Finite Group Databasebeta

← #11520

Subgroups — #11520

Order 256.

209 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 55 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#2317C211yesmaximal, characteristic
2#1464C211yesmaximal, characteristic
3#1464C211yesmaximal, characteristic
4#1326C214yesmaximal
5C2×#117C411yescharacteristic
6C2×#117C2211yescharacteristic
7C22×C4⋊C4C411yescharacteristic
8#432—21no—
9#432—21no—
10#389—21no—
11#389—21no—
12#385C412yes—
13#385C2212yes—
14#384—21no—
15#382—21no—
16#324—21no—
17#324—21no—
18#323C2212yes—
19#323C2212yes—
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C412yes—
22#117—24no—
23#117—41no—
24#117—41no—
25C2×C4⋊C4—21no—
26C2×C4⋊C4—21no—
27C2×C4⋊C4—21no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—41no—
31C2×C4⋊C4—41no—
32C2×C4⋊C4D411yescharacteristic
33C2×C4⋊C4D411yescharacteristic
34C2×C22⋊C4C2×C412yes—
35C2×C22⋊C4C2×C411yescharacteristic
36C2×C22⋊C4C2311yescharacteristic
37C2×C42—21no—
38C2×C42D411yescharacteristic
39C2×C42D411yescharacteristic
40C4⋊C8—41no—
41C4⋊C8—41no—
42C22⋊C8—22no—
43C22⋊C8—22no—
44#97—41no—
45#97—41no—
46#95—22no—
47#95—22no—
48C24C22×C411yescharacteristic
49C22×C4—22no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—22no—
53C22×C4—22no—
54C22×C4—22no—
55C22×C4—42no—
56C22×C4—82no—
57C22×C4C22⋊C411yescharacteristic
58C22×C4C22⋊C411yescharacteristic
59C22×C4C22⋊C411yescharacteristic
60C22×C4C22⋊C411yescharacteristic
61C22×C4C2×D411yescharacteristic
62C22×C4C2×D411yescharacteristic
63C2×C8—42no—
64C2×C8—42no—
65C4⋊C4—24no—
66C4⋊C4—41no—
67C4⋊C4—41no—
68C4⋊C4—42no—
69C4⋊C4—41no—
70C4⋊C4—41no—
71C4⋊C4—82no—
72C4⋊C4—82no—
73C22⋊C4—22no—
74C22⋊C4—41no—
75C22⋊C4—41no—
76C22⋊C4M1614yes—
77C42—24no—
78C42—21no—
79C42—21no—
80C42—21no—
81C42—21no—
82C23—21no—
83C23—21no—
84C23—41no—
85C23—41no—
86C23C2×C22⋊C411yescharacteristic
87C23C2×M1612yes—
88C2×C4—24no—
89C2×C4—24no—
90C2×C4—21no—
91C2×C4—21no—
92C2×C4—21no—
93C2×C4—21no—
94C2×C4—24no—
95C2×C4—21no—
96C2×C4—21no—
97C2×C4—21no—
98C2×C4—21no—
99C2×C4—42no—
100C2×C4—42no—
101C2×C4—41no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—42no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—84no—
109C2×C4#9911yescharacteristic
110C2×C4#9911yescharacteristic
111C2×C4#10112yes—
112C8—82no—
113C8—82no—
114C22—21no—
115C22—21no—
116C22—22no—
117C22—42no—
118C22—42no—
119C22—41no—
120C22—41no—
121C22#40711yescharacteristic
122C22C2×#9911yescharacteristic
123C22C2×#10111yescharacteristic
124C4—21no—
125C4—21no—
126C4—22no—
127C4—44no—
128C4—41no—
129C4—41no—
130C4—41no—
131C4—41no—
132C4—82no—
133C2—21no—
134C2—21no—
135C2—42no—
136C2#146311yescharacteristic
137C2#212211yescharacteristic
138C2#212311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutator, prime radicalC24C22×C4
Frattini, cosocleC2×C22⋊C4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#23172
#14642
#13262