Finite Group Databasebeta

← #3532

Subgroups — #3532

Order 128.

668 conjugacy classes of subgroups; only the 384 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 384 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
1H(3,𝔽4)C211yesmaximal, characteristic, complemented
2#560C213yesmaximal, complemented
3#551C213yesmaximal, complemented
4#546C216yesmaximal, complemented
5D42C213yesmaximal, complemented
6#538C216yesmaximal, complemented
7#535C213yesmaximal, complemented
8#534C213yesmaximal, complemented
9C2×#120C213yesmaximal, complemented
10C25C2211yescharacteristic, complemented
11C2×#41C2213yescomplemented
12C22×D4C2213yescomplemented
13C22×D4C2216yescomplemented
14C22×D4C2213yescomplemented
15#127C2211yescharacteristic, complemented
16#127C2213yescomplemented
17#126C2212yescomplemented
18#126C2216yescomplemented
19#124C2213yescomplemented
20#124C2213yescomplemented
21#124C2216yescomplemented
22#123C2213yescomplemented
23#123C2213yescomplemented
24#123C2216yescomplemented
25#123C2216yescomplemented
26C22⋊Q8C2216yescomplemented
27#121C2216yescomplemented
28#121C2216yescomplemented
29#121C2216yescomplemented
30#121C2216yescomplemented
31#121C2216yescomplemented
32#120C2213yes—
33#120C2216yescomplemented
34#120C2216yescomplemented
35#120C2213yescomplemented
36#120C2213yescomplemented
37#120C2213yescomplemented
38#120C2216yescomplemented
39#120C2211yescharacteristic, complemented
40#120C2213yescomplemented
41C4×D4C2216yescomplemented
42C4×D4C2216yescomplemented
43#117C2213yescomplemented
44C2×C22⋊C4C2213yescomplemented
45C2×C22⋊C4C2216yescomplemented
46C2×C22⋊C4C2213yescomplemented
47C24—46no—
48C24C2313yes—
49C24C2313yes—
50C24C2313yes—
51C24C2311yescharacteristic
52C24C2311yescharacteristic, complemented
53#41—46no—
54C2×Q8C2313yescomplemented
55C2×D4—46no—
56C2×D4—412no—
57C2×D4—46no—
58C2×D4C2316yes—
59C2×D4C2316yes—
60C2×D4C2313yes—
61C2×D4C2313yes—
62C2×D4C2316yes—
63C2×D4C2313yes—
64C2×D4C2316yes—
65C2×D4C2313yes—
66C2×D4C2313yescomplemented
67C2×D4C2316yescomplemented
68C22×C4C2313yes—
69C22×C4C2316yes—
70C22×C4C2316yescomplemented
71C22×C4C2313yescomplemented
72C4⋊C4C2316yescomplemented
73C4⋊C4C2316yescomplemented
74C4⋊C4C2316yescomplemented
75C22⋊C4C2316yes—
76C22⋊C4C2316yes—
77C22⋊C4C2313yes—
78C22⋊C4C2316yes—
79C22⋊C4C2316yes—
80C22⋊C4C2316yes—
81C22⋊C4C2313yes—
82C22⋊C4C2313yescomplemented
83C22⋊C4C2313yescomplemented
84C22⋊C4C2316yescomplemented
85C22⋊C4C2316yescomplemented
86C42C2311yescharacteristic, complemented
87C42C2313yescomplemented
88C42C2312yescomplemented
89C23—212no—
90C23—26no—
91C23—26no—
92C23—43no—
93C23—46no—
94C23—43no—
95C23—46no—
96C23—43no—
97C23—416no—
98C23C2413yes—
99C23C2411yescharacteristic
100C23C2413yes—
101C23C2413yes—
102C23C2413yes—
103Q8—43no—
104D4—212no—
105D4—212no—
106D4—43no—
107D4—43no—
108D4—43no—
109D4—43no—
110D4—46no—
111D4—46no—
112D4—46no—
113D4—43no—
114C2×C4—26no—
115C2×C4—46no—
116C2×C4—43no—
117C2×C4—46no—
118C2×C4C2413yes—
119C2×C4C2416yes—
120C2×C4C2416yes—
121C2×C4C2413yes—
122C22—26no—
123C22—26no—
124C22—23no—
125C22—26no—
126C22—23no—
127C22—23no—
128C22—26no—
129C22—23no—
130C22—412no—
131C22—412no—
132C22—44no—
133C22—86no—
134C22—82no—
135C22H(5,𝔽2)16yes—
136C22C2511yescharacteristic
137C4—23no—
138C4—23no—
139C4—26no—
140C4—26no—
141C2—26no—
142C2—43no—
143C2—41no—
144C2—43no—
145C2—43no—
146C2C2×H(5,𝔽2)13yes—
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. H(3,𝔽4)C2semidirect
2. #560C2semidirect
3. #551C2semidirect
4. #546C2semidirect
5. D42C2semidirect
6. #538C2semidirect
7. #535C2semidirect
8. #534C2semidirect
9. C2×#120C2semidirect
10. C25C22semidirect
11. C2×#41C22semidirect
12. C22×D4C22semidirect
13. C22×D4C22semidirect
14. C22×D4C22semidirect
15. #127C22semidirect
16. #127C22semidirect
17. #126C22semidirect
18. #126C22semidirect
19. #124C22semidirect
20. #124C22semidirect
21. #124C22semidirect
22. #123C22semidirect
23. #123C22semidirect
24. #123C22semidirect
25. #123C22semidirect
26. C22⋊Q8C22semidirect
27. #121C22semidirect
28. #121C22semidirect
29. #121C22semidirect
30. #121C22semidirect
31. #121C22semidirect
33. #120C22semidirect
34. #120C22semidirect
35. #120C22semidirect
36. #120C22semidirect
37. #120C22semidirect
38. #120C22semidirect
39. #120C22semidirect
40. #120C22semidirect
41. C4×D4C22semidirect
42. C4×D4C22semidirect
43. #117C22semidirect
44. C2×C22⋊C4C22semidirect
45. C2×C22⋊C4C22semidirect
46. C2×C22⋊C4C22semidirect
52. C24C23semidirect
54. C2×Q8C23semidirect
66. C2×D4C23semidirect
67. C2×D4C23semidirect
70. C22×C4C23semidirect
71. C22×C4C23semidirect
72. C4⋊C4C23semidirect
73. C4⋊C4C23semidirect
74. C4⋊C4C23semidirect
82. C22⋊C4C23semidirect
83. C22⋊C4C23semidirect
84. C22⋊C4C23semidirect
85. C22⋊C4C23semidirect
86. C42C23semidirect
87. C42C23semidirect
88. C42C23semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽4)2
#5602
#5512
#5462
D422
#5382
#5352
#5342
C2×#1202