Finite Group Databasebeta

← #1639

Subgroups — #1639

Order 128.

154 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 38 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
1D42C211yesmaximal, characteristic
2#486C211yesmaximal, characteristic, complemented
3#474C211yesmaximal, characteristic, complemented
4#461C211yesmaximal, characteristic, complemented
5#331C211yesmaximal, characteristic, complemented
6D4⋊C8C211yesmaximal, characteristic, complemented
7D4⋊C8C211yesmaximal, characteristic, complemented
8C22×D4—21no—
9C22×D4—21no—
10C2×SD16—21no—
11C4⋊Q8C2211yescharacteristic, complemented
12#127C2211yescharacteristic
13#121—21no—
14#121—21no—
15#120—22no—
16C4×D4C2211yescharacteristic
17C4×D4C2211yescharacteristic
18#107—21no—
19C4⋊C8C2211yescharacteristic, complemented
20C4⋊C8C2211yescharacteristic, complemented
21#103—21no—
22#102—21no—
23#102—22no—
24C4×C8C2211yescharacteristic, complemented
25C24—22no—
26C2×Q8—21no—
27C2×D4—22no—
28C2×D4—22no—
29C2×D4—21no—
30C2×D4—21no—
31C2×D4—42no—
32C2×D4—42no—
33C2×D4—42no—
34C2×D4—42no—
35C2×D4D411yescharacteristic
36C2×D4D411yescharacteristic
37C2×D4D412yes—
38C22×C4—21no—
39C22×C4—21no—
40SD16—42no—
41C2×C8—21no—
42C2×C8—21no—
43C2×C8—21no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C4⋊C4D411yescharacteristic
47C4⋊C4D411yescharacteristic
48C22⋊C4—22no—
49C22⋊C4—21no—
50C22⋊C4—21no—
51C42C2311yescharacteristic
52C23—21no—
53C23—21no—
54C23—22no—
55C23—44no—
56C23—44no—
57C23—82no—
58Q8—41no—
59Q8—41no—
60D4—21no—
61D4—21no—
62D4—42no—
63D4—44no—
64D4—44no—
65D4—41no—
66D4—41no—
67D4—41no—
68D4—41no—
69D4D812yes—
70D4SD1612yescomplemented
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—41no—
74C2×C4—41no—
75C2×C4—41no—
76C2×C4—41no—
77C2×C4—41no—
78C2×C4C2×D411yescharacteristic
79C2×C4C2×D411yescharacteristic
80C2×C4C2×D411yescharacteristic
81C8—41no—
82C8—41no—
83C8—41no—
84C22—22no—
85C22—22no—
86C22—41no—
87C22—41no—
88C22—41no—
89C22—41no—
90C22—42no—
91C22—42no—
92C22—42no—
93C22—88no—
94C22#12011yescharacteristic
95C4—21no—
96C4—41no—
97C4—41no—
98C4—81no—
99C4C2×D811yescharacteristic
100C4C2×SD1611yescharacteristic
101C4AGL(1,ℤ/8)11yescharacteristic
102C4AGL(1,ℤ/8)11yescharacteristic
103C2—42no—
104C2—42no—
105C2—82no—
106C2#44711yescharacteristic
107C2#45011yescharacteristic
108C2#45511yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC42C23
prime radicalD42C2
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
2. #486C2semidirect
3. #474C2semidirect
4. #461C2semidirect
5. #331C2semidirect
6. D4⋊C8C2semidirect
7. D4⋊C8C2semidirect
11. C4⋊Q8C22semidirect
19. C4⋊C8C22semidirect
20. C4⋊C8C22semidirect
24. C4×C8C22semidirect
70. D4SD16semidirect

Maximal subgroups

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

subgroupindex
D422
#4862
#4742
#4612
#3312
D4⋊C82