Finite Group Databasebeta

← #3003

Subgroups — #3003

Order 128.

497 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C23×D4C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)C212yesmaximal, complemented
3#525C211yesmaximal, characteristic, complemented
4#450C214yesmaximal, complemented
5#447C214yesmaximal, complemented
6#418C212yesmaximal, complemented
7#407C211yesmaximal, characteristic, complemented
8C25—21no—
9C2×#41C2211yescharacteristic, complemented
10C22×D4—28no—
11C22×D4—22no—
12C22×D4—21no—
13C22×D4C2214yes—
14C22×D4C2212yes—
15C23×C4C2211yescharacteristic
16AGL(1,ℤ/8)—28no—
17C2×SD16C2214yescomplemented
18C2×D8C2214yescomplemented
19C2×M16C2212yescomplemented
20#123—21no—
21C22⋊Q8C2212yescomplemented
22#121C2212yescomplemented
23#120—21no—
24C4×D4—22no—
25#117C2211yescharacteristic, complemented
26#102C2218yescomplemented
27C22⋊C8C2214yescomplemented
28C24—28no—
29C24—24no—
30C24—22no—
31C24—48no—
32C24D411yescharacteristic
33#41—24no—
34C2×Q8C2311yescharacteristic, complemented
35C2×D4—28no—
36C2×D4—216no—
37C2×D4—216no—
38C2×D4—22no—
39C2×D4—21no—
40C2×D4—24no—
41C2×D4—22no—
42C2×D4—24no—
43C2×D4D418yescomplemented
44C2×D4C2314yes—
45C2×D4C2311yescharacteristic, complemented
46C22×C4—24no—
47C22×C4—21no—
48C22×C4D412yes—
49C22×C4D411yescharacteristic
50C22×C4C2312yes—
51C22×C4C2311yescharacteristic
52SD16—28no—
53D8—28no—
54M16—24no—
55C2×C8C2314yescomplemented
56C4⋊C4—22no—
57C4⋊C4C2312yescomplemented
58C22⋊C4—21no—
59C22⋊C4—22no—
60C22⋊C4—22no—
61C42—21no—
62C23—22no—
63C23—24no—
64C23—216no—
65C23—28no—
66C23—24no—
67C23—21no—
68C23—432no—
69C23—48no—
70C23—48no—
71C23—48no—
72C23C2×D411yescharacteristic
73C23C2×D412yes—
74Q8—22no—
75D4—216no—
76D4—28no—
77D4—24no—
78D4—22no—
79D4—42no—
80D4—44no—
81D4C2×D418yes—
82C2×C4—22no—
83C2×C4—24no—
84C2×C4—24no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—42no—
88C2×C4C2×D412yes—
89C2×C4C2×D411yescharacteristic
90C2×C4C2×D414yes—
91C2×C4C2411yescharacteristic
92C8—24no—
93C22—28no—
94C22—22no—
95C22—22no—
96C22—22no—
97C22—28no—
98C22—416no—
99C22—432no—
100C22—48no—
101C22—41no—
102C22—42no—
103C22#12012yes—
104C22AGL(1,ℤ/8)14yescomplemented
105C22C22×D411yescharacteristic
106C4—22no—
107C4—41no—
108C4—42no—
109C4#12012yes—
110C4C22×D412yes—
111C2—22no—
112C2—24no—
113C2—48no—
114C2—81no—
115C2C2×#12011yescharacteristic
116C2C2×AGL(1,ℤ/8)12yes—
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×D4C2semidirect
2. C2×AGL(1,ℤ/8)C2semidirect
3. #525C2semidirect
4. #450C2semidirect
5. #447C2semidirect
6. #418C2semidirect
7. #407C2semidirect
9. C2×#41C22semidirect
17. C2×SD16C22semidirect
18. C2×D8C22semidirect
19. C2×M16C22semidirect
21. C22⋊Q8C22semidirect
22. #121C22semidirect
25. #117C22semidirect
26. #102C22semidirect
27. C22⋊C8C22semidirect
34. C2×Q8C23semidirect
43. C2×D4D4semidirect
45. C2×D4C23semidirect
55. C2×C8C23semidirect
57. C4⋊C4C23semidirect
104. C22AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C23×D42
C2×AGL(1,ℤ/8)2
#5252
#4502
#4472
#4182
#4072