Finite Group Databasebeta

← #17392

Subgroups — #17392

Order 256.

725 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 725 conjugacy classes of subgroups — too many to draw. Only its 187 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×#123C211yesmaximal, characteristic
2C2×#407C212yesmaximal, complemented
3#1468C218yesmaximal, complemented
4#1463C212yesmaximal, complemented
5#1462C212yesmaximal, complemented
6C23×D4C411yescharacteristic
7C24×C4C2211yescharacteristic
8C22×M16—22no—
9C2×#123—22no—
10C2×#123—22no—
11C2×#123—22no—
12C2×#123—22no—
13C2×#123C414yes—
14C2×#123C2214yes—
15C22×C4⋊C4C411yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C411yescharacteristic
18C22×C22⋊C4C411yescharacteristic
19C22×C22⋊C4C2211yescharacteristic
20#407—22no—
21#407—22no—
22#407C2214yes—
23#407C2214yescomplemented
24C2×C22⋊C8—22no—
25C2×C22⋊C8C2214yescomplemented
26#324C2218yescomplemented
27#323C2218yescomplemented
28C25C2×C411yescharacteristic
29C25C2×C411yescharacteristic
30C22×D4—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4—44no—
34C22×D4C2×C414yes—
35C23×C4—44no—
36C23×C4C2×C411yescharacteristic
37C23×C4C2×C411yescharacteristic
38C23×C4C2×C411yescharacteristic
39C23×C4C2×C411yescharacteristic
40C23×C4C2×C412yes—
41C23×C4D412yes—
42C23×C4D412yes—
43C23×C4D412yes—
44C23×C4D412yes—
45C23×C4C2311yescharacteristic
46C23×C4C2312yes—
47C2×M16—24no—
48C2×M16—24no—
49C2×M16—48no—
50C22×C8—24no—
51#123—24no—
52#123—24no—
53#123—24no—
54#123—24no—
55#123—24no—
56#123—24no—
57#123—24no—
58#123—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2×C414yes—
65C2×C4⋊C4C2×C412yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C22⋊C4C2×C414yes—
76C2×C22⋊C4C2×C414yes—
77C2×C22⋊C4C2×C412yes—
78C2×C22⋊C4C2312yes—
79C22⋊C8—24no—
80C22⋊C8—24no—
81C22⋊C8—24no—
82C22⋊C8C2318yes—
83C24—24no—
84C24—21no—
85C24—21no—
86C24—42no—
87C24—42no—
88C24—42no—
89C24—42no—
90C24—42no—
91C24C22⋊C412yes—
92C24C22⋊C411yescharacteristic
93C24C22⋊C411yescharacteristic
94C24C22×C411yescharacteristic
95C24C22×C412yes—
96C24C22×C412yes—
97C2×D4—24no—
98C2×D4—24no—
99C2×D4—48no—
100C2×D4—44no—
101C2×D4—44no—
102C2×D4—48no—
103C22×C4—28no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—44no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4—44no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4—42no—
119C22×C4—42no—
120C22×C4C22⋊C412yes—
121C22×C4C22⋊C412yes—
122C22×C4C22⋊C412yes—
123C22×C4C22⋊C411yescharacteristic
124C22×C4C22⋊C411yescharacteristic
125C22×C4C22⋊C412yes—
126C22×C4C22⋊C412yes—
127C22×C4C22×C412yes—
128C22×C4C22×C412yes—
129C22×C4C22×C412yes—
130C22×C4C22×C412yes—
131C22×C4C22×C411yescharacteristic
132C22×C4C2×D412yes—
133C22×C4C2×D412yes—
134C22×C4C2×D412yes—
135C22×C4C2×D412yes—
136C22×C4C2×D412yes—
137C22×C4C2×D412yes—
138C22×C4C2411yescharacteristic
139M16—48no—
140M16—48no—
141C2×C8—28no—
142C2×C8—48no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—24no—
151C22⋊C4—24no—
152C22⋊C4—24no—
153C23—22no—
154C23—22no—
155C23—22no—
156C23—24no—
157C23—24no—
158C23—22no—
159C23—24no—
160C23—24no—
161C23—22no—
162C23—24no—
163C23—41no—
164C23—42no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—42no—
169C23—416no—
170C23—42no—
171C23—42no—
172C23—42no—
173C23—42no—
174C23—42no—
175C23—42no—
176C23—42no—
177C23—42no—
178C23—88no—
179C23#9914yes—
180C23C2×C22⋊C412yes—
181C23C2×C22⋊C412yes—
182C23C2×C22⋊C411yescharacteristic
183C23C2×C22⋊C411yescharacteristic
184C23C23×C411yescharacteristic
185D4—48no—
186D4—48no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4C2×C22⋊C412yes—
216C2×C4C2×C22⋊C412yes—
217C2×C4C2×C22⋊C412yes—
218C2×C4C22×D412yes—
219C8—48no—
220C22—28no—
221C22—21no—
222C22—21no—
223C22—22no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—44no—
233C22—48no—
234C22—48no—
235C22—44no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—84no—
240C22—84no—
241C22—84no—
242C22C2×#9912yes—
243C22C2×#9914yes—
244C22#41312yes—
245C22C22×C22⋊C411yescharacteristic
246C22#56812yes—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—42no—
253C4—42no—
254C2—22no—
255C2—22no—
256C2—22no—
257C2—24no—
258C2—42no—
259C2—82no—
260C2#287911yescharacteristic
261C2C22×#9911yescharacteristic
262C2C2×#41311yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC22×C4C24
prime radicalC23×D4C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#407C2semidirect
3. #1468C2semidirect
4. #1463C2semidirect
5. #1462C2semidirect
23. #407C22semidirect
25. C2×C22⋊C8C22semidirect
26. #324C22semidirect
27. #323C22semidirect

Maximal subgroups

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

subgroupindex
C22×#1232
C2×#4072
#14682
#14632
#14622