Finite Group Databasebeta

← #7077

Subgroups — #7077

Order 256.

109 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 109 conjugacy classes of subgroups — too many to draw. Only its 27 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#2076C211yesmaximal, characteristic
2#1314C211yesmaximal, characteristic
3#1289C211yesmaximal, characteristic
4C2×#107C411yescharacteristic, complemented
5C2×C22⋊C8C2211yescharacteristic
6#380—21no—
7#380—21no—
8C22⋊C16—21no—
9#340—21no—
10#340C411yescharacteristic, complemented
11C23×C4D411yescharacteristic
12C22×C8C2×C411yescharacteristic
13C22×C8D411yescharacteristic
14C2×C4⋊C4—21no—
15C2×C4⋊C4—21no—
16C2×C4⋊C4—41no—
17C2×C22⋊C4—41no—
18C2×C22⋊C4—41no—
19C2×C16—41no—
20#107—41no—
21C22⋊C8—41no—
22#95—22no—
23#95—41no—
24C24D811yescharacteristic
25C22×C4—21no—
26C22×C4—21no—
27C22×C4—41no—
28C22×C4—41no—
29C22×C4—42no—
30C22×C4—41no—
31C22×C4—41no—
32C22×C4C22⋊C411yescharacteristic
33C22×C4SD1611yescharacteristic
34C2×C8—21no—
35C2×C8—21no—
36C2×C8—41no—
37C2×C8—41no—
38C4⋊C4—41no—
39C4⋊C4—41no—
40C4⋊C4—41no—
41C4⋊C4—41no—
42C4⋊C4—81no—
43C4⋊C4—81no—
44C22⋊C4—81no—
45C22⋊C4—82no—
46C22⋊C4—81no—
47C16—81no—
48C23—21no—
49C23—21no—
50C23—41no—
51C23—41no—
52C23#10211yescharacteristic
53C23SD3211yescharacteristic
54C23Q3211yescharacteristic
55C2×C4—21no—
56C2×C4—21no—
57C2×C4—21no—
58C2×C4—41no—
59C2×C4—41no—
60C2×C4—41no—
61C2×C4—81no—
62C2×C4—81no—
63C2×C4—82no—
64C2×C4—81no—
65C2×C4—81no—
66C2×C4—81no—
67C2×C4—82no—
68C2×C4—82no—
69C2×C4—81no—
70C2×C4—81no—
71C2×C4#9911yescharacteristic
72C2×C4#10411yescharacteristic
73C8—41no—
74C8—41no—
75C22—21no—
76C22—21no—
77C22—21no—
78C22—21no—
79C22—41no—
80C22—41no—
81C22—41no—
82C22—41no—
83C22—81no—
84C22#32711yescharacteristic
85C22#35811yescharacteristic
86C22#36011yescharacteristic
87C4—21no—
88C4—41no—
89C4—161no—
90C4—162no—
91C4—161no—
92C4#35211yescharacteristic
93C4#35411yescharacteristic
94C2—21no—
95C2—21no—
96C2—41no—
97C2—41no—
98C2#133911yescharacteristic
99C2#134411yescharacteristic
100C2#134811yescharacteristic
subgroupG/H
center, socleC22#327
commutatorC22×C8C2×C4
Frattini, cosocleC2×C22⋊C8C22
prime radicalC24D8
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#107C4semidirect
10. #340C4semidirect

Maximal subgroups

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

subgroupindex
#20762
#13142
#12892