no | — |
| 10 | #338 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#41 | — | 2 | 1 | no | — |
| 12 | C2×#41 | D4 | 1 | 1 | yes | characteristic |
| 13 | #137 | — | 4 | 1 | no | — |
| 14 | #137 | — | 4 | 1 | no | — |
| 15 | #137 | — | 4 | 1 | no | — |
| 16 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 17 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 18 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 19 | #135 | — | 4 | 1 | no | — |
| 20 | #135 | — | 4 | 2 | no | — |
| 21 | C2×M16 | — | 2 | 1 | no | — |
| 22 | C2×M16 | — | 4 | 1 | no | — |
| 23 | C2×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 25 | C2×C16 | — | 4 | 1 | no | — |
| 26 | #101 | — | 4 | 1 | no | — |
| 27 | #100 | — | 2 | 2 | no | — |
| 28 | C22⋊C8 | — | 4 | 1 | no | — |
| 29 | C22⋊C8 | — | 4 | 2 | no | — |
| 30 | #41 | — | 4 | 1 | no | — |
| 31 | #41 | — | 4 | 1 | no | — |
| 32 | #41 | — | 4 | 1 | no | — |
| 33 | #41 | — | 4 | 1 | no | — |
| 34 | #41 | — | 4 | 1 | no | — |
| 35 | #41 | — | 4 | 1 | no | — |
| 36 | C2×Q8 | — | 2 | 1 | no | — |
| 37 | C2×Q8 | D8 | 1 | 1 | yes | characteristic |
| 38 | C2×D4 | — | 2 | 1 | no | — |
| 39 | C2×D4 | — | 2 | 1 | no | — |
| 40 | C2×D4 | — | 4 | 1 | no | — |
| 41 | C2×D4 | SD16 | 1 | 1 | yes | characteristic |
| 42 | C22×C4 | — | 2 | 1 | no | — |
| 43 | C22×C4 | — | 4 | 1 | no | — |
| 44 | C22×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | Q16 | — | 4 | 1 | no | — |
| 46 | Q16 | — | 4 | 2 | no | — |
| 47 | SD16 | — | 4 | 2 | no | — |
| 48 | SD16 | — | 4 | 2 | no | — |
| 49 | SD16 | — | 4 | 2 | no | — |
| 50 | D8 | — | 4 | 1 | no | — |
| 51 | D8 | — | 4 | 2 | no | — |
| 52 | M16 | — | 4 | 1 | no | — |
| 53 | M16 | — | 4 | 1 | no | — |
| 54 | M16 | — | 4 | 1 | no | — |
| 55 | M16 | — | 4 | 2 | no | — |
| 56 | M16 | — | 8 | 1 | no | — |
| 57 | C2×C8 | — | 2 | 2 | no | — |
| 58 | C2×C8 | — | 2 | 1 | no | — |
| 59 | C2×C8 | — | 2 | 1 | no | — |
| 60 | C2×C8 | — | 4 | 1 | no | — |
| 61 | C2×C8 | — | 8 | 1 | no | — |
| 62 | C16 | — | 8 | 1 | no | — |
| 63 | C23 | — | 2 | 1 | no | — |
| 64 | C23 | — | 4 | 1 | no | — |
| 65 | C23 | #104 | 1 | 1 | yes | characteristic |
| 66 | Q8 | — | 4 | 1 | no | — |
| 67 | Q8 | — | 4 | 1 | no | — |
| 68 | Q8 | — | 4 | 1 | no | — |
| 69 | D4 | — | 4 | 1 | no | — |
| 70 | D4 | — | 4 | 1 | no | — |
| 71 | D4 | — | 4 | 1 | no | — |
| 72 | D4 | — | 8 | 1 | no | — |
| 73 | D4 | — | 8 | 1 | no | — |
| 74 | D4 | — | 8 | 1 | no | — |
| 75 | C2×C4 | — | 2 | 1 | no | — |
| 76 | C2×C4 | — | 2 | 1 | no | — |
| 77 | C2×C4 | — | 2 | 1 | no | — |
| 78 | C2×C4 | — | 4 | 1 | no | — |
| 79 | C2×C4 | — | 4 | 1 | no | — |
| 80 | C2×C4 | — | 4 | 1 | no | — |
| 81 | C2×C4 | — | 8 | 1 | no | — |
| 82 | C2×C4 | — | 8 | 1 | no | — |
| 83 | C2×C4 | #99 | 1 | 1 | yes | characteristic |
| 84 | C2×C4 | #102 | 1 | 1 | yes | characteristic |
| 85 | C8 | — | 4 | 2 | no | — |
| 86 | C8 | — | 4 | 1 | no | — |
| 87 | C8 | — | 4 | 1 | no | — |
| 88 | C8 | — | 8 | 2 | no | — |
| 89 | C22 | — | 2 | 1 | no | — |
| 90 | C22 | — | 4 | 1 | no | — |
| 91 | C22 | — | 4 | 1 | no | — |
| 92 | C22 | — | 8 | 1 | no | — |
| 93 | C22 | — | 8 | 2 | no | — |
| 94 | C22 | — | 8 | 1 | no | — |
| 95 | C22 | #327 | 1 | 1 | yes | characteristic |
| 96 | C4 | — | 2 | 1 | no | — |
| 97 | C4 | — | 4 | 1 | no | — |
| 98 | C4 | — | 8 | 1 | no | — |
| 99 | C4 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 100 | C4 | #353 | 1 | 1 | yes | characteristic |
| 101 | C2 | — | 2 | 1 | no | — |
| 102 | C2 | — | 4 | 1 | no | — |
| 103 | C2 | — | 8 | 1 | no | — |
| 104 | C2 | — | 16 | 1 | no | — |
| 105 | C2 | #1343 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #353 |
| commutator | C2×M16 | C2×C4 |
| Frattini, cosocle | #408 | C22 |
| socle | C2 | #1343 |
| prime radical | #2017 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #408 ◁ G | |
| LECS | G ▷ #408 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1392 | C2 | semidirect |
| 3. #1315 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2017 | 2 |
| #1392 | 2 |
| #1315 | 2 |