Finite Group Databasebeta

← #1913

Subgroups — #1913

Order 128.

190 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 54 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#534C211yesmaximal, characteristic
2#518C211yesmaximal, characteristic, complemented
3C2×#99C211yesmaximal, characteristic, complemented
4C2×#99C212yesmaximal, complemented
5#342C212yesmaximal, complemented
6C22×D4C2211yescharacteristic
7C22×D4C2211yescharacteristic
8#123—21no—
9#123C412yescomplemented
10#121—22no—
11#120—21no—
12#120C412yescomplemented
13C4×D4—24no—
14#117—21no—
15C2×C22⋊C4—21no—
16C2×C22⋊C4C2212yes—
17C2×C22⋊C4C2211yescharacteristic
18C2×C22⋊C4C2212yes—
19#99—22no—
20#99—24no—
21C24C2×C411yescharacteristic, complemented
22C24D411yescharacteristic
23C24C2311yescharacteristic
24C2×D4—22no—
25C2×D4—22no—
26C2×D4—22no—
27C2×D4—21no—
28C2×D4—21no—
29C2×D4—22no—
30C2×D4—22no—
31C2×D4C2×C412yes—
32C2×D4D412yes—
33C22×C4—22no—
34C22×C4—22no—
35C22×C4—21no—
36C22×C4—22no—
37C22×C4C2×C411yescharacteristic, complemented
38C22×C4D411yescharacteristic
39C4⋊C4—22no—
40C4⋊C4—22no—
41C22⋊C4—22no—
42C22⋊C4—22no—
43C22⋊C4—22no—
44C22⋊C4—21no—
45C22⋊C4—22no—
46C22⋊C4—22no—
47C22⋊C4—22no—
48C22⋊C4C2×C412yescomplemented
49C22⋊C4D414yescomplemented
50C42—22no—
51C23—21no—
52C23—21no—
53C23—21no—
54C23—22no—
55C23—81no—
56C23—81no—
57C23—81no—
58C23C22⋊C412yes—
59C23C22×C411yescharacteristic
60C23C2×D411yescharacteristic
61C23C2×D411yescharacteristic
62C23C2×D412yes—
63C23#4112yes—
64D4—24no—
65D4—24no—
66D4—41no—
67D4—42no—
68D4—41no—
69D4—82no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—22no—
73C2×C4—21no—
74C2×C4—22no—
75C2×C4—24no—
76C2×C4—42no—
77C2×C4—82no—
78C2×C4—82no—
79C2×C4—81no—
80C2×C4—82no—
81C2×C4C22⋊C412yes—
82C22—22no—
83C22—42no—
84C22—41no—
85C22—42no—
86C22—41no—
87C22—41no—
88C22—42no—
89C22—42no—
90C22—42no—
91C22—41no—
92C22—81no—
93C22—82no—
94C22C2×C22⋊C411yescharacteristic
95C22C4×D412yes—
96C22#12011yescharacteristic
97C22#12112yes—
98C22#12311yescharacteristic
99C4—22no—
100C4—44no—
101C4—42no—
102C4—41no—
103C4—82no—
104C2—21no—
105C2—21no—
106C2—22no—
107C2—22no—
108C2—21no—
109C2—42no—
110C2—81no—
111C2#38611yescharacteristic
subgroupG/H
center, socleC2#386
commutatorC23C22×C4
Frattini, cosocleC24C23
prime radical#534C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #518C2semidirect
3. C2×#99C2semidirect
4. C2×#99C2semidirect
5. #342C2semidirect
9. #123C4semidirect
12. #120C4semidirect
21. C24C2×C4semidirect
37. C22×C4C2×C4semidirect
48. C22⋊C4C2×C4semidirect
49. C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#5342
#5182
C2×#992
#3422