Finite Group Databasebeta

← #215630

Subgroups — #215630

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic
2#21916C212yesmaximal
3#9883C214yesmaximal
4#9881C214yesmaximal
5#8701C212yesmaximal
6#8699C212yesmaximal
7C22×#125C411yescharacteristic
8#2330—28no—
9C2×#389C412yes—
10C2×#389C411yescharacteristic
11C2×#389C411yescharacteristic
12C2×#389C2212yes—
13C2×#387C412yes—
14C2×#387C2212yes—
15C2×#384C411yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1917—28no—
20#1817C2214yes—
21#1797C2214yes—
22#1749C2214yes—
23#1280C2214yes—
24#1280C2218yes—
25#1280C2214yes—
26C2×#125—24no—
27C2×#125—24no—
28C2×#125—24no—
29C22×C4⋊C4C2×C412yes—
30C22×C4⋊C4C2×C412yes—
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C412yes—
35C22×C4⋊C4C2×C412yes—
36C22×C4⋊C4C2×C412yes—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C412yes—
42C22×C42C2311yescharacteristic
43C22×C42C2312yes—
44C2×C4⋊C8—24no—
45C2×C22⋊C8C2318yes—
46C2×#97—24no—
47#389—24no—
48#389—24no—
49#389—24no—
50#389—24no—
51#389—28no—
52#387—24no—
53#387—24no—
54#387—28no—
55#384—24no—
56C4×C4⋊C4—24no—
57C4×C4⋊C4D418yes—
58#376—24no—
59C2×#95C2×C411yescharacteristic
60C2×#95C2×C411yescharacteristic
61C2×#95C2×C412yes—
62C2×#95C2×C412yes—
63C2×#95C2×C412yes—
64C2×#95C2311yescharacteristic
65C2×#95C2311yescharacteristic
66C2×#95C2312yes—
67#336—24no—
68#336—28no—
69#336—24no—
70C23×C4C22×C411yescharacteristic
71C23×C4C22×C411yescharacteristic
72C23×C4C22×C412yes—
73C23×C4C22×C411yescharacteristic
74C23×C4C22×C412yes—
75C23×C4C22×C412yes—
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C22×C412yes—
80C23×C4C2411yescharacteristic
81C22×C8—28no—
82#125—232no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—24no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—24no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—42no—
99C2×C4⋊C4—42no—
100C2×C4⋊C4—82no—
101C2×C4⋊C4—82no—
102C2×C4⋊C4—82no—
103C2×C4⋊C4—82no—
104C2×C4⋊C4C22⋊C414yes—
105C2×C4⋊C4C22⋊C414yes—
106C2×C4⋊C4M1614yes—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—24no—
110C2×C42—22no—
111C2×C42—24no—
112C2×C42—42no—
113C2×C42—82no—
114C2×C42—82no—
115C2×C42M1614yes—
116C2×C42C2×D414yes—
117C2×C42C2×D414yes—
118C2×C42C2×D414yes—
119C4⋊C8—84no—
120C22⋊C8—216no—
121#97—84no—
122#95—22no—
123#95—22no—
124#95—24no—
125#95—24no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95C22⋊C418yes—
131C24C23×C411yescharacteristic
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—24no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—42no—
163C22×C4—41no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4C2×C22⋊C412yes—
171C22×C4C2×C22⋊C414yes—
172C22×C4C2×C22⋊C414yes—
173C22×C4C2×C22⋊C412yes—
174C22×C4C2×M1612yes—
175C22×C4C2×M1612yes—
176C22×C4C2×M1612yes—
177C22×C4C2×M1612yes—
178C22×C4C2×M1614yes—
179C22×C4C22×D412yes—
180C22×C42−1+414yes—
181C2×C8—48no—
182C2×C8—48no—
183C2×C8—48no—
184C4⋊C4—216no—
185C4⋊C4—216no—
186C4⋊C4—28no—
187C4⋊C4—28no—
188C4⋊C4—44no—
189C4⋊C4—44no—
190C4⋊C4—84no—
191C4⋊C4—84no—
192C4⋊C4—84no—
193C4⋊C4—84no—
194C42—28no—
195C42—28no—
196C42—44no—
197C42—44no—
198C42—44no—
199C42—84no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23C22×C22⋊C411yescharacteristic
205C23#51912yes—
206C23C22×M1611yescharacteristic
207C23C22×M1611yescharacteristic
208C23#56812yes—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—24no—
219C2×C4—24no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—84no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4#40714yes—
257C2×C4#53614yes—
258C8—88no—
259C22—22no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22#231011yescharacteristic
268C22C2×#40711yescharacteristic
269C22#287911yescharacteristic
270C22#288311yescharacteristic
271C22#288411yescharacteristic
272C22#288811yescharacteristic
273C22#288911yescharacteristic
274C22#298312yes—
275C22#298312yes—
276C4—24no—
277C4—24no—
278C4—44no—
279C4—42no—
280C4—44no—
281C4—42no—
282C4—42no—
283C4—44no—
284C4—81no—
285C4—82no—
286C4—81no—
287C2—22no—
288C2—22no—
289C2#1737011yescharacteristic
290C2#1737211yescharacteristic
291C2#1740811yescharacteristic
292C2#1741011yescharacteristic
293C2#2182211yescharacteristic
294C2#2182811yescharacteristic
295C2#2183211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊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
C2×#23302
#219162
#98832
#98812
#87012
#86992