Finite Group Databasebeta

← C2×#575 #3580

Subgroups — C2×#575 #3580

Order 128.

742 conjugacy classes of subgroups; only the 428 normal subgroups are shown.

normal subgroups only

This group has 742 conjugacy classes of subgroups — too many to draw. Only its 428 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
1C22×#41C212yesmaximal, complemented
2#575C2116yesmaximal, direct product factor
3C2×#137C214yesmaximal, complemented
4C2×AGL(1,ℤ/8)C214yesmaximal, complemented
5C2×#135C214yesmaximal, complemented
6C22×M16C211yesmaximal, characteristic, complemented
7C2×#41—22no—
8C2×#41—28no—
9C2×#41—22no—
10C2×#41C2214yes—
11C2×#41C2214yescomplemented
12C2×#41C22116yescomplemented
13C22×Q8C2212yescomplemented
14C22×D4—22no—
15C22×D4C2212yescomplemented
16C23×C4—22no—
17C23×C4C2211yescharacteristic
18#137C22132yescomplemented
19AGL(1,ℤ/8)C22132yescomplemented
20#135C22132yescomplemented
21C2×Q16C2214yescomplemented
22C2×SD16C2218yescomplemented
23C2×D8C2214yescomplemented
24C2×M16C2214yescomplemented
25C2×M16C2218yescomplemented
26C22×C8C2212yescomplemented
27C24—22no—
28C24D411yescharacteristic
29#41—28no—
30#41—216no—
31#41—216no—
32#41—28no—
33#41C23116yes—
34#41C23116yescomplemented
35C2×Q8—22no—
36C2×Q8—24no—
37C2×Q8C2314yes—
38C2×Q8C2318yescomplemented
39C2×D4—24no—
40C2×D4—24no—
41C2×D4—216no—
42C2×D4—22no—
43C2×D4—24no—
44C2×D4C2314yes—
45C2×D4C2318yescomplemented
46C22×C4—22no—
47C22×C4—24no—
48C22×C4—216no—
49C22×C4—24no—
50C22×C4D411yescharacteristic
51C22×C4D414yes—
52C22×C4D412yes—
53C22×C4C2312yes—
54C22×C4C2314yes—
55C22×C4C2311yescharacteristic
56Q16C23116yescomplemented
57SD16C23132yescomplemented
58D8C23116yescomplemented
59M16C23116yescomplemented
60C2×C8C2314yes—
61C2×C8C2318yescomplemented
62C23—24no—
63C23—28no—
64C23—42no—
65C23—48no—
66C23C2×D412yes—
67C23C2×D411yescharacteristic
68C23C2×D414yes—
69Q8—24no—
70Q8—28no—
71Q8C2418yes—
72D4—216no—
73D4—216no—
74D4—24no—
75D4—28no—
76D4C2418yes—
77C2×C4—24no—
78C2×C4—28no—
79C2×C4—216no—
80C2×C4—216no—
81C2×C4C2×D412yes—
82C2×C4C2×D418yes—
83C2×C4C2×D414yes—
84C2×C4C2×D412yes—
85C2×C4C2×D414yes—
86C2×C4C2×D411yescharacteristic
87C2×C4C2411yescharacteristic
88C2×C4C2412yes—
89C2×C4C2414yes—
90C8C2418yes—
91C22—24no—
92C22—22no—
93C22—28no—
94C22—44no—
95C22—416no—
96C22—48no—
97C22C22×D414yes—
98C22C22×D412yes—
99C22C22×D411yescharacteristic
100C4—28no—
101C4C22×D412yes—
102C4C22×D414yes—
103C4C22×D411yescharacteristic
104C4C2511yescharacteristic
105C2—24no—
106C2—22no—
107C2—48no—
108C2#57512yesdirect product factor
109C2C23×D411yescharacteristic
subgroupG/H
centerC2×C4C2×D4
commutator, Frattini, cosocleC4C25
socleC22C22×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#41C2semidirect
2. #575C2direct (N × H)
3. C2×#137C2semidirect
4. C2×AGL(1,ℤ/8)C2semidirect
5. C2×#135C2semidirect
6. C22×M16C2semidirect
11. C2×#41C22semidirect
12. C2×#41C22semidirect
13. C22×Q8C22semidirect
15. C22×D4C22semidirect
18. #137C22semidirect
19. AGL(1,ℤ/8)C22semidirect
20. #135C22semidirect
21. C2×Q16C22semidirect
22. C2×SD16C22semidirect
23. C2×D8C22semidirect
24. C2×M16C22semidirect
25. C2×M16C22semidirect
26. C22×C8C22semidirect
34. #41C23semidirect
38. C2×Q8C23semidirect
45. C2×D4C23semidirect
56. Q16C23semidirect
57. SD16C23semidirect
58. D8C23semidirect
59. M16C23semidirect
61. C2×C8C23semidirect
108. C2#575direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#412
#5752
C2×#1372
C2×AGL(1,ℤ/8)2
C2×#1352
C22×M162