Finite Group Databasebeta

← #3585

Subgroups — #3585

Order 128.

732 conjugacy classes of subgroups; only the 426 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 426 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#585C211yesmaximal, characteristic, complemented
2C2×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3#577C218yesmaximal, complemented
4#576C218yesmaximal, complemented
5#575C216yesmaximal, complemented
6C2×AGL(1,ℤ/8)C216yesmaximal, complemented
7#568C211yesmaximal, characteristic, complemented
82−1+4—21no—
92−1+4C2214yescomplemented
10H(5,𝔽2)—24no—
11H(5,𝔽2)—23no—
12H(5,𝔽2)C2218yescomplemented
13H(5,𝔽2)C2214yescomplemented
14C2×#41—24no—
15C2×#41—21no—
16C2×#41C2211yescharacteristic
17C2×#41C2213yescomplemented
18C2×#41C2216yescomplemented
19C22×D4—23no—
20C22×D4C2213yescomplemented
21#137C2214yescomplemented
22#137C22112yescomplemented
23AGL(1,ℤ/8)C22112yescomplemented
24AGL(1,ℤ/8)C22124yescomplemented
25AGL(1,ℤ/8)C22112yescomplemented
26#135C22124yescomplemented
27C2×SD16C22112yescomplemented
28C2×D8C22112yescomplemented
29#131C2218yescomplemented
30C2×M16C2216yescomplemented
31C24—23no—
32#41—28no—
33#41—24no—
34#41—212no—
35#41—26no—
36#41—26no—
37#41—26no—
38#41D414yes—
39#41C23112yes—
40#41C2314yes—
41#41C2314yes—
42#41C23112yescomplemented
43C2×Q8—24no—
44C2×Q8D411yescharacteristic
45C2×Q8C2311yescharacteristic
46C2×Q8C2316yescomplemented
47C2×D4—23no—
48C2×D4—26no—
49C2×D4—224no—
50C2×D4—212no—
51C2×D4—23no—
52C2×D4—26no—
53C2×D4D413yes—
54C2×D4C23112yes—
55C2×D4C2313yes—
56C2×D4C2316yescomplemented
57C22×C4—23no—
58C22×C4—26no—
59C22×C4C2313yes—
60Q16C2318yescomplemented
61SD16C23124yescomplemented
62SD16C2318yescomplemented
63D8C23124yescomplemented
64M16C2314yes—
65M16C23112yescomplemented
66C2×C8C23112yescomplemented
67C23—23no—
68C23—26no—
69C23—212no—
70C23—86no—
71C23C2×D413yes—
72Q8—26no—
73Q8—21no—
74Q8C2×D414yes—
75Q8C2414yes—
76Q8C2412yes—
77D4—212no—
78D4—24no—
79D4—212no—
80D4—23no—
81D4—212no—
82D4—26no—
83D4C2×D4112yes—
84D4C2414yes—
85D4C2416yes—
86C2×C4—21no—
87C2×C4—26no—
88C2×C4—24no—
89C2×C4—212no—
90C2×C4—212no—
91C2×C4C2×D413yes—
92C2×C4C2×D416yes—
93C2×C4C2416yes—
94C2×C4C2411yescharacteristic
95C8C2418yes—
96C22—24no—
97C22—26no—
98C22—43no—
99C22—46no—
100C22—86no—
101C22—812no—
102C22C22×D411yescharacteristic
103C22C22×D416yes—
104C4—22no—
105C4—24no—
106C4C22×D411yescharacteristic
107C4C22×D416yes—
108C4C2511yescharacteristic
109C2—21no—
110C2—26no—
111C2—46no—
112C2—44no—
113C2C23×D411yescharacteristic
subgroupG/H
center, socleC2C23×D4
commutator, Frattini, cosocleC4C25
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×#41 ◁ G
LECSG ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #585C2semidirect
2. C2×H(5,𝔽2)C2semidirect
3. #577C2semidirect
4. #576C2semidirect
5. #575C2semidirect
6. C2×AGL(1,ℤ/8)C2semidirect
7. #568C2semidirect
9. 2−1+4C22semidirect
12. H(5,𝔽2)C22semidirect
13. H(5,𝔽2)C22semidirect
17. C2×#41C22semidirect
18. C2×#41C22semidirect
20. C22×D4C22semidirect
21. #137C22semidirect
22. #137C22semidirect
23. AGL(1,ℤ/8)C22semidirect
24. AGL(1,ℤ/8)C22semidirect
25. AGL(1,ℤ/8)C22semidirect
26. #135C22semidirect
27. C2×SD16C22semidirect
28. C2×D8C22semidirect
29. #131C22semidirect
30. C2×M16C22semidirect
42. #41C23semidirect
46. C2×Q8C23semidirect
56. C2×D4C23semidirect
60. Q16C23semidirect
61. SD16C23semidirect
62. SD16C23semidirect
63. D8C23semidirect
65. M16C23semidirect
66. C2×C8C23semidirect

Maximal subgroups

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

subgroupindex
#5852
C2×H(5,𝔽2)2
#5772
#5762
#5752
C2×AGL(1,ℤ/8)2
#5682