Finite Group Databasebeta

← #215619

Subgroups — #215619

Order 512.

930 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 234 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×C4×#125C211yesmaximal, characteristic
2#20778C212yesmaximal
3#9881C218yesmaximal
4#8701C214yesmaximal
5C22×#125C411yescharacteristic
6C4×#125—22no—
7C4×#125—22no—
8C4×#125—22no—
9C4×#125—22no—
10C2×#389C412yes—
11C2×#387C2214yes—
12C2×#382C411yescharacteristic
13C2×C4×C4⋊C4C414yes—
14C2×C4×C4⋊C4C2212yes—
15C2×C43C2211yescharacteristic
16#1839—22no—
17#1831—22no—
18#1817C2214yes—
19#1797C2214yes—
20#1767—22no—
21#1749C2214yes—
22#1745—22no—
23#1280C22116yes—
24C2×#125—28no—
25C2×#125—21no—
26C2×#125—22no—
27C2×#125—21no—
28C22×C4⋊C4C2×C414yes—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4C2×C414yes—
31C22×C4⋊C4C2×C412yes—
32C22×C42C2×C411yescharacteristic
33C22×C42C2×C414yes—
34C22×C42C2×C412yes—
35C22×C42C2×C411yescharacteristic
36C22×C42C2311yescharacteristic
37C22×C42C2312yes—
38C2×C4⋊C8—24no—
39C2×C22⋊C8C2318yes—
40C2×#97—24no—
41#389—22no—
42#389—22no—
43#389—22no—
44#389—22no—
45#387—24no—
46#387—24no—
47#387—24no—
48#387—24no—
49#382—21no—
50#382—22no—
51#382—21no—
52C4×C4⋊C4—24no—
53C4×C4⋊C4—24no—
54C4×C4⋊C4—24no—
55C4×C4⋊C4—24no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4—22no—
58C4×C4⋊C4—22no—
59C4×C4⋊C4—22no—
60C2×#95C2×C414yes—
61C2×#95C2×C414yes—
62C2×#95C2314yes—
63C43D412yes—
64C43D412yes—
65C43D412yes—
66C43D412yes—
67#336—24no—
68#336—24no—
69#336—24no—
70#336—24no—
71C23×C4C22×C411yescharacteristic
72C23×C4C22×C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C22×C414yes—
75C23×C4C22×C412yes—
76C23×C4C22×C414yes—
77C23×C4C2411yescharacteristic
78C22×C8—28no—
79#125—28no—
80#125—216no—
81#125—28no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—24no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—28no—
87C2×C4⋊C4—28no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—44no—
94C2×C4⋊C4—48no—
95C2×C4⋊C4M1618yes—
96C2×C42—24no—
97C2×C42—24no—
98C2×C42—24no—
99C2×C42—24no—
100C2×C42—44no—
101C2×C42—44no—
102C2×C42—42no—
103C2×C42—48no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22⋊C412yes—
106C2×C42C22⋊C411yescharacteristic
107C2×C42C22⋊C412yes—
108C2×C42C22⋊C412yes—
109C2×C42C22⋊C412yes—
110C2×C42C22⋊C412yes—
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C412yes—
113C2×C42C22⋊C411yescharacteristic
114C2×C42C2×D412yes—
115C2×C42C2×D412yes—
116C2×C42C2×D412yes—
117C2×C42C2×D412yes—
118C2×C42C2×D412yes—
119C2×C42C2×D412yes—
120C4⋊C8—48no—
121C22⋊C8—216no—
122#97—48no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—24no—
127#95—24no—
128#95—24no—
129C24C23×C411yescharacteristic
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—28no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—44no—
147C22×C4—44no—
148C22×C4—44no—
149C22×C4—44no—
150C22×C4C2×C22⋊C412yes—
151C22×C4C2×C22⋊C412yes—
152C22×C4C2×C22⋊C412yes—
153C22×C4C2×C22⋊C411yescharacteristic
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4C2×C22⋊C412yes—
156C22×C4C2×C22⋊C412yes—
157C22×C4C2×M1614yes—
158C22×C4C2×M1618yes—
159C22×C4C22×D412yes—
160C22×C42−1+414yes—
161C2×C8—48no—
162C2×C8—48no—
163C2×C8—48no—
164C4⋊C4—216no—
165C4⋊C4—216no—
166C4⋊C4—28no—
167C4⋊C4—28no—
168C4⋊C4—44no—
169C4⋊C4—44no—
170C4⋊C4—48no—
171C4⋊C4—48no—
172C42—24no—
173C42—24no—
174C42—22no—
175C42—24no—
176C42—22no—
177C42—24no—
178C42—24no—
179C42—24no—
180C42—24no—
181C42—41no—
182C42—42no—
183C42—41no—
184C42—42no—
185C42—42no—
186C42—42no—
187C42—42no—
188C42—48no—
189C42—48no—
190C23—22no—
191C23—22no—
192C23C22×C22⋊C411yescharacteristic
193C23#51912yes—
194C23C22×M1612yes—
195C23#56812yes—
196C2×C4—28no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—28no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—28no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—44no—
219C2×C4—48no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—48no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—88no—
230C2×C4#40712yes—
231C2×C4#40712yes—
232C2×C4#41014yes—
233C2×C4#41314yes—
234C2×C4#53614yes—
235C8—88no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—24no—
241C22#231011yescharacteristic
242C22C2×#40711yescharacteristic
243C22#287911yescharacteristic
244C22C2×#41012yes—
245C22C2×#41312yes—
246C22#298314yes—
247C4—22no—
248C4—22no—
249C4—24no—
250C4—24no—
251C4—24no—
252C4—42no—
253C4—42no—
254C4—48no—
255C4—42no—
256C4—84no—
257C2—24no—
258C2#1736612yes—
259C2#1740212yes—
260C2#2182211yescharacteristic
261C2#2182611yescharacteristic
262C2#2183011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ 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
C2×C4×#1252
#207782
#98812
#87012