Finite Group Databasebeta

← #1597

Subgroups — #1597

Order 128.

136 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 32 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#539C211yesmaximal, characteristic, complemented
2#450C211yesmaximal, characteristic, complemented
3#447C211yesmaximal, characteristic
4#394C211yesmaximal, characteristic, complemented
5#328C211yesmaximal, characteristic, complemented
6#327C211yesmaximal, characteristic, complemented
7#324C211yesmaximal, characteristic, complemented
8C22×D4C2211yescharacteristic
9C2×SD16—21no—
10C2×D8—21no—
11#126—21no—
12#125—21no—
13#123—21no—
14#123—21no—
15C22⋊Q8—21no—
16C22⋊Q8C2211yescharacteristic
17#121C2211yescharacteristic
18C4×D4—21no—
19C2×C4⋊C4C2211yescharacteristic
20C2×C22⋊C4—21no—
21C2×C22⋊C4—21no—
22#102—21no—
23#102—21no—
24C22⋊C8C2211yescharacteristic
25C22⋊C8C2211yescharacteristic
26#95C2211yescharacteristic
27C24—21no—
28C2×Q8D411yescharacteristic
29C2×D4—21no—
30C2×D4—21no—
31C2×D4—21no—
32C2×D4—42no—
33C2×D4D411yescharacteristic
34C22×C4—21no—
35C22×C4—21no—
36C22×C4D411yescharacteristic
37C22×C4D411yescharacteristic
38C22×C4C2311yescharacteristic
39SD16—42no—
40D8—42no—
41C2×C8—21no—
42C2×C8—21no—
43C4⋊C4—21no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C4⋊C4—21no—
47C4⋊C4—21no—
48C4⋊C4—21no—
49C4⋊C4D411yescharacteristic
50C4⋊C4D411yescharacteristic
51C22⋊C4—21no—
52C22⋊C4—21no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—42no—
57C22⋊C4—42no—
58C42—21no—
59C23—22no—
60C23—21no—
61C23—41no—
62C23—41no—
63C23—41no—
64C23—41no—
65C23—41no—
66C23—41no—
67C23C2×D411yescharacteristic
68Q8—41no—
69D4—22no—
70D4—22no—
71D4—41no—
72D4—41no—
73D4—41no—
74D4—41no—
75D4—41no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—81no—
89C2×C4C2×D411yescharacteristic
90C2×C4C2×D411yescharacteristic
91C8—41no—
92C8—41no—
93C22—21no—
94C22—21no—
95C22—42no—
96C22—42no—
97C22—41no—
98C22—41no—
99C22—41no—
100C22—42no—
101C22—81no—
102C22—81no—
103C22—81no—
104C22—81no—
105C22#12011yescharacteristic
106C22#13511yescharacteristic
107C22AGL(1,ℤ/8)11yescharacteristic
108C4—21no—
109C4—21no—
110C4—41no—
111C4—41no—
112C4—41no—
113C4—41no—
114C4—41no—
115C4—81no—
116C2—21no—
117C2—21no—
118C2—81no—
119C2—82no—
120C2#44911yescharacteristic
121C2#45411yescharacteristic
122C2B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC22×C4C23
prime radical#447C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #539C2semidirect
2. #450C2semidirect
4. #394C2semidirect
5. #328C2semidirect
6. #327C2semidirect
7. #324C2semidirect

Maximal subgroups

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

subgroupindex
#5392
#4502
#4472
#3942
#3282
#3272
#3242