Finite Group Databasebeta

← #1606

Subgroups — #1606

Order 128.

130 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 32 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#538C211yesmaximal, characteristic
2#451C211yesmaximal, characteristic, complemented
3#448C211yesmaximal, characteristic, complemented
4#393C211yesmaximal, characteristic, complemented
5#328C211yesmaximal, characteristic, complemented
6#327C211yesmaximal, characteristic, complemented
7#323C211yesmaximal, characteristic, complemented
8C22×Q8C2211yescharacteristic
9C2×Q16—21no—
10C2×SD16—21no—
11#126—21no—
12#124—21no—
13#123—21no—
14C22⋊Q8C2211yescharacteristic
15#121—21no—
16#121C2211yescharacteristic
17#120—21no—
18C4×D4—21no—
19C2×C22⋊C4—21no—
20C2×C22⋊C4C2211yescharacteristic
21#103—21no—
22#103—21no—
23C22⋊C8C2211yescharacteristic
24C22⋊C8C2211yescharacteristic
25#95—21no—
26#95C2211yescharacteristic
27C24D411yescharacteristic
28C2×Q8—21no—
29C2×Q8—21no—
30C2×Q8—42no—
31C2×Q8D411yescharacteristic
32C2×D4—21no—
33C2×D4—21no—
34C2×D4—21no—
35C2×D4D411yescharacteristic
36C22×C4—21no—
37C22×C4—21no—
38C22×C4—21no—
39C22×C4D411yescharacteristic
40C22×C4C2311yescharacteristic
41Q16—42no—
42SD16—42no—
43C2×C8—21no—
44C2×C8—21no—
45C4⋊C4—21no—
46C4⋊C4—21no—
47C4⋊C4D411yescharacteristic
48C4⋊C4D411yescharacteristic
49C22⋊C4—21no—
50C22⋊C4—21no—
51C22⋊C4—21no—
52C22⋊C4—21no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—42no—
57C42—21no—
58C23—21no—
59C23—21no—
60C23—21no—
61C23—21no—
62C23—41no—
63C23—41no—
64C23C2×D411yescharacteristic
65Q8—22no—
66Q8—22no—
67Q8—41no—
68Q8—41no—
69Q8—41no—
70D4—41no—
71D4—41no—
72D4—41no—
73D4—81no—
74D4—81no—
75C2×C4—22no—
76C2×C4—21no—
77C2×C4—21no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—41no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—81no—
88C2×C4C2×D411yescharacteristic
89C2×C4C2×D411yescharacteristic
90C8—41no—
91C8—41no—
92C22—21no—
93C22—21no—
94C22—42no—
95C22—41no—
96C22—41no—
97C22—41no—
98C22—41no—
99C22—41no—
100C22—41no—
101C22—81no—
102C22#12011yescharacteristic
103C22#13511yescharacteristic
104C22#13711yescharacteristic
105C4—21no—
106C4—21no—
107C4—41no—
108C4—41no—
109C4—41no—
110C4—42no—
111C4—41no—
112C4—81no—
113C2—21no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2#45211yescharacteristic
118C2#45511yescharacteristic
119C2B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC22×C4C23
prime radical#538C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #451C2semidirect
3. #448C2semidirect
4. #393C2semidirect
5. #328C2semidirect
6. #327C2semidirect
7. #323C2semidirect

Maximal subgroups

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

subgroupindex
#5382
#4512
#4482
#3932
#3282
#3272
#3232