Finite Group Databasebeta

← #7395

Subgroups — #7395

Order 256.

142 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 21 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#2022C211yesmaximal, characteristic
2#1317C211yesmaximal, characteristic, complemented
3#1306C211yesmaximal, characteristic, complemented
4#534C411yescharacteristic
5C2×#123—21no—
6C2×#99—21no—
7#407—21no—
8#381C2211yescharacteristic
9#342C411yescharacteristic
10#323—21no—
11C22×D4—21no—
12C23×C4D411yescharacteristic
13C2×M16—41no—
14#123—42no—
15#123—41no—
16#123—41no—
17#123—41no—
18#121—41no—
19#120—41no—
20C2×C4⋊C4—41no—
21C2×C22⋊C4—21no—
22C2×C22⋊C4—41no—
23C2×C22⋊C4C2×C411yescharacteristic
24C2×C22⋊C4D411yescharacteristic
25#99—42no—
26C22⋊C8—22no—
27C22⋊C8—41no—
28C22⋊C8—41no—
29#95—41no—
30C24—21no—
31C24C22⋊C411yescharacteristic
32C2×D4—41no—
33C2×D4—41no—
34C2×D4—41no—
35C2×D4—41no—
36C2×D4—41no—
37C2×D4—41no—
38C2×D4—41no—
39C22×C4—21no—
40C22×C4—21no—
41C22×C4—21no—
42C22×C4—21no—
43C22×C4—21no—
44C22×C4—41no—
45C22×C4—41no—
46C22×C4—41no—
47C22×C4D811yescharacteristic
48C22×C4SD1611yescharacteristic
49M16—81no—
50M16—81no—
51C2×C8—42no—
52C2×C8—81no—
53C4⋊C4—41no—
54C4⋊C4—42no—
55C4⋊C4—42no—
56C22⋊C4—22no—
57C22⋊C4—42no—
58C22⋊C4—42no—
59C22⋊C4—42no—
60C22⋊C4—41no—
61C22⋊C4—41no—
62C22⋊C4—41no—
63C23—21no—
64C23—21no—
65C23—41no—
66C23—41no—
67C23—41no—
68C23—81no—
69C23—82no—
70C23#9911yescharacteristic
71C23#10211yescharacteristic
72C23#10411yescharacteristic
73D4—81no—
74D4—81no—
75D4—81no—
76D4—81no—
77D4—161no—
78C2×C4—21no—
79C2×C4—21no—
80C2×C4—41no—
81C2×C4—41no—
82C2×C4—41no—
83C2×C4—41no—
84C2×C4—42no—
85C2×C4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—81no—
91C2×C4—82no—
92C2×C4—81no—
93C2×C4—81no—
94C8—82no—
95C8—161no—
96C22—21no—
97C22—21no—
98C22—41no—
99C22—41no—
100C22—41no—
101C22—42no—
102C22—81no—
103C22—82no—
104C22—82no—
105C22—81no—
106C22—161no—
107C22#32711yescharacteristic
108C22AΣL(1,𝔽16)11yescharacteristic
109C22#35311yescharacteristic
110C4—41no—
111C4—41no—
112C4—81no—
113C4—81no—
114C4—82no—
115C4—81no—
116C2—21no—
117C2—21no—
118C2—21no—
119C2—41no—
120C2—41no—
121C2—161no—
122C2#134311yescharacteristic
subgroupG/H
center, socleC2#1343
commutatorC2×C22⋊C4C2×C4
Frattini, cosocle#381C22
prime radical#534C4
LCSG ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ G
LECSG ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1317C2semidirect
3. #1306C2semidirect

Maximal subgroups

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

subgroupindex
#20222
#13172
#13062