Finite Group Databasebeta

← #1504

Subgroups — #1504

Order 128.

167 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 48 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#525C211yesmaximal, characteristic
2#407C211yesmaximal, characteristic, complemented
3C2×#95C211yesmaximal, characteristic, complemented
4#328C212yesmaximal, complemented
5#327C212yesmaximal, complemented
6C2×#41C411yescharacteristic, complemented
7C23×C4—21no—
8C23×C4C2211yescharacteristic
9C2×M16—21no—
10#123—21no—
11C22⋊Q8C411yescharacteristic, complemented
12C22⋊Q8C2211yescharacteristic
13#121C411yescharacteristic, complemented
14#121C2211yescharacteristic
15#120—21no—
16C4×D4—21no—
17C4×D4—21no—
18#117C411yescharacteristic, complemented
19C22⋊C8—21no—
20C22⋊C8C2212yes—
21#95—21no—
22#95C2212yes—
23C24D411yescharacteristic
24#41—42no—
25C2×Q8C2×C411yescharacteristic, complemented
26C2×D4—21no—
27C2×D4—21no—
28C2×D4—21no—
29C2×D4C2×C411yescharacteristic, complemented
30C22×C4—24no—
31C22×C4—22no—
32C22×C4—21no—
33C22×C4—42no—
34C22×C4—42no—
35C22×C4—42no—
36C22×C4C2×C411yescharacteristic
37C22×C4C2×C411yescharacteristic
38C22×C4D411yescharacteristic
39C22×C4D411yescharacteristic
40C22×C4D411yescharacteristic
41C22×C4C2311yescharacteristic
42M16—42no—
43C2×C8—22no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C4⋊C4C2×C411yescharacteristic, complemented
47C4⋊C4C2×C411yescharacteristic, complemented
48C22⋊C4—21no—
49C22⋊C4—21no—
50C22⋊C4—21no—
51C22⋊C4—21no—
52C22⋊C4—21no—
53C42—21no—
54C23—22no—
55C23—22no—
56C23—21no—
57C23—21no—
58C23—21no—
59C23C22⋊C411yescharacteristic
60C23C22⋊C411yescharacteristic
61C23C2×D411yescharacteristic
62Q8—41no—
63D4—41no—
64D4—42no—
65D4—42no—
66D4—41no—
67D4—41no—
68C2×C4—22no—
69C2×C4—22no—
70C2×C4—24no—
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—21no—
74C2×C4—41no—
75C2×C4—41no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4—42no—
79C2×C4—48no—
80C2×C4—42no—
81C2×C4—42no—
82C2×C4C22⋊C411yescharacteristic
83C2×C4C22⋊C411yescharacteristic
84C2×C4C22×C411yescharacteristic
85C2×C4C2×D411yescharacteristic
86C8—42no—
87C22—24no—
88C22—24no—
89C22—21no—
90C22—21no—
91C22—21no—
92C22—21no—
93C22—21no—
94C22—21no—
95C22—41no—
96C22—41no—
97C22—41no—
98C22#9912yes—
99C22#10412yes—
100C22C2×C22⋊C411yescharacteristic
101C22AGL(1,ℤ/8)11yescharacteristic
102C22#13711yescharacteristic
103C4—21no—
104C4—21no—
105C4—21no—
106C4—21no—
107C4—44no—
108C4—41no—
109C4—41no—
110C4—41no—
111C2—21no—
112C2—22no—
113C2—22no—
114C2—21no—
115C2—81no—
116C2C2×#9911yescharacteristic
117C2#41711yescharacteristic
118C2C2×#10411yescharacteristic
subgroupG/H
center, socleC22C2×C22⋊C4
commutatorC2×C4C22×C4
Frattini, cosocleC22×C4C23
prime radical#525C2
LCSG ▷ C2×C4 ▷ C22 ▷ 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×#95C2semidirect
4. #328C2semidirect
5. #327C2semidirect
6. C2×#41C4semidirect
11. C22⋊Q8C4semidirect
13. #121C4semidirect
18. #117C4semidirect
25. C2×Q8C2×C4semidirect
29. C2×D4C2×C4semidirect
46. C4⋊C4C2×C4semidirect
47. C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#5252
#4072
C2×#952
#3282
#3272