Finite Group Databasebeta

← #7428

Subgroups — #7428

Order 256.

113 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 113 conjugacy classes of subgroups — too many to draw. Only its 23 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#2025C211yesmaximal, characteristic
2#1393C211yesmaximal, characteristic
3#1318C211yesmaximal, characteristic
4C2×#125C411yescharacteristic, complemented
5C2×#102—21no—
6C2×#99—21no—
7#390C2211yescharacteristic
8#340—21no—
9#336C411yescharacteristic, complemented
10#324—21no—
11C22×D4D411yescharacteristic
12C22×C8—21no—
13#125—42no—
14C2×C4⋊C4—21no—
15C2×C4⋊C4—41no—
16C2×C4⋊C4D411yescharacteristic
17C2×C22⋊C4—41no—
18C2×C22⋊C4—41no—
19C2×C42C2×C411yescharacteristic
20#102—42no—
21#99—22no—
22#99—41no—
23C22⋊C8—41no—
24C24—21no—
25C2×D4—21no—
26C2×D4—41no—
27C2×D4—41no—
28C2×D4SD1611yescharacteristic
29C2×D4Q1611yescharacteristic
30C22×C4—21no—
31C22×C4—21no—
32C22×C4—41no—
33C22×C4—81no—
34C22×C4C22⋊C411yescharacteristic
35C2×C8—41no—
36C2×C8—41no—
37C2×C8—41no—
38C2×C8—81no—
39C4⋊C4—22no—
40C4⋊C4—41no—
41C4⋊C4—41no—
42C4⋊C4—42no—
43C4⋊C4—42no—
44C22⋊C4—42no—
45C22⋊C4—82no—
46C22⋊C4—81no—
47C42—41no—
48C23—21no—
49C23—21no—
50C23—41no—
51C23—41no—
52C23—81no—
53C23—81no—
54C23#9911yescharacteristic
55D4—41no—
56D4—41no—
57D4—81no—
58C2×C4—21no—
59C2×C4—21no—
60C2×C4—41no—
61C2×C4—41no—
62C2×C4—41no—
63C2×C4—42no—
64C2×C4—41no—
65C2×C4—81no—
66C2×C4—81no—
67C2×C4—81no—
68C2×C4—82no—
69C2×C4—161no—
70C2×C4—161no—
71C2×C4#10311yescharacteristic
72C2×C4#10411yescharacteristic
73C8—81no—
74C8—161no—
75C22—21no—
76C22—21no—
77C22—41no—
78C22—41no—
79C22—81no—
80C22—81no—
81C22—81no—
82C22—81no—
83C22—161no—
84C22#32811yescharacteristic
85C22AΣL(1,𝔽16)11yescharacteristic
86C22#35411yescharacteristic
87C4—21no—
88C4—21no—
89C4—81no—
90C4—82no—
91C4—81no—
92C4—162no—
93C2—21no—
94C2—21no—
95C2—81no—
96C2—81no—
97C2#135111yescharacteristic
98C2#141211yescharacteristic
99C2#141411yescharacteristic
subgroupG/H
center, socleC22#354
commutatorC2×C42C2×C4
Frattini, cosocle#390C22
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ G
LECSG ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#125C4semidirect
9. #336C4semidirect

Maximal subgroups

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

subgroupindex
#20252
#13932
#13182