Finite Group Databasebeta

← #2122

Subgroups — #2122

Order 128.

133 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 133 conjugacy classes of subgroups — too many to draw. Only its 42 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
1#535C211yesmaximal, characteristic
2#413C211yesmaximal, characteristic, complemented
3#410C211yesmaximal, characteristic, complemented
4AΣL(1,𝔽16)C214yesmaximal, complemented
5C2×#41C2211yescharacteristic
6C22×D4C411yescharacteristic, complemented
7C2×M16—21no—
8#127—21no—
9#124—21no—
10#121—21no—
11#121—21no—
12#120C412yescomplemented
13#120C2212yes—
14#117—21no—
15#117C411yescharacteristic, complemented
16#101—21no—
17#100C2212yes—
18#99—21no—
19#99C2212yes—
20C24C2×C412yes—
21#41—41no—
22#41—41no—
23C2×Q8D411yescharacteristic
24C2×D4—22no—
25C2×D4—22no—
26C2×D4—21no—
27C2×D4—21no—
28C2×D4—41no—
29C2×D4—41no—
30C2×D4C2×C411yescharacteristic
31C2×D4D411yescharacteristic
32C2×D4C2311yescharacteristic
33C22×C4C2×C411yescharacteristic
34C22×C4D411yescharacteristic
35C22×C4D411yescharacteristic
36M16—22no—
37M16—41no—
38C2×C8—41no—
39C4⋊C4—21no—
40C4⋊C4—41no—
41C22⋊C4—22no—
42C22⋊C4—22no—
43C22⋊C4—22no—
44C22⋊C4C2×C412yes—
45C42—21no—
46C42—41no—
47C23—22no—
48C23—42no—
49C23—82no—
50C23C22⋊C411yescharacteristic
51C23C22×C411yescharacteristic
52C23C2×D411yescharacteristic
53Q8—41no—
54D4—41no—
55D4—41no—
56D4—41no—
57D4—42no—
58D4—41no—
59D4—41no—
60D4—82no—
61D4—82no—
62C2×C4—21no—
63C2×C4—21no—
64C2×C4—21no—
65C2×C4—21no—
66C2×C4—21no—
67C2×C4—21no—
68C2×C4—22no—
69C2×C4—42no—
70C2×C4C22⋊C411yescharacteristic
71C2×C4C22⋊C411yescharacteristic
72C2×C4C22⋊C411yescharacteristic
73C2×C4C2×D411yescharacteristic
74C8—42no—
75C22—21no—
76C22—21no—
77C22—21no—
78C22—41no—
79C22—41no—
80C22—41no—
81C22—42no—
82C22—42no—
83C22—42no—
84C22—84no—
85C22C2×C22⋊C411yescharacteristic
86C4—21no—
87C4—21no—
88C4—21no—
89C4—42no—
90C4—82no—
91C4#9911yescharacteristic
92C4#9911yescharacteristic
93C2—21no—
94C2—41no—
95C2—41no—
96C2—41no—
97C2—82no—
98C2C2×#9911yescharacteristic
subgroupG/H
center, socleC2C2×#99
commutatorC23C22×C4
Frattini, cosocleC2×D4C23
prime radical#535C2
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G
LECSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #413C2semidirect
3. #410C2semidirect
4. AΣL(1,𝔽16)C2semidirect
6. C22×D4C4semidirect
12. #120C4semidirect
15. #117C4semidirect

Maximal subgroups

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

subgroupindex
#5352
#4132
#4102
AΣL(1,𝔽16)2