Finite Group Databasebeta

← #7490

Subgroups — #7490

Order 256.

137 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 137 conjugacy classes of subgroups — too many to draw. Only its 33 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#2126C211yesmaximal, characteristic
2#1884C211yesmaximal, characteristic
3#1543C211yesmaximal, characteristic
4#568C411yescharacteristic, complemented
5#532C2211yescharacteristic
6#431—21no—
7#421—21no—
8#410—21no—
9#410C411yescharacteristic, complemented
10#354C412yescomplemented
11#353—21no—
12#339—21no—
13#333C412yescomplemented
14#331—21no—
15C2×#41C2×C411yescharacteristic
16C2×#41D411yescharacteristic
17#131—42no—
18C2×M16—21no—
19C2×M16—41no—
20C4⋊Q8C2×C411yescharacteristic
21#127D411yescharacteristic
22#124C2×C411yescharacteristic
23#124Q811yescharacteristic
24#121—41no—
25C4×D4—41no—
26#117—21no—
27#117—41no—
28C2×C42D411yescharacteristic
29C4⋊C8—22no—
30#104—42no—
31#99—22no—
32#99—41no—
33#99—41no—
34C22⋊C8—41no—
35C4×C8—41no—
36#41—41no—
37#41—41no—
38#41—41no—
39#41—41no—
40C2×Q8C4211yescharacteristic
41C2×Q8C4⋊C411yescharacteristic
42C2×D4—21no—
43C2×D4—21no—
44C2×D4—41no—
45C2×D4C22⋊C411yescharacteristic
46C2×D4C4⋊C411yescharacteristic
47C22×C4—21no—
48C22×C4—21no—
49C22×C4—21no—
50C22×C4C22⋊C411yescharacteristic
51M16—22no—
52M16—41no—
53M16—42no—
54C2×C8—41no—
55C2×C8—42no—
56C2×C8—42no—
57C4⋊C4—41no—
58C4⋊C4—41no—
59C4⋊C4—81no—
60C22⋊C4—41no—
61C22⋊C4—41no—
62C22⋊C4—42no—
63C22⋊C4—41no—
64C42—21no—
65C42—41no—
66C42—81no—
67C42C22⋊C411yescharacteristic
68C42C4⋊C411yescharacteristic
69C23—21no—
70C23—21no—
71C23#9911yescharacteristic
72Q8—41no—
73Q8—41no—
74D4—41no—
75D4—41no—
76D4—42no—
77D4—82no—
78D4—81no—
79C2×C4—21no—
80C2×C4—21no—
81C2×C4—21no—
82C2×C4—21no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—42no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—41no—
89C2×C4—41no—
90C2×C4—81no—
91C2×C4—82no—
92C2×C4#9511yescharacteristic
93C2×C4#9911yescharacteristic
94C8—42no—
95C8—82no—
96C22—21no—
97C22—41no—
98C22—41no—
99C22—41no—
100C22—81no—
101C22—81no—
102C22#34211yescharacteristic
103C4—22no—
104C4—21no—
105C4—41no—
106C4—41no—
107C4—41no—
108C4—161no—
109C4—162no—
110C4AΣL(1,𝔽16)11yescharacteristic
111C4#35211yescharacteristic
112C2—21no—
113C2—41no—
114C2—81no—
115C2—81no—
116C2#139011yescharacteristic
subgroupG/H
centerC4AΣL(1,𝔽16)
commutatorC2×Q8C42
Frattini, prime radical, cosocle#532C22
socleC2#1390
LCSG ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G
LECSG ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #568C4semidirect
9. #410C4semidirect
10. #354C4semidirect
13. #333C4semidirect

Maximal subgroups

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

subgroupindex
#21262
#18842
#15432