Finite Group Databasebeta

← #1468

Subgroups — #1468

Order 128.

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

normal subgroups only

This group has 140 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×#123C211yesmaximal, characteristic
2#407C211yesmaximal, characteristic, complemented
3C2×C22⋊C8C211yesmaximal, characteristic, complemented
4#324C212yesmaximal, complemented
5#323C212yesmaximal, complemented
6C22×D4C411yescharacteristic
7C23×C4C2211yescharacteristic
8C2×M16—21no—
9C22×C8—21no—
10#123—21no—
11#123—21no—
12#123—21no—
13#123—21no—
14C2×C4⋊C4C411yescharacteristic
15C2×C4⋊C4C2211yescharacteristic
16C2×C22⋊C4C411yescharacteristic
17C2×C22⋊C4C411yescharacteristic
18C2×C22⋊C4C2211yescharacteristic
19C22⋊C8—21no—
20C22⋊C8—21no—
21C22⋊C8C2212yes—
22C22⋊C8C2212yes—
23C24C2×C411yescharacteristic
24C24C2×C411yescharacteristic
25C2×D4—21no—
26C2×D4—21no—
27C2×D4—42no—
28C22×C4—42no—
29C22×C4C2×C411yescharacteristic
30C22×C4C2×C411yescharacteristic
31C22×C4C2×C411yescharacteristic
32C22×C4C2×C411yescharacteristic
33C22×C4D411yescharacteristic
34C22×C4D411yescharacteristic
35C22×C4D411yescharacteristic
36C22×C4D411yescharacteristic
37C22×C4C2311yescharacteristic
38M16—42no—
39C2×C8—22no—
40C2×C8—22no—
41C2×C8—42no—
42C4⋊C4—21no—
43C4⋊C4—21no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C22⋊C4—21no—
47C22⋊C4—21no—
48C22⋊C4—21no—
49C22⋊C4—21no—
50C22⋊C4—21no—
51C22⋊C4—21no—
52C23—22no—
53C23—22no—
54C23—21no—
55C23—22no—
56C23—41no—
57C23—41no—
58C23—41no—
59C23C22⋊C411yescharacteristic
60C23C22⋊C411yescharacteristic
61C23C22×C411yescharacteristic
62D4—42no—
63D4—42no—
64C2×C4—22no—
65C2×C4—22no—
66C2×C4—21no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—22no—
70C2×C4—41no—
71C2×C4—41no—
72C2×C4—41no—
73C2×C4—41no—
74C2×C4—41no—
75C2×C4—41no—
76C2×C4C22⋊C411yescharacteristic
77C2×C4C22⋊C411yescharacteristic
78C2×C4C2×D411yescharacteristic
79C2×C4C2×D411yescharacteristic
80C8—42no—
81C8—42no—
82C22—21no—
83C22—21no—
84C22—21no—
85C22—21no—
86C22—21no—
87C22—21no—
88C22—21no—
89C22—44no—
90C22—41no—
91C22—41no—
92C22—41no—
93C22—82no—
94C22#9912yes—
95C22C2×C22⋊C411yescharacteristic
96C22#13112yes—
97C4—21no—
98C4—21no—
99C4—21no—
100C4—21no—
101C4—41no—
102C4—41no—
103C4—41no—
104C2—22no—
105C2—22no—
106C2—41no—
107C2—81no—
108C2#40811yescharacteristic
109C2C2×#9911yescharacteristic
110C2#41311yescharacteristic
subgroupG/H
center, socleC22C2×C22⋊C4
commutatorC23C22×C4
Frattini, cosocleC22×C4C23
prime radicalC22×D4C4
LCSG ▷ C23 ▷ C2 ▷ 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. C2×C22⋊C8C2semidirect
4. #324C2semidirect
5. #323C2semidirect

Maximal subgroups

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

subgroupindex
C2×#1232
#4072
C2×C22⋊C82
#3242
#3232