Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
89 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #342 | C2 | 1 | 7 | yes | maximal |
| 2 | C2×C22⋊C4 | C22 | 1 | 7 | yes | — |
| 3 | C24 | C23 | 1 | 1 | yes | characteristic |
| 4 | C22×C4 | — | 2 | 7 | no | — |
| 5 | C22⋊C4 | — | 2 | 14 | no | — |
| 6 | C23 | — | 8 | 1 | no | — |
| 7 | C23 | #41 | 1 | 7 | yes | — |
| 8 | C2×C4 | — | 4 | 7 | no | — |
| 9 | C2×C4 | — | 8 | 7 | no | — |
| 10 | C22 | — | 4 | 7 | no | — |
| 11 | C22 | #126 | 1 | 7 | yes | — |
| 12 | C4 | — | 8 | 7 | no | — |
| 13 | C2 | — | 2 | 7 | no | — |
| 14 | C2 | #401 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #401 |
| commutator, Frattini, prime radical, cosocle | C24 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #342 | 2 |