Finite Group Databasebeta

← #8555

Subgroups — #8555

Order 256.

551 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 83 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×C22⋊Q8C211yesmaximal, characteristic
2#1821C212yesmaximal, complemented
3#1463C212yesmaximal, complemented
4#1462C212yesmaximal
5C23×Q8C411yescharacteristic
6C24×C4C411yescharacteristic
7C2×C22⋊Q8—24no—
8C2×C22⋊Q8—24no—
9C2×C22⋊Q8—24no—
10C22×C4⋊C4C411yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12C22×C4⋊C4C2211yescharacteristic
13C22×C22⋊C4C411yescharacteristic
14C22×C22⋊C4C2211yescharacteristic
15#407C2214yes—
16#343—24no—
17#324—24no—
18#323—24no—
19C25C2×C411yescharacteristic
20C22×Q8—22no—
21C22×Q8—22no—
22C22×Q8—22no—
23C22×Q8—44no—
24C23×C4—22no—
25C23×C4—22no—
26C23×C4—22no—
27C23×C4—44no—
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2×C411yescharacteristic
31C23×C4C2×C411yescharacteristic
32C23×C4C2×C411yescharacteristic
33C23×C4C2311yescharacteristic
34C2×M16—24no—
35C22⋊Q8—232no—
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×C22⋊C4—22no—
46C2×C22⋊C4—22no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C22⋊C4—22no—
50C2×C22⋊C4D414yes—
51C22⋊C8—24no—
52C22⋊C8—24no—
53C24—24no—
54C24—21no—
55C24—21no—
56C24—42no—
57C24—42no—
58C24C22⋊C412yes—
59C24C22×C411yescharacteristic
60C2×Q8—28no—
61C2×Q8—48no—
62C2×Q8—48no—
63C2×Q8—44no—
64C2×Q8—44no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—24no—
68C22×C4—28no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—42no—
78C22×C4—48no—
79C22×C4—48no—
80C22×C4—44no—
81C22×C4—44no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4C22⋊C412yes—
87C22×C4C22⋊C412yes—
88C22×C4C22⋊C412yes—
89C22×C4C22⋊C412yes—
90C22×C4C22⋊C412yes—
91C22×C4C2×D412yes—
92C22×C4C2×D412yes—
93C22×C4C2×D412yes—
94M16—48no—
95C2×C8—44no—
96C4⋊C4—28no—
97C4⋊C4—28no—
98C4⋊C4—28no—
99C22⋊C4—28no—
100C22⋊C4—28no—
101C23—22no—
102C23—22no—
103C23—22no—
104C23—24no—
105C23—24no—
106C23—22no—
107C23—22no—
108C23—42no—
109C23—42no—
110C23—42no—
111C23—416no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23#10114yes—
116C23C2×C22⋊C411yescharacteristic
117C23C2×C22⋊C411yescharacteristic
118C23C2×C22⋊C411yescharacteristic
119Q8—416no—
120C2×C4—24no—
121C2×C4—24no—
122C2×C4—22no—
123C2×C4—24no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—416no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4#10014yes—
145C2×C4#12014yes—
146C8—84no—
147C22—28no—
148C22—21no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—21no—
153C22—21no—
154C22—24no—
155C22—44no—
156C22—48no—
157C22—48no—
158C22—44no—
159C22—42no—
160C22—42no—
161C22#37911yescharacteristic
162C22#41012yes—
163C22C2×#10012yes—
164C22C2×#10112yes—
165C4—24no—
166C4—24no—
167C4—42no—
168C4—42no—
169C4—42no—
170C4—42no—
171C2—22no—
172C2—22no—
173C2—22no—
174C2—44no—
175C2#178211yescharacteristic
176C2#178311yescharacteristic
177C2#178411yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC24C22×C4
Frattini, cosocleC23×C4C23
prime radicalC25C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1821C2semidirect
3. #1463C2semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
#18212
#14632
#14622