Finite Group Databasebeta

← #215632

Subgroups — #215632

Order 512.

818 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 202 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
1C2×#2667C211yesmaximal, characteristic
2#21906C212yesmaximal
3#9883C214yesmaximal, complemented
4#9881C214yesmaximal, complemented
5#8704C212yesmaximal
6#8702C212yesmaximal
7#2667—28no—
8C2×#399C412yes—
9C2×#398C412yes—
10C2×#398C412yes—
11C2×#396C412yes—
12C2×#389C2212yes—
13C2×#387C2212yes—
14C2×#381C2211yescharacteristic
15C2×#380C2211yescharacteristic
16C2×C4×C22⋊C4C2211yescharacteristic
17#1924—28no—
18#1797C2214yes—
19#1779C2214yes—
20#1749C2214yes—
21#1280C2218yes—
22#1280C2214yes—
23#1280C2214yes—
24C24×C4C2311yescharacteristic
25C22×C4⋊C4C2×C412yes—
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4C2×C412yes—
28C22×C4⋊C4C2×C412yes—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C412yes—
32C22×C22⋊C4C2×C411yescharacteristic
33C22×C22⋊C4C2×C412yes—
34C22×C22⋊C4C2×C411yescharacteristic
35C22×C22⋊C4C2×C412yes—
36C22×C42C2312yes—
37C2×C4⋊C8—24no—
38C2×C22⋊C8—24no—
39C2×C22⋊C8—24no—
40C2×C22⋊C8C2318yes—
41C2×#97—24no—
42#399—28no—
43#398—28no—
44#398—28no—
45#396—28no—
46#389—24no—
47#389—24no—
48#387—24no—
49#387—24no—
50#381—24no—
51#380—24no—
52C4×C22⋊C4D418yes—
53C2×#95C2×C412yes—
54C2×#95C2×C412yes—
55C2×#95C2×C412yes—
56C2×#95C2×C411yescharacteristic
57C2×#95C2×C411yescharacteristic
58C2×#95C2312yes—
59C2×#95C2311yescharacteristic
60C2×#95C2311yescharacteristic
61#336—24no—
62#336—24no—
63C25C22×C411yescharacteristic
64C23×C4—24no—
65C23×C4—82no—
66C23×C4C22×C412yes—
67C23×C4C22×C412yes—
68C23×C4C22×C412yes—
69C23×C4C22×C412yes—
70C23×C4C22×C412yes—
71C23×C4C22×C412yes—
72C23×C4C22×C411yescharacteristic
73C23×C4C2×D414yes—
74C23×C4C2411yescharacteristic
75C22×C8—28no—
76C2×C4⋊C4—24no—
77C2×C4⋊C4—24no—
78C2×C4⋊C4—24no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—24no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—44no—
84C2×C4⋊C4—44no—
85C2×C4⋊C4—82no—
86C2×C4⋊C4—82no—
87C2×C4⋊C4—82no—
88C2×C4⋊C4—82no—
89C2×C4⋊C4—82no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—44no—
94C2×C22⋊C4—81no—
95C2×C22⋊C4—81no—
96C2×C22⋊C4—82no—
97C2×C22⋊C4C22⋊C418yes—
98C2×C42—82no—
99C2×C42C2×D414yes—
100C2×C42C2×D414yes—
101C4⋊C8—84no—
102C22⋊C8—216no—
103C22⋊C8—84no—
104C22⋊C8—84no—
105#97—84no—
106#95—24no—
107#95—24no—
108#95—24no—
109#95—22no—
110#95—22no—
111#95—22no—
112#95—22no—
113#95C22⋊C414yes—
114#95C22⋊C414yes—
115C24—22no—
116C24—24no—
117C24—41no—
118C24—42no—
119C24—41no—
120C24C2×C22⋊C412yes—
121C24C23×C411yescharacteristic
122C22×C4—22no—
123C22×C4—24no—
124C22×C4—24no—
125C22×C4—28no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—24no—
132C22×C4—24no—
133C22×C4—24no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—44no—
139C22×C4—44no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—88no—
151C22×C4C2×C22⋊C412yes—
152C22×C4C2×C22⋊C414yes—
153C22×C4C2×C22⋊C414yes—
154C22×C4C22×D412yes—
155C22×C42−1+414yes—
156C2×C8—48no—
157C2×C8—48no—
158C2×C8—48no—
159C4⋊C4—48no—
160C4⋊C4—48no—
161C4⋊C4—84no—
162C4⋊C4—84no—
163C4⋊C4—84no—
164C4⋊C4—84no—
165C4⋊C4—84no—
166C22⋊C4—48no—
167C22⋊C4—84no—
168C22⋊C4—82no—
169C22⋊C4—82no—
170C42—44no—
171C42—44no—
172C23—22no—
173C23—21no—
174C23—21no—
175C23—22no—
176C23—24no—
177C23—24no—
178C23—22no—
179C23—22no—
180C23—44no—
181C23—44no—
182C23—44no—
183C23—84no—
184C23—84no—
185C23C22×C22⋊C411yescharacteristic
186C23#51912yes—
187C23#56814yes—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—44no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—88no—
214C2×C4—84no—
215C2×C4—84no—
216C2×C4—84no—
217C2×C4—84no—
218C2×C4—84no—
219C2×C4#53614yes—
220C8—88no—
221C22—24no—
222C22—22no—
223C22—22no—
224C22—24no—
225C22—44no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—44no—
230C22—88no—
231C22#231011yescharacteristic
232C22#287912yes—
233C22#288312yes—
234C22#288811yescharacteristic
235C22#288911yescharacteristic
236C22#298714yes—
237C4—24no—
238C4—44no—
239C4—42no—
240C4—44no—
241C4—82no—
242C4—82no—
243C4—82no—
244C4—82no—
245C2—24no—
246C2—82no—
247C2#1740612yes—
248C2#1740812yes—
249C2#2182311yescharacteristic
250C2#2182711yescharacteristic
251C2#2183211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9883C2semidirect
4. #9881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26672
#219062
#98832
#98812
#87042
#87022