Finite Group Databasebeta

← #3500

Subgroups — #3500

Order 128.

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

normal subgroups only

This group has 770 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#548C214yesmaximal, complemented
4#546C2112yesmaximal, complemented
5#536C213yesmaximal, complemented
6#535C213yesmaximal, complemented
7#532C213yesmaximal, complemented
8C2×#124C213yesmaximal, complemented
9C4×#41C211yesmaximal, characteristic, complemented
102−1+4—28no—
11H(5,𝔽2)—28no—
12C2×#41C2213yes—
13C2×#41C2212yes—
14C2×#41C2211yescharacteristic
15C2×#41C2213yescomplemented
16C2×#41C2216yescomplemented
17C22×Q8C2213yescomplemented
18C22×Q8C2212yescomplemented
19C22×D4C2213yes—
20C22×D4C2216yes—
21C4⋊Q8C2213yescomplemented
22#127C2213yescomplemented
23#124C2213yescomplemented
24#124C2216yescomplemented
25#124C22112yescomplemented
26#124C2211yescharacteristic, complemented
27#123C22112yescomplemented
28C22⋊Q8C22112yescomplemented
29#121C22112yescomplemented
30#121C22112yescomplemented
31#120C22112yescomplemented
32C4×Q8C2212yescomplemented
33C4×D4C2216yescomplemented
34C4×D4C22112yescomplemented
35#117C2213yescomplemented
36C2×C22⋊C4C22112yescomplemented
37C2×C42C2213yescomplemented
38C24C2316yes—
39#41—28no—
40#41—224no—
41#41—212no—
42#41—212no—
43#41D418yescomplemented
44C2×Q8—28no—
45C2×Q8—212no—
46C2×Q8C2316yes—
47C2×Q8C2313yes—
48C2×Q8C2311yescharacteristic
49C2×Q8C2311yescharacteristic
50C2×D4—224no—
51C2×D4—212no—
52C2×D4C2316yes—
53C2×D4C2316yes—
54C2×D4C2316yes—
55C2×D4C2313yes—
56C2×D4C2311yescharacteristic
57C2×D4C23112yes—
58C2×D4C2313yes—
59C22×C4C2316yes—
60C22×C4C2316yes—
61C22×C4C2316yes—
62C22×C4C2316yes—
63C22×C4C2313yes—
64C4⋊C4C2316yes—
65C4⋊C4C23112yes—
66C22⋊C4C23112yes—
67C22⋊C4C23124yes—
68C22⋊C4C2316yes—
69C22⋊C4C2314yes—
70C42C2311yescharacteristic
71C42C2313yes—
72C42C2316yes—
73C23—212no—
74C23—412no—
75C23C2416yes—
76C23C2412yes—
77C23C2413yes—
78Q8—212no—
79Q8—26no—
80Q8—22no—
81Q8C2×D414yes—
82D4—212no—
83D4—212no—
84D4—212no—
85D4—26no—
86D4—22no—
87D4—412no—
88D4C2×D4112yes—
89C2×C4—212no—
90C2×C4—212no—
91C2×C4—212no—
92C2×C4—46no—
93C2×C4C2×D4112yes—
94C2×C4C2412yes—
95C2×C4C2416yes—
96C2×C4C2416yes—
97C2×C4C2412yes—
98C2×C4C2411yescharacteristic
99C2×C4C2413yes—
100C22—26no—
101C22—22no—
102C22—26no—
103C22—24no—
104C22—212no—
105C22—812no—
106C22C22×D416yes—
107C22C2511yescharacteristic
108C4—22no—
109C4—26no—
110C4—22no—
111C4—26no—
112C4C22×D416yes—
113C4C22×D412yes—
114C2—26no—
115C2—42no—
116C2—46no—
117C2C23×D411yescharacteristic
118C2#58512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC22C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ G
LECSG ▷ C22 ▷ 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. #548C2semidirect
4. #546C2semidirect
5. #536C2semidirect
6. #535C2semidirect
7. #532C2semidirect
8. C2×#124C2semidirect
9. C4×#41C2semidirect
15. C2×#41C22semidirect
16. C2×#41C22semidirect
17. C22×Q8C22semidirect
18. C22×Q8C22semidirect
21. C4⋊Q8C22semidirect
22. #127C22semidirect
23. #124C22semidirect
24. #124C22semidirect
25. #124C22semidirect
26. #124C22semidirect
27. #123C22semidirect
28. C22⋊Q8C22semidirect
29. #121C22semidirect
30. #121C22semidirect
31. #120C22semidirect
32. C4×Q8C22semidirect
33. C4×D4C22semidirect
34. C4×D4C22semidirect
35. #117C22semidirect
36. C2×C22⋊C4C22semidirect
37. C2×C42C22semidirect
43. #41D4semidirect

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
#5482
#5462
#5362
#5352
#5322
C2×#1242
C4×#412