Finite Group Databasebeta

← #2200

Subgroups — #2200

Order 128.

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

normal subgroups only

This group has 130 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#560C211yesmaximal, characteristic, complemented
2#458C211yesmaximal, characteristic, complemented
3B(4,𝔽2)C211yesmaximal, characteristic, complemented
4#453C211yesmaximal, characteristic, complemented
5#353C211yesmaximal, characteristic, complemented
6#352C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
8H(5,𝔽2)C2211yescharacteristic
9AGL(1,ℤ/8)—21no—
10#127C2211yescharacteristic
11#126—21no—
12#123—21no—
13#123—21no—
14#123C2211yescharacteristic
15#121—21no—
16#121—21no—
17#121—21no—
18#120—21no—
19#120—21no—
20#120C2211yescharacteristic
21#104—21no—
22#100C2211yescharacteristic
23#99—21no—
24#99—21no—
25#99C2211yescharacteristic
26#99C2211yescharacteristic
27C24D411yescharacteristic
28#41—21no—
29#41—41no—
30C2×D4—21no—
31C2×D4—21no—
32C2×D4—21no—
33C2×D4—21no—
34C2×D4—21no—
35C2×D4—21no—
36C2×D4—21no—
37C2×D4—41no—
38C2×D4—41no—
39C2×D4D411yescharacteristic, complemented
40C2×D4C2311yescharacteristic
41C22×C4—21no—
42C22×C4D411yescharacteristic, complemented
43SD16—41no—
44D8—41no—
45M16—21no—
46C4⋊C4—21no—
47C4⋊C4—21no—
48C4⋊C4—21no—
49C4⋊C4—41no—
50C22⋊C4—21no—
51C22⋊C4—21no—
52C22⋊C4—21no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—21no—
57C22⋊C4—41no—
58C22⋊C4—41no—
59C22⋊C4D411yescharacteristic
60C22⋊C4D411yescharacteristic, complemented
61C42D411yescharacteristic, complemented
62C23—21no—
63C23—21no—
64C23—21no—
65C23—21no—
66C23—41no—
67C23—81no—
68C23C2×D411yescharacteristic
69C23C2×D411yescharacteristic
70Q8—21no—
71D4—21no—
72D4—41no—
73D4—41no—
74D4—41no—
75D4—41no—
76D4—41no—
77D4—81no—
78D4—81no—
79D4—81no—
80D4—81no—
81D4—81no—
82D4—82no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—41no—
90C2×C4—41no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—81no—
94C2×C4C2×D411yescharacteristic
95C8—41no—
96C22—21no—
97C22—21no—
98C22—41no—
99C22—41no—
100C22—41no—
101C22—41no—
102C22—41no—
103C22—41no—
104C22—42no—
105C22—41no—
106C22—41no—
107C22—41no—
108C22—81no—
109C22—81no—
110C22—81no—
111C22#12011yescharacteristic
112C4—21no—
113C4—41no—
114C4—41no—
115C4—41no—
116C4—41no—
117C4—41no—
118C4—81no—
119C4—81no—
120C2—21no—
121C2—41no—
122C2—41no—
123C2—81no—
124C2—81no—
125C2—81no—
126C2B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23
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. #560C2semidirect
2. #458C2semidirect
3. B(4,𝔽2)C2semidirect
4. #453C2semidirect
5. #353C2semidirect
6. #352C2semidirect
7. AΣL(1,𝔽16)C2semidirect
39. C2×D4D4semidirect
42. C22×C4D4semidirect
60. C22⋊C4D4semidirect
61. C42D4semidirect

Maximal subgroups

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

subgroupindex
#5602
#4582
B(4,𝔽2)2
#4532
#3532
#3522
AΣL(1,𝔽16)2