Finite Group Databasebeta

← #8554

Subgroups — #8554

Order 256.

389 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C22×#125C211yesmaximal, characteristic
2#1821C212yesmaximal
3#1463C212yesmaximal
4#1463C212yesmaximal
5C2×#125—24no—
6C2×#125—24no—
7C2×#125—24no—
8C22×C4⋊C4C411yescharacteristic
9C22×C4⋊C4C411yescharacteristic
10C22×C4⋊C4C411yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C42C411yescharacteristic
15#407C2214yes—
16#343—24no—
17#324—24no—
18#324—24no—
19C23×C4C2×C411yescharacteristic
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C411yescharacteristic
22C23×C4C2×C411yescharacteristic
23C23×C4C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4C2311yescharacteristic
26C2×M16—24no—
27#125—232no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—22no—
32C2×C4⋊C4—22no—
33C2×C4⋊C4—22no—
34C2×C4⋊C4—22no—
35C2×C4⋊C4—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4—22no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4D414yes—
44C2×C4⋊C4D414yes—
45C2×C4⋊C4D414yes—
46C2×C42—22no—
47C2×C42—22no—
48C2×C42—22no—
49C22⋊C8—24no—
50C22⋊C8—24no—
51C24C22×C411yescharacteristic
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—24no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—42no—
66C22×C4—42no—
67C22×C4—42no—
68C22×C4—42no—
69C22×C4—42no—
70C22×C4—42no—
71C22×C4C22⋊C412yes—
72C22×C4C22⋊C412yes—
73C22×C4C22⋊C412yes—
74C22×C4C22⋊C412yes—
75C22×C4C22⋊C412yes—
76C22×C4C22⋊C412yes—
77C22×C4C2×D412yes—
78C22×C4C2×D412yes—
79C22×C4C2×D412yes—
80M16—48no—
81C2×C8—44no—
82C4⋊C4—28no—
83C4⋊C4—28no—
84C4⋊C4—28no—
85C4⋊C4—28no—
86C4⋊C4—28no—
87C4⋊C4—28no—
88C42—28no—
89C23—22no—
90C23—22no—
91C23—22no—
92C23C2×C22⋊C411yescharacteristic
93C23C2×C22⋊C411yescharacteristic
94C23C2×C22⋊C411yescharacteristic
95C2×C4—24no—
96C2×C4—24no—
97C2×C4—22no—
98C2×C4—24no—
99C2×C4—22no—
100C2×C4—22no—
101C2×C4—22no—
102C2×C4—22no—
103C2×C4—22no—
104C2×C4—22no—
105C2×C4—24no—
106C2×C4—42no—
107C2×C4—42no—
108C2×C4—42no—
109C2×C4—42no—
110C2×C4—42no—
111C2×C4—42no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—42no—
115C2×C4—42no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—42no—
119C2×C4—42no—
120C2×C4—42no—
121C2×C4—42no—
122C2×C4#10114yes—
123C2×C4#12014yes—
124C8—84no—
125C22—28no—
126C22—21no—
127C22—21no—
128C22—21no—
129C22—21no—
130C22—21no—
131C22—21no—
132C22#37911yescharacteristic
133C22#41012yes—
134C22C2×#10112yes—
135C22#41312yes—
136C4—24no—
137C4—24no—
138C4—42no—
139C4—42no—
140C4—42no—
141C4—42no—
142C4—42no—
143C2—22no—
144C2—22no—
145C2—22no—
146C2#178211yescharacteristic
147C2#178411yescharacteristic
148C2#178511yescharacteristic
subgroupG/H
center, socleC22#379
commutator, prime radicalC24C22×C4
Frattini, cosocleC23×C4C23
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C22×#1252
#18212
#14632