Finite Group Databasebeta

← #7078

Subgroups — #7078

Order 256.

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

normal subgroups only

This group has 143 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#2001C211yesmaximal, characteristic
2#1314C211yesmaximal, characteristic
3#1289C211yesmaximal, characteristic
4C22×Q16C411yescharacteristic, complemented
5C2×C22⋊Q8—21no—
6C2×#103—21no—
7C2×C22⋊C8C2211yescharacteristic
8#380—21no—
9C22⋊C16—21no—
10#340C411yescharacteristic, complemented
11C22×Q8—21no—
12C23×C4D411yescharacteristic
13C2×Q16—41no—
14C2×Q16—42no—
15C22×C8C2×C411yescharacteristic
16C22×C8D411yescharacteristic
17C22⋊Q8—42no—
18C22⋊Q8—42no—
19C2×C4⋊C4—21no—
20C2×C4⋊C4—41no—
21C2×C4⋊C4—41no—
22C2×C22⋊C4—41no—
23C2×C22⋊C4—41no—
24C2×C16—41no—
25#103—42no—
26C22⋊C8—41no—
27#95—22no—
28C24D811yescharacteristic
29C2×Q8—22no—
30C2×Q8—41no—
31C2×Q8—41no—
32C2×Q8—41no—
33C2×Q8—41no—
34C2×Q8—41no—
35C22×C4—21no—
36C22×C4—21no—
37C22×C4—41no—
38C22×C4—41no—
39C22×C4—41no—
40C22×C4—41no—
41C22×C4—42no—
42C22×C4C22⋊C411yescharacteristic
43C22×C4SD1611yescharacteristic
44Q16—42no—
45Q16—42no—
46C2×C8—21no—
47C2×C8—21no—
48C2×C8—41no—
49C2×C8—41no—
50C4⋊C4—22no—
51C4⋊C4—41no—
52C4⋊C4—44no—
53C4⋊C4—81no—
54C4⋊C4—81no—
55C22⋊C4—44no—
56C22⋊C4—81no—
57C22⋊C4—81no—
58C16—81no—
59C23—21no—
60C23—21no—
61C23—41no—
62C23—41no—
63C23#10211yescharacteristic
64C23SD3211yescharacteristic
65C23Q3211yescharacteristic
66Q8—42no—
67Q8—42no—
68Q8—81no—
69Q8—81no—
70C2×C4—21no—
71C2×C4—21no—
72C2×C4—21no—
73C2×C4—41no—
74C2×C4—41no—
75C2×C4—41no—
76C2×C4—42no—
77C2×C4—42no—
78C2×C4—81no—
79C2×C4—81no—
80C2×C4—81no—
81C2×C4—81no—
82C2×C4—81no—
83C2×C4—82no—
84C2×C4—82no—
85C2×C4—82no—
86C2×C4#9911yescharacteristic
87C2×C4#10411yescharacteristic
88C8—41no—
89C8—41no—
90C22—21no—
91C22—21no—
92C22—21no—
93C22—21no—
94C22—41no—
95C22—41no—
96C22—41no—
97C22—41no—
98C22—81no—
99C22#32711yescharacteristic
100C22#35811yescharacteristic
101C22#36011yescharacteristic
102C4—21no—
103C4—41no—
104C4—82no—
105C4—82no—
106C4—162no—
107C4AΣL(1,𝔽16)11yescharacteristic
108C4#35311yescharacteristic
109C2—21no—
110C2—21no—
111C2—41no—
112C2—41no—
113C2#134011yescharacteristic
114C2#134311yescharacteristic
115C2#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. C22×Q16C4semidirect
10. #340C4semidirect

Maximal subgroups

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

subgroupindex
#20012
#13142
#12892