Finite Group Databasebeta

← #1603

Subgroups — #1603

Order 128.

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

normal subgroups only

This group has 122 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#539C211yesmaximal, characteristic
2#451C211yesmaximal, characteristic, complemented
3#448C211yesmaximal, characteristic
4#395C211yesmaximal, characteristic, complemented
5#328C211yesmaximal, characteristic, complemented
6#327C211yesmaximal, characteristic
7#324C211yesmaximal, characteristic, complemented
8C22×Q8C2211yescharacteristic
9C2×Q16—21no—
10C2×SD16—21no—
11#126—21no—
12#125—21no—
13#123—21no—
14#123—21no—
15C22⋊Q8—21no—
16C22⋊Q8C2211yescharacteristic
17#121C2211yescharacteristic
18C4×D4—21no—
19C2×C4⋊C4—21no—
20C2×C4⋊C4C2211yescharacteristic
21#103—21no—
22#103—21no—
23C22⋊C8C2211yescharacteristic
24C22⋊C8C2211yescharacteristic
25#95—21no—
26#95C2211yescharacteristic
27C2×Q8—21no—
28C2×Q8—21no—
29C2×Q8—42no—
30C2×Q8D411yescharacteristic
31C2×D4—21no—
32C2×D4D411yescharacteristic
33C22×C4—21no—
34C22×C4—21no—
35C22×C4—21no—
36C22×C4D411yescharacteristic
37C22×C4D411yescharacteristic
38C22×C4C2311yescharacteristic
39Q16—42no—
40SD16—42no—
41C2×C8—21no—
42C2×C8—21no—
43C4⋊C4—21no—
44C4⋊C4—21no—
45C4⋊C4—21no—
46C4⋊C4—21no—
47C4⋊C4—21no—
48C4⋊C4—21no—
49C4⋊C4—42no—
50C4⋊C4D411yescharacteristic
51C4⋊C4D411yescharacteristic
52C22⋊C4—21no—
53C22⋊C4—21no—
54C22⋊C4—21no—
55C22⋊C4—21no—
56C22⋊C4—21no—
57C42—21no—
58C23—21no—
59C23C2×D411yescharacteristic
60Q8—22no—
61Q8—22no—
62Q8—41no—
63Q8—41no—
64Q8—41no—
65D4—41no—
66D4—41no—
67D4—41no—
68C2×C4—22no—
69C2×C4—22no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—21no—
74C2×C4—21no—
75C2×C4—21no—
76C2×C4—21no—
77C2×C4—41no—
78C2×C4—41no—
79C2×C4—41no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4—81no—
84C2×C4C2×D411yescharacteristic
85C2×C4C2×D411yescharacteristic
86C8—41no—
87C8—41no—
88C22—21no—
89C22—21no—
90C22—41no—
91C22—41no—
92C22—41no—
93C22#12011yescharacteristic
94C22#13511yescharacteristic
95C22#13711yescharacteristic
96C4—21no—
97C4—21no—
98C4—42no—
99C4—41no—
100C4—41no—
101C4—41no—
102C4—42no—
103C4—41no—
104C4—41no—
105C2—21no—
106C2—21no—
107C2—81no—
108C2#45211yescharacteristic
109C2#45611yescharacteristic
110C2B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22#120
commutator, Frattini, cosocleC22×C4C23
prime radical#121C22
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
4. #395C2semidirect
5. #328C2semidirect
7. #324C2semidirect

Maximal subgroups

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

subgroupindex
#5392
#4512
#4482
#3952
#3282
#3272
#3242