Finite Group Databasebeta

← #3081

Subgroups — #3081

Order 128.

199 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 100 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#544C211yesmaximal, characteristic, complemented
2#518C211yesmaximal, characteristic
3#478C212yesmaximal, complemented
4#478C212yesmaximal, complemented
5#476C212yesmaximal, complemented
6#474C212yesmaximal, complemented
7#428C211yesmaximal, characteristic, complemented
8#427C211yesmaximal, characteristic, complemented
9#424C211yesmaximal, characteristic, complemented
10#418C211yesmaximal, characteristic
11C2×#102C211yesmaximal, characteristic
12C22×D4C2211yescharacteristic
13C2×M16C2211yescharacteristic
14C22×C8C2211yescharacteristic
15C4⋊Q8—21no—
16C4⋊Q8C2212yescomplemented
17#125—21no—
18#125C2212yescomplemented
19C22⋊Q8—22no—
20C4×D4—21no—
21C4×D4—21no—
22C4×D4C2212yes—
23C4×D4C2212yes—
24#117C2211yescharacteristic
25#117C2211yescharacteristic
26#117C2211yescharacteristic
27C2×C4⋊C4C2211yescharacteristic
28C2×C22⋊C4—22no—
29#107C2211yescharacteristic, complemented
30#107C2211yescharacteristic, complemented
31#107C2212yescomplemented
32#106C2211yescharacteristic, complemented
33#106C2211yescharacteristic, complemented
34#106C2212yescomplemented
35C4⋊C8C2212yescomplemented
36C4⋊C8C2212yescomplemented
37#102C2212yes—
38#102C2212yes—
39#102C2212yes—
40#102C2212yes—
41C24—21no—
42C2×Q8—22no—
43C2×D4—21no—
44C2×D4—22no—
45C2×D4Q814yescomplemented
46C2×D4C2312yes—
47C22×C4—21no—
48C22×C4—22no—
49C22×C4—22no—
50C22×C4C2311yescharacteristic
51M16—22no—
52C2×C8—22no—
53C2×C8C2311yescharacteristic
54C2×C8C2312yes—
55C2×C8C2311yescharacteristic
56C4⋊C4—21no—
57C4⋊C4—22no—
58C4⋊C4—22no—
59C4⋊C4—22no—
60C4⋊C4D411yescharacteristic
61C4⋊C4D411yescharacteristic
62C4⋊C4C2312yes—
63C4⋊C4C2311yescharacteristic
64C4⋊C4C2311yescharacteristic
65C4⋊C4C2311yescharacteristic
66C4⋊C4C2311yescharacteristic
67C22⋊C4—21no—
68C22⋊C4—21no—
69C22⋊C4—22no—
70C22⋊C4—22no—
71C22⋊C4D412yes—
72C42—21no—
73C42—21no—
74C42C2312yes—
75C23—22no—
76C23—24no—
77C23—81no—
78C23—81no—
79C23C2×D411yescharacteristic
80Q8—82no—
81D4—24no—
82D4—22no—
83D4C2×Q814yes—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—22no—
89C2×C4—82no—
90C2×C4—82no—
91C2×C4—81no—
92C2×C4C2×D412yes—
93C2×C4C2×D412yes—
94C2×C4C2×D411yescharacteristic
95C2×C4C2×Q812yes—
96C2×C4#4112yes—
97C2×C4C2411yescharacteristic
98C8—21no—
99C8—21no—
100C8—22no—
101C22—21no—
102C22—21no—
103C22—24no—
104C22—42no—
105C22—42no—
106C22—84no—
107C22C22⋊Q812yes—
108C22C22×D411yescharacteristic
109C4—22no—
110C4—22no—
111C4—41no—
112C4—42no—
113C4—41no—
114C4—41no—
115C4—41no—
116C4C22⋊Q812yes—
117C4C22×Q811yescharacteristic
118C4C2×#4111yescharacteristic
119C2—22no—
120C2—44no—
121C2C2×C22⋊Q811yescharacteristic
122C2#57611yescharacteristic
123C2#57711yescharacteristic
subgroupG/H
center, socleC22C22×D4
commutator, Frattini, cosocleC2×C4C24
prime radicalC22×D4C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #544C2semidirect
3. #478C2semidirect
4. #478C2semidirect
5. #476C2semidirect
6. #474C2semidirect
7. #428C2semidirect
8. #427C2semidirect
9. #424C2semidirect
16. C4⋊Q8C22semidirect
18. #125C22semidirect
29. #107C22semidirect
30. #107C22semidirect
31. #107C22semidirect
32. #106C22semidirect
33. #106C22semidirect
34. #106C22semidirect
35. C4⋊C8C22semidirect
36. C4⋊C8C22semidirect
45. C2×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#5442
#5182
#4782
#4762
#4742
#4282
#4272
#4242
#4182
C2×#1022