Finite Group Databasebeta

← C2×#577 #3582

Subgroups — C2×#577 #3582

Order 128.

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

normal subgroups only

This group has 730 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
1C2×2−1+4C211yesmaximal, characteristic, complemented
2C2×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3#577C2116yesmaximal, direct product factor
4C2×#137C213yesmaximal, complemented
5C2×AGL(1,ℤ/8)C213yesmaximal, complemented
6C2×#135C213yesmaximal, complemented
7C22×SD16C213yesmaximal, complemented
8C2×#131C211yesmaximal, characteristic, complemented
92−1+4—24no—
102−1+4C2218yescomplemented
11H(5,𝔽2)—24no—
12H(5,𝔽2)C2218yescomplemented
13C2×#41—21no—
14C2×#41—23no—
15C2×#41C2213yes—
16C2×#41C2211yescharacteristic
17C2×#41C2213yescomplemented
18C22×Q8—21no—
19C22×Q8C2213yescomplemented
20C22×D4—23no—
21C22×D4C2213yes—
22#137C22124yescomplemented
23AGL(1,ℤ/8)C22124yescomplemented
24#135C22124yescomplemented
25C2×Q16C2213yescomplemented
26C2×SD16C2213yescomplemented
27C2×SD16C2216yescomplemented
28C2×SD16C2211yescharacteristic, complemented
29C2×SD16C22124yescomplemented
30C2×D8C2213yescomplemented
31#131C2218yescomplemented
32C2×M16C2213yescomplemented
33C22×C8C2213yescomplemented
34C24—23no—
35#41—28no—
36#41—224no—
37#41—26no—
38#41—26no—
39#41D414yes—
40#41C23112yes—
41#41C2314yes—
42#41C23112yescomplemented
43C2×Q8—28no—
44C2×Q8—23no—
45C2×Q8—26no—
46C2×Q8D411yescharacteristic
47C2×Q8C2313yes—
48C2×Q8C2311yescharacteristic
49C2×Q8C23112yescomplemented
50C2×D4—23no—
51C2×D4—23no—
52C2×D4—224no—
53C2×D4—23no—
54C2×D4—26no—
55C2×D4D413yes—
56C2×D4C2311yescharacteristic
57C2×D4C2313yes—
58C2×D4C23112yes—
59C22×C4—23no—
60C22×C4—23no—
61C22×C4—23no—
62C22×C4C2313yes—
63Q16C23112yescomplemented
64SD16C23112yescomplemented
65SD16C23124yescomplemented
66SD16C2314yescomplemented
67D8C23112yescomplemented
68M16C23112yescomplemented
69C2×C8C2311yescharacteristic
70C2×C8C2313yes—
71C2×C8C23112yescomplemented
72C23—21no—
73C23—23no—
74C23—212no—
75C23—86no—
76C23C2×D413yes—
77Q8—23no—
78Q8—212no—
79Q8—21no—
80Q8C2×D414yes—
81Q8C2416yes—
82Q8C2412yes—
83D4—212no—
84D4—212no—
85D4—212no—
86D4—23no—
87D4—21no—
88D4C2×D4112yes—
89D4C2412yes—
90D4C2416yes—
91C2×C4—212no—
92C2×C4—21no—
93C2×C4—23no—
94C2×C4—212no—
95C2×C4—212no—
96C2×C4C2×D416yes—
97C2×C4C2×D413yes—
98C2×C4C2416yes—
99C2×C4C2411yescharacteristic
100C8C2412yes—
101C8C2416yes—
102C22—26no—
103C22—22no—
104C22—26no—
105C22—42no—
106C22—46no—
107C22—812no—
108C22C22×D416yes—
109C22C22×D411yescharacteristic
110C4—22no—
111C4—26no—
112C4C22×D416yes—
113C4C22×D411yescharacteristic
114C4C2511yescharacteristic
115C2—26no—
116C2—42no—
117C2—46no—
118C2#57712yesdirect product factor
119C2C23×D411yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC4C25
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#41 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4C2semidirect
2. C2×H(5,𝔽2)C2semidirect
3. #577C2direct (N × H)
4. C2×#137C2semidirect
5. C2×AGL(1,ℤ/8)C2semidirect
6. C2×#135C2semidirect
7. C22×SD16C2semidirect
8. C2×#131C2semidirect
10. 2−1+4C22semidirect
12. H(5,𝔽2)C22semidirect
17. C2×#41C22semidirect
19. C22×Q8C22semidirect
22. #137C22semidirect
23. AGL(1,ℤ/8)C22semidirect
24. #135C22semidirect
25. C2×Q16C22semidirect
26. C2×SD16C22semidirect
27. C2×SD16C22semidirect
28. C2×SD16C22semidirect
29. C2×SD16C22semidirect
30. C2×D8C22semidirect
31. #131C22semidirect
32. C2×M16C22semidirect
33. C22×C8C22semidirect
42. #41C23semidirect
49. C2×Q8C23semidirect
63. Q16C23semidirect
64. SD16C23semidirect
65. SD16C23semidirect
66. SD16C23semidirect
67. D8C23semidirect
68. M16C23semidirect
71. C2×C8C23semidirect
118. C2#577direct (N × H)

Maximal subgroups

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

subgroupindex
C2×2−1+42
C2×H(5,𝔽2)2
#5772
C2×#1372
C2×AGL(1,ℤ/8)2
C2×#1352
C22×SD162
C2×#1312