Finite Group Databasebeta

← #1609

Subgroups — #1609

Order 128.

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

normal subgroups only

This group has 108 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
2#483C211yesmaximal, characteristic, complemented
3#480C211yesmaximal, characteristic
4#398C211yesmaximal, characteristic, complemented
5#328C211yesmaximal, characteristic, complemented
6#327C211yesmaximal, characteristic
7#324C211yesmaximal, characteristic, complemented
8#126—21no—
9#125—21no—
10#123—21no—
11#123—21no—
12C22⋊Q8—21no—
13C22⋊Q8C2211yescharacteristic
14#121C2211yescharacteristic
15C4×D4—21no—
16C2×C4⋊C4—21no—
17C2×C4⋊C4C2211yescharacteristic
18C2×C4⋊C4C2211yescharacteristic
19#107—21no—
20#106—21no—
21#103—21no—
22#102—21no—
23C22⋊C8C2211yescharacteristic
24C22⋊C8C2211yescharacteristic
25#95—21no—
26#95C2211yescharacteristic
27C2×Q8D411yescharacteristic
28C2×D4—21no—
29C2×D4D411yescharacteristic
30C22×C4—21no—
31C22×C4—21no—
32C22×C4—21no—
33C22×C4D411yescharacteristic
34C22×C4D411yescharacteristic
35C22×C4C2311yescharacteristic
36C2×C8—21no—
37C2×C8—21no—
38C4⋊C4—21no—
39C4⋊C4—21no—
40C4⋊C4—21no—
41C4⋊C4—21no—
42C4⋊C4—21no—
43C4⋊C4—21no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C4⋊C4—42no—
47C4⋊C4D411yescharacteristic
48C4⋊C4D411yescharacteristic
49C22⋊C4—21no—
50C22⋊C4—21no—
51C22⋊C4—21no—
52C22⋊C4—21no—
53C22⋊C4—21no—
54C42—21no—
55C23—21no—
56C23C2×D411yescharacteristic
57Q8—41no—
58D4—41no—
59D4—41no—
60D4—41no—
61C2×C4—22no—
62C2×C4—21no—
63C2×C4—21no—
64C2×C4—21no—
65C2×C4—21no—
66C2×C4—21no—
67C2×C4—21no—
68C2×C4—21no—
69C2×C4—41no—
70C2×C4—41no—
71C2×C4—41no—
72C2×C4—41no—
73C2×C4—41no—
74C2×C4—41no—
75C2×C4—41no—
76C2×C4—81no—
77C2×C4C2×D411yescharacteristic
78C2×C4C2×D411yescharacteristic
79C8—41no—
80C8—41no—
81C22—21no—
82C22—21no—
83C22—41no—
84C22—41no—
85C22—41no—
86C22#12011yescharacteristic
87C22#13511yescharacteristic
88C22AGL(1,ℤ/8)11yescharacteristic
89C4—21no—
90C4—21no—
91C4—42no—
92C4—41no—
93C4—41no—
94C4—41no—
95C4—41no—
96C4—41no—
97C4—81no—
98C2—21no—
99C2—21no—
100C2—81no—
101C2#44911yescharacteristic
102C2#45611yescharacteristic
103C2#45811yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC22×C4C23
prime radical#121C22
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
2. #483C2semidirect
4. #398C2semidirect
5. #328C2semidirect
7. #324C2semidirect

Maximal subgroups

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

subgroupindex
#5392
#4832
#4802
#3982
#3282
#3272
#3242