Finite Group Databasebeta

← #1899

Subgroups — #1899

Order 128.

189 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 56 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
2C2×#117C211yesmaximal, characteristic, complemented
3C2×#99C212yesmaximal
4#381C211yesmaximal, characteristic, complemented
5#342C212yesmaximal, complemented
6C22×D4C2211yescharacteristic
7C23×C4C2211yescharacteristic
8#123—22no—
9#123—21no—
10#123—21no—
11#117—24no—
12C2×C4⋊C4—21no—
13C2×C4⋊C4C412yescomplemented
14C2×C22⋊C4C412yescomplemented
15C2×C22⋊C4C2212yes—
16C2×C22⋊C4C2211yescharacteristic
17C2×C22⋊C4C2212yes—
18C2×C42—21no—
19#99—24no—
20#95—22no—
21C24C2×C411yescharacteristic
22C24C2311yescharacteristic
23C2×D4—22no—
24C2×D4—21no—
25C2×D4—21no—
26C2×D4—22no—
27C22×C4—22no—
28C22×C4—22no—
29C22×C4—22no—
30C22×C4—22no—
31C22×C4C2×C411yescharacteristic
32C22×C4C2×C412yes—
33C22×C4C2×C412yes—
34C22×C4D412yes—
35C22×C4D411yescharacteristic
36C22×C4D411yescharacteristic
37C4⋊C4—24no—
38C4⋊C4—22no—
39C4⋊C4—22no—
40C22⋊C4—22no—
41C22⋊C4—21no—
42C22⋊C4—21no—
43C22⋊C4—22no—
44C22⋊C4—22no—
45C22⋊C4—22no—
46C22⋊C4—22no—
47C22⋊C4D414yes—
48C42—24no—
49C23—21no—
50C23—22no—
51C23—42no—
52C23—42no—
53C23C22×C411yescharacteristic
54C23C2×D411yescharacteristic
55C23C2×D411yescharacteristic
56C23C2×D412yes—
57C23#4112yes—
58D4—41no—
59D4—42no—
60D4—41no—
61C2×C4—24no—
62C2×C4—21no—
63C2×C4—21no—
64C2×C4—24no—
65C2×C4—21no—
66C2×C4—21no—
67C2×C4—22no—
68C2×C4—21no—
69C2×C4—22no—
70C2×C4—24no—
71C2×C4—44no—
72C2×C4—42no—
73C2×C4—44no—
74C2×C4—42no—
75C2×C4C22⋊C412yes—
76C2×C4C22⋊C412yes—
77C22—22no—
78C22—22no—
79C22—22no—
80C22—42no—
81C22—42no—
82C22—41no—
83C22—42no—
84C22—82no—
85C22C2×C22⋊C411yescharacteristic
86C22C4×D412yes—
87C22#12011yescharacteristic
88C22#12112yes—
89C22#12311yescharacteristic
90C4—22no—
91C4—22no—
92C4—44no—
93C4—41no—
94C4—42no—
95C4—82no—
96C2—21no—
97C2—22no—
98C2—22no—
99C2—21no—
100C2—81no—
101C2#38611yescharacteristic
102C2#41012yes—
subgroupG/H
center, socleC22C2×C22⋊C4
commutatorC23C22×C4
Frattini, cosocleC24C23
prime radicalC22×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#117C2semidirect
4. #381C2semidirect
5. #342C2semidirect
13. C2×C4⋊C4C4semidirect
14. C2×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#1232
C2×#1172
C2×#992
#3812
#3422