Finite Group Databasebeta

← #3289

Subgroups — #3289

Order 128.

224 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 92 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#578C212yesmaximal, complemented
2#577C212yesmaximal, complemented
3#497C211yesmaximal, characteristic, complemented
4#473C212yesmaximal, complemented
5#471C212yesmaximal, complemented
6#456C212yesmaximal, complemented
7#455C211yesmaximal, characteristic, complemented
8#454C211yesmaximal, characteristic, complemented
9#444C212yesmaximal, complemented
102−1+4C2212yes—
112−1+4C2211yescharacteristic, complemented
12H(5,𝔽2)C2211yescharacteristic
13#137—22no—
14#137—22no—
15#137—22no—
16#137C2212yes—
17#137C2212yes—
18#137C2212yescomplemented
19AGL(1,ℤ/8)—22no—
20AGL(1,ℤ/8)C2212yes—
21#135—22no—
22#135—22no—
23#135C2212yes—
24C2×Q16—22no—
25C2×Q16C2212yescomplemented
26C2×SD16—22no—
27C2×SD16C2212yes—
28#131C2212yes—
29#131C2212yescomplemented
30C4⋊Q8C2211yescharacteristic, complemented
31#124C2211yescharacteristic
32#108C2212yescomplemented
33#104C2212yes—
34#104C2212yescomplemented
35#101C2211yescharacteristic, complemented
36#101C2212yescomplemented
37#100C2211yescharacteristic
38#97C2211yescharacteristic, complemented
39#41—22no—
40#41—22no—
41#41—22no—
42#41—22no—
43#41—42no—
44#41—41no—
45#41C2312yes—
46#41C2312yes—
47C2×Q8—22no—
48C2×Q8—22no—
49C2×Q8—22no—
50C2×Q8C2311yescharacteristic
51C2×Q8C2312yes—
52C2×D4—22no—
53C2×D4—41no—
54C2×D4C2311yescharacteristic
55Q16—22no—
56Q16—22no—
57Q16—22no—
58Q16—22no—
59Q16—22no—
60Q16D412yes—
61SD16—22no—
62SD16—22no—
63SD16—22no—
64SD16—22no—
65SD16—22no—
66SD16—22no—
67SD16D412yes—
68SD16D412yes—
69D8—22no—
70D8D412yes—
71M16—22no—
72M16—22no—
73M16C2312yes—
74M16C2312yes—
75C2×C8—22no—
76C2×C8—22no—
77C2×C8C2312yes—
78C4⋊C4—41no—
79C22⋊C4—41no—
80C42C2311yescharacteristic
81C23—21no—
82C23—41no—
83Q8—22no—
84Q8—22no—
85Q8—22no—
86Q8—42no—
87Q8—41no—
88Q8C2×D412yes—
89Q8C2×D412yes—
90D4—22no—
91D4—22no—
92D4—22no—
93D4—42no—
94D4—41no—
95D4—42no—
96D4C2×D412yes—
97D4C2×D412yes—
98C2×C4—22no—
99C2×C4—22no—
100C2×C4—21no—
101C2×C4—21no—
102C2×C4—22no—
103C2×C4—41no—
104C2×C4—42no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4C2411yescharacteristic
108C8—22no—
109C8—22no—
110C8C2×D412yes—
111C8C2×D412yes—
112C22—22no—
113C22—22no—
114C22—41no—
115C22—81no—
116C22—161no—
117C22H(5,𝔽2)11yescharacteristic
118C4—22no—
119C4—22no—
120C4—41no—
121C4—41no—
122C4—42no—
123C4C22×D412yes—
124C2—21no—
125C2—42no—
126C2—42no—
127C2—81no—
128C2D4211yescharacteristic
subgroupG/H
center, socleC2D42
commutator, Frattini, cosocleC2×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #578C2semidirect
2. #577C2semidirect
3. #497C2semidirect
4. #473C2semidirect
5. #471C2semidirect
6. #456C2semidirect
7. #455C2semidirect
8. #454C2semidirect
9. #444C2semidirect
11. 2−1+4C22semidirect
18. #137C22semidirect
25. C2×Q16C22semidirect
29. #131C22semidirect
30. C4⋊Q8C22semidirect
32. #108C22semidirect
34. #104C22semidirect
35. #101C22semidirect
36. #101C22semidirect
38. #97C22semidirect

Maximal subgroups

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

subgroupindex
#5782
#5772
#4972
#4732
#4712
#4562
#4552
#4542
#4442