Finite Group Databasebeta

← #1871

Subgroups — #1871

Order 128.

188 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C22×M16C211yesmaximal, characteristic, complemented
2#525C211yesmaximal, characteristic
3C2×#104C212yesmaximal, complemented
4C4×C22⋊C4C211yesmaximal, characteristic, complemented
5#339C212yesmaximal, complemented
6C2×#41C2211yescharacteristic
7C23×C4C2211yescharacteristic
8C2×M16—21no—
9C2×M16—24no—
10C2×M16C2212yes—
11C22×C8—21no—
12#123—21no—
13C22⋊Q8C411yescharacteristic, complemented
14C22⋊Q8C411yescharacteristic, complemented
15#121C411yescharacteristic, complemented
16#121C411yescharacteristic, complemented
17#120—21no—
18C4×D4—21no—
19C4×D4—21no—
20#117C2211yescharacteristic
21C2×C22⋊C4—21no—
22C2×C42C2212yes—
23#104—24no—
24#95—21no—
25C24D411yescharacteristic
26#41—22no—
27#41—22no—
28C2×Q8C2×C411yescharacteristic
29C2×D4—21no—
30C2×D4—21no—
31C2×D4—21no—
32C2×D4C2×C411yescharacteristic
33C22×C4—22no—
34C22×C4—22no—
35C22×C4—22no—
36C22×C4—21no—
37C22×C4C2×C411yescharacteristic
38C22×C4C2×C411yescharacteristic
39C22×C4D411yescharacteristic
40C22×C4D411yescharacteristic
41C22×C4D411yescharacteristic
42C22×C4C2311yescharacteristic
43M16—24no—
44M16—22no—
45M16D414yescomplemented
46C2×C8—24no—
47C2×C8—22no—
48C4⋊C4—21no—
49C4⋊C4—21no—
50C4⋊C4C2×C411yescharacteristic
51C4⋊C4C2×C411yescharacteristic
52C22⋊C4—24no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—21no—
57C22⋊C4—21no—
58C42—21no—
59C42—22no—
60C42—22no—
61C23—22no—
62C23—22no—
63C23—21no—
64C23—41no—
65C23C22⋊C411yescharacteristic
66C23C22⋊C411yescharacteristic
67C23C2×D411yescharacteristic
68Q8—22no—
69D4—22no—
70D4—42no—
71D4—42no—
72D4—42no—
73C2×C4—22no—
74C2×C4—22no—
75C2×C4—22no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—24no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4C22⋊C411yescharacteristic
89C2×C4C22⋊C411yescharacteristic
90C2×C4C22×C411yescharacteristic
91C2×C4C2×D412yes—
92C2×C4C2×D411yescharacteristic
93C2×C4#4112yes—
94C8—24no—
95C22—21no—
96C22—21no—
97C22—21no—
98C22—21no—
99C22—21no—
100C22—21no—
101C22—21no—
102C22—44no—
103C22—41no—
104C22—41no—
105C22—41no—
106C22#10412yescomplemented
107C22C2×C22⋊C411yescharacteristic
108C22#12112yes—
109C4—21no—
110C4—21no—
111C4—44no—
112C4—41no—
113C4—41no—
114C4—41no—
115C4C4×D412yes—
116C4#12011yescharacteristic
117C4#12311yescharacteristic
118C2—22no—
119C2—22no—
120C2—41no—
121C2—81no—
122C2#38611yescharacteristic
123C2C2×#10411yescharacteristic
124C2#42111yescharacteristic
subgroupG/H
centerC2×C4C2×D4
commutatorC2×C4C22×C4
Frattini, cosocleC22×C4C23
socleC22C2×C22⋊C4
prime radical#525C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×M16C2semidirect
3. C2×#104C2semidirect
4. C4×C22⋊C4C2semidirect
5. #339C2semidirect
13. C22⋊Q8C4semidirect
14. C22⋊Q8C4semidirect
15. #121C4semidirect
16. #121C4semidirect
45. M16D4semidirect
106. C22#104semidirect

Maximal subgroups

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

subgroupindex
C22×M162
#5252
C2×#1042
C4×C22⋊C42
#3392