Finite Group Databasebeta

← #9442

Subgroups — #9442

Order 256.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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#3142C211yesmaximal, characteristic
2#2371C211yesmaximal, characteristic, complemented
3#1534C211yesmaximal, characteristic, complemented
4#1505C212yesmaximal
5#1300C212yesmaximal, complemented
6#532C2211yescharacteristic
7C2×#125C2211yescharacteristic
8#491C412yescomplemented
9#490C412yescomplemented
10#488—22no—
11#432C2212yes—
12#417—22no—
13C4×M16C2211yescharacteristic
14#387—22no—
15#387—22no—
16C4×C4⋊C4—22no—
17#376C2212yes—
18#330—22no—
19#327—22no—
20#327—22no—
21C2×#41—21no—
22C2×M16—21no—
23C4⋊Q8C2×C411yescharacteristic
24#127C2×C411yescharacteristic
25#125—22no—
26#125C2×C412yes—
27#125D412yes—
28#124—21no—
29#121—22no—
30C4×D4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4D412yes—
34C2×C42—22no—
35C2×C42—22no—
36C2×C42C2311yescharacteristic
37C4⋊C8—22no—
38#103—24no—
39#102—24no—
40C22⋊C8—22no—
41#97—22no—
42#97C2×C412yes—
43C4×C8—21no—
44#95—22no—
45#95—22no—
46#95—22no—
47#95D412yes—
48#95D412yes—
49#41—44no—
50C2×Q8—21no—
51C2×D4—21no—
52C2×D4—22no—
53C2×D4—41no—
54C22×C4—22no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—41no—
58C22×C4C2×D411yescharacteristic
59C22×C4C2×D412yes—
60M16—42no—
61C2×C8—22no—
62C2×C8—42no—
63C4⋊C4—24no—
64C4⋊C4—24no—
65C4⋊C4—22no—
66C4⋊C4—22no—
67C4⋊C4—44no—
68C4⋊C4C22⋊C414yes—
69C22⋊C4—42no—
70C42—44no—
71C42—44no—
72C42C22×C411yescharacteristic
73C42C2×D411yescharacteristic
74C42#4112yes—
75C23—41no—
76C23#12011yescharacteristic
77Q8—42no—
78D4—42no—
79D4—82no—
80D4—82no—
81C2×C4—22no—
82C2×C4—24no—
83C2×C4—24no—
84C2×C4—42no—
85C2×C4—42no—
86C2×C4—42no—
87C2×C4—42no—
88C2×C4—42no—
89C2×C4—42no—
90C2×C4—42no—
91C2×C4—41no—
92C2×C4—42no—
93C2×C4—82no—
94C2×C4C2×C22⋊C411yescharacteristic
95C2×C4C4×D412yes—
96C2×C4#12112yes—
97C2×C4#12311yescharacteristic
98C8—42no—
99C8—82no—
100C22—22no—
101C22—81no—
102C22—82no—
103C22#38611yescharacteristic
104C22#45412yes—
105C4—21no—
106C4—21no—
107C4—22no—
108C4—22no—
109C4—44no—
110C4—44no—
111C4—82no—
112C4—81no—
113C2—22no—
114C2—161no—
115C2#188811yescharacteristic
116C2#190612yes—
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2371C2semidirect
3. #1534C2semidirect
5. #1300C2semidirect
8. #491C4semidirect
9. #490C4semidirect

Maximal subgroups

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

subgroupindex
#31422
#23712
#15342
#15052
#13002