Finite Group Databasebeta

← #3232

Subgroups — #3232

Order 128.

167 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 167 conjugacy classes of subgroups — too many to draw. Only its 86 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#529C211yesmaximal, characteristic
2C2×#125C211yesmaximal, characteristic, complemented
3#499C212yesmaximal, complemented
4#488C212yesmaximal, complemented
5#484C211yesmaximal, characteristic, complemented
6#483C211yesmaximal, characteristic, complemented
7#481C211yesmaximal, characteristic
8#480C211yesmaximal, characteristic
9#479C212yesmaximal, complemented
10#478C212yesmaximal, complemented
11#431C211yesmaximal, characteristic, complemented
12#126—21no—
13#125—21no—
14#125—21no—
15#125C2212yes—
16#125C2212yes—
17#125C2211yescharacteristic
18#124C2211yescharacteristic
19#123—21no—
20C22⋊Q8C2211yescharacteristic
21#121C2211yescharacteristic
22C4×Q8C2211yescharacteristic
23C4×D4—21no—
24C4×D4C2211yescharacteristic
25#117—21no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4C2211yescharacteristic
29C2×C4⋊C4C2211yescharacteristic
30C2×C42C2211yescharacteristic
31#107C2212yes—
32#107C2212yes—
33#106C2212yes—
34#106C2212yes—
35C4⋊C8C2212yes—
36#103C2212yes—
37#103C2212yes—
38#102C2212yes—
39#102C2212yes—
40C22⋊C8C2211yescharacteristic
41C22⋊C8C2211yescharacteristic
42C4×C8C2212yes—
43C2×Q8C2311yescharacteristic
44C2×D4—21no—
45C2×D4C2311yescharacteristic
46C22×C4—21no—
47C22×C4—21no—
48C22×C4—21no—
49C22×C4D411yescharacteristic
50C22×C4D411yescharacteristic
51C22×C4C2311yescharacteristic
52C2×C8C2312yes—
53C2×C8C2312yes—
54C4⋊C4—22no—
55C4⋊C4—22no—
56C4⋊C4—22no—
57C4⋊C4—22no—
58C4⋊C4—21no—
59C4⋊C4—21no—
60C4⋊C4—21no—
61C4⋊C4—21no—
62C4⋊C4—21no—
63C4⋊C4—21no—
64C4⋊C4C2312yes—
65C4⋊C4C2312yes—
66C4⋊C4C2311yescharacteristic
67C4⋊C4C2311yescharacteristic
68C22⋊C4—21no—
69C22⋊C4—21no—
70C22⋊C4—21no—
71C22⋊C4—21no—
72C22⋊C4—21no—
73C42—21no—
74C42D411yescharacteristic
75C42D411yescharacteristic
76C42C2311yescharacteristic
77C42C2311yescharacteristic
78C23—21no—
79C23C2×D411yescharacteristic
80Q8—22no—
81D4—22no—
82D4—42no—
83C2×C4—22no—
84C2×C4—22no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—22no—
90C2×C4—41no—
91C2×C4—42no—
92C2×C4—42no—
93C2×C4—42no—
94C2×C4C2×D411yescharacteristic
95C2×C4C2×D411yescharacteristic
96C2×C4C2×D411yescharacteristic
97C2×C4C2×D411yescharacteristic
98C2×C4C2×D411yescharacteristic
99C2×C4C2411yescharacteristic
100C8—22no—
101C8—22no—
102C22—21no—
103C22—21no—
104C22—41no—
105C22—41no—
106C22—41no—
107C22#13512yes—
108C22C22×D411yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—21no—
113C4—21no—
114C4—41no—
115C4—42no—
116C4—41no—
117C4—41no—
118C4—42no—
119C42−1+412yes—
120C2—22no—
121C2—81no—
122C2#53611yescharacteristic
123C2C2×#13511yescharacteristic
124C2#57511yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
prime radical#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#125C2semidirect
3. #499C2semidirect
4. #488C2semidirect
5. #484C2semidirect
6. #483C2semidirect
9. #479C2semidirect
10. #478C2semidirect
11. #431C2semidirect

Maximal subgroups

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

subgroupindex
#5292
C2×#1252
#4992
#4882
#4842
#4832
#4812
#4802
#4792
#4782
#4312