Finite Group Databasebeta

← #2197

Subgroups — #2197

Order 128.

129 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 129 conjugacy classes of subgroups — too many to draw. Only its 28 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#551C211yesmaximal, characteristic, complemented
2#458C211yesmaximal, characteristic, complemented
3B(4,𝔽2)C211yesmaximal, characteristic
4#455C211yesmaximal, characteristic, complemented
5#354C211yesmaximal, characteristic, complemented
6#352C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8H(5,𝔽2)C2211yescharacteristic
9#137—21no—
10#126—21no—
11#124C2211yescharacteristic
12#123—21no—
13#123—21no—
14#123C2211yescharacteristic
15C22⋊Q8—21no—
16#120—21no—
17#120C2211yescharacteristic
18C4×D4—21no—
19#117—21no—
20C2×C22⋊C4—21no—
21#104—21no—
22#100C2211yescharacteristic
23#99—21no—
24#99—21no—
25#99C2211yescharacteristic
26#99C2211yescharacteristic
27C24D411yescharacteristic
28#41—21no—
29#41—41no—
30C2×Q8D411yescharacteristic, complemented
31C2×D4—21no—
32C2×D4—21no—
33C2×D4—21no—
34C2×D4—41no—
35C2×D4—41no—
36C2×D4C2311yescharacteristic
37C22×C4—21no—
38C22×C4—21no—
39C22×C4D411yescharacteristic, complemented
40Q16—41no—
41SD16—41no—
42M16—21no—
43C4⋊C4—21no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C4⋊C4—21no—
47C4⋊C4—41no—
48C22⋊C4—21no—
49C22⋊C4—21no—
50C22⋊C4—21no—
51C22⋊C4—21no—
52C22⋊C4—21no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—41no—
57C22⋊C4—41no—
58C22⋊C4D411yescharacteristic
59C22⋊C4D411yescharacteristic, complemented
60C42—21no—
61C42D411yescharacteristic, complemented
62C23—21no—
63C23—21no—
64C23—21no—
65C23—41no—
66C23—42no—
67C23C2×D411yescharacteristic
68C23C2×D411yescharacteristic
69Q8—21no—
70Q8—41no—
71D4—21no—
72D4—41no—
73D4—42no—
74D4—41no—
75D4—41no—
76D4—41no—
77D4—82no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—41no—
86C2×C4—42no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—81no—
91C2×C4C2×D411yescharacteristic
92C8—41no—
93C22—22no—
94C22—21no—
95C22—21no—
96C22—41no—
97C22—41no—
98C22—41no—
99C22—42no—
100C22—41no—
101C22—41no—
102C22—81no—
103C22—81no—
104C22—81no—
105C22#12011yescharacteristic
106C4—21no—
107C4—41no—
108C4—41no—
109C4—41no—
110C4—41no—
111C4—41no—
112C4—41no—
113C4—81no—
114C4—81no—
115C2—21no—
116C2—41no—
117C2—41no—
118C2—42no—
119C2—81no—
120C2B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23
prime radicalB(4,𝔽2)C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G
LECSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #551C2semidirect
2. #458C2semidirect
4. #455C2semidirect
5. #354C2semidirect
6. #352C2semidirect
7. AΣL(1,𝔽16)C2semidirect
30. C2×Q8D4semidirect
39. C22×C4D4semidirect
59. C22⋊C4D4semidirect
61. C42D4semidirect

Maximal subgroups

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

subgroupindex
#5512
#4582
B(4,𝔽2)2
#4552
#3542
#3522
AΣL(1,𝔽16)2