Finite Group Databasebeta

← #1625

Subgroups — #1625

Order 128.

128 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 36 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#547C211yesmaximal, characteristic, complemented
2#486C211yesmaximal, characteristic, complemented
3#459C211yesmaximal, characteristic, complemented
4#459C211yesmaximal, characteristic, complemented
5#332C211yesmaximal, characteristic, complemented
6D4⋊C8C211yesmaximal, characteristic, complemented
7D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#41—21no—
9C2×D8—21no—
10C2×D8—21no—
11C4⋊Q8C2211yescharacteristic
12#127C2211yescharacteristic
13#123—22no—
14C22⋊Q8—21no—
15#121—21no—
16C4×D4C2211yescharacteristic
17C4×D4C2211yescharacteristic
18C2×C4⋊C4—21no—
19C4⋊C8C2211yescharacteristic
20C4⋊C8C2211yescharacteristic
21#102—21no—
22#102—21no—
23#102—22no—
24C4×C8C2211yescharacteristic
25#41—42no—
26#41—42no—
27C2×Q8—21no—
28C2×D4—22no—
29C2×D4—21no—
30C2×D4—21no—
31C2×D4—21no—
32C2×D4D411yescharacteristic
33C2×D4D411yescharacteristic
34C22×C4—22no—
35C22×C4—21no—
36C22×C4—21no—
37D8—42no—
38D8—42no—
39C2×C8—21no—
40C2×C8—21no—
41C2×C8—21no—
42C4⋊C4—22no—
43C4⋊C4—21no—
44C4⋊C4D411yescharacteristic
45C4⋊C4D411yescharacteristic
46C4⋊C4D412yes—
47C22⋊C4—22no—
48C22⋊C4—21no—
49C22⋊C4—21no—
50C42C2311yescharacteristic
51C23—21no—
52C23—21no—
53C23—41no—
54Q8—41no—
55Q8—41no—
56D4—21no—
57D4—21no—
58D4—21no—
59D4—42no—
60D4—44no—
61D4—41no—
62D4—41no—
63D4—41no—
64D4—41no—
65D4D812yescomplemented
66C2×C4—21no—
67C2×C4—21no—
68C2×C4—22no—
69C2×C4—44no—
70C2×C4—41no—
71C2×C4—41no—
72C2×C4—41no—
73C2×C4—41no—
74C2×C4C2×D411yescharacteristic
75C2×C4C2×D411yescharacteristic
76C2×C4C2×D411yescharacteristic
77C8—41no—
78C8—41no—
79C8—41no—
80C22—22no—
81C22—41no—
82C22—41no—
83C22—41no—
84C22—41no—
85C22—41no—
86C22—81no—
87C22—81no—
88C22—81no—
89C22#12011yescharacteristic
90C4—21no—
91C4—41no—
92C4—42no—
93C4—41no—
94C4C2×D811yescharacteristic
95C4#13511yescharacteristic
96C4AGL(1,ℤ/8)11yescharacteristic
97C4AGL(1,ℤ/8)11yescharacteristic
98C2—42no—
99C2—81no—
100C2—161no—
101C2#44711yescharacteristic
102C2#44911yescharacteristic
103C2#45411yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC42C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #547C2semidirect
2. #486C2semidirect
3. #459C2semidirect
4. #459C2semidirect
5. #332C2semidirect
6. D4⋊C8C2semidirect
7. D4⋊C8C2semidirect
65. D4D8semidirect

Maximal subgroups

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

subgroupindex
#5472
#4862
#4592
#3322
D4⋊C82