Finite Group Databasebeta

← #1510

Subgroups — #1510

Order 128.

163 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#121C211yesmaximal, characteristic
2#407C211yesmaximal, characteristic, complemented
3#381C211yesmaximal, characteristic, complemented
4#327C214yesmaximal, complemented
5C22×D4—21no—
6C22×D4C411yescharacteristic, complemented
7C23×C4C2211yescharacteristic
8C2×M16—21no—
9#121—21no—
10#121—21no—
11#121C412yescomplemented
12#121C2212yes—
13C2×C4⋊C4C411yescharacteristic, complemented
14C2×C22⋊C4—21no—
15C2×C22⋊C4—21no—
16C22⋊C8—21no—
17C22⋊C8C2212yescomplemented
18#95—21no—
19#95C2212yescomplemented
20C24—21no—
21C24D411yescharacteristic
22C2×D4—22no—
23C2×D4—22no—
24C2×D4—21no—
25C2×D4—42no—
26C2×D4—44no—
27C2×D4C2×C412yes—
28C22×C4—21no—
29C22×C4—22no—
30C22×C4—42no—
31C22×C4C2×C411yescharacteristic
32C22×C4C2×C411yescharacteristic
33C22×C4D411yescharacteristic
34C22×C4D411yescharacteristic
35C22×C4D411yescharacteristic
36C22×C4C2311yescharacteristic
37M16—42no—
38C2×C8—22no—
39C4⋊C4—21no—
40C4⋊C4C2×C412yes—
41C22⋊C4—22no—
42C22⋊C4—22no—
43C22⋊C4—42no—
44C23—22no—
45C23—22no—
46C23—22no—
47C23—41no—
48C23—42no—
49C23—42no—
50C23—42no—
51C23C22⋊C411yescharacteristic
52C23C22⋊C411yescharacteristic
53C23C2×D411yescharacteristic
54D4—42no—
55D4—42no—
56D4—44no—
57D4—44no—
58C2×C4—22no—
59C2×C4—22no—
60C2×C4—22no—
61C2×C4—41no—
62C2×C4—41no—
63C2×C4—41no—
64C2×C4—41no—
65C2×C4—42no—
66C2×C4—42no—
67C2×C4—42no—
68C2×C4—42no—
69C2×C4C22⋊C411yescharacteristic
70C2×C4C22⋊C411yescharacteristic
71C2×C4C22×C411yescharacteristic
72C2×C4C2×D411yescharacteristic
73C8—42no—
74C22—21no—
75C22—21no—
76C22—21no—
77C22—21no—
78C22—21no—
79C22—21no—
80C22—21no—
81C22—44no—
82C22—42no—
83C22—42no—
84C22—42no—
85C22—84no—
86C22#9912yes—
87C22C2×C22⋊C411yescharacteristic
88C22AGL(1,ℤ/8)12yes—
89C4—21no—
90C4—21no—
91C4—21no—
92C4—21no—
93C4—42no—
94C4—82no—
95C2—22no—
96C2—22no—
97C2—41no—
98C2—82no—
99C2C2×#9911yescharacteristic
100C2#41811yescharacteristic
101C2#42111yescharacteristic
subgroupG/H
center, socleC22C2×C22⋊C4
commutatorC2×C4C22×C4
Frattini, cosocleC22×C4C23
prime radicalC2×#121C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #407C2semidirect
3. #381C2semidirect
4. #327C2semidirect
6. C22×D4C4semidirect
11. #121C4semidirect
13. C2×C4⋊C4C4semidirect
17. C22⋊C8C22semidirect
19. #95C22semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#1212
#4072
#3812
#3272