Finite Group Databasebeta

← #215629

Subgroups — #215629

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic
2#21916C212yesmaximal
3#9882C214yesmaximal
4#9881C214yesmaximal
5#8701C212yesmaximal
6#8699C212yesmaximal
7C22×C4⋊Q8C411yescharacteristic
8#2331—28no—
9C2×#391C412yes—
10C2×#391C2212yes—
11C2×#389C411yescharacteristic
12C2×#389C411yescharacteristic
13C2×#387C412yes—
14C2×#387C2212yes—
15C2×#382C411yescharacteristic
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—
26C23×Q8C2×C412yes—
27C2×C4⋊Q8—24no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8—24no—
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×C411yescharacteristic
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C412yes—
40C22×C42C2311yescharacteristic
41C22×C42C2312yes—
42C2×C4⋊C8—24no—
43C2×C22⋊C8C2318yes—
44C2×#97—24no—
45#391—24no—
46#391—24no—
47#391—28no—
48#389—24no—
49#389—24no—
50#387—24no—
51#387—24no—
52#387—28no—
53#382—24no—
54C4×C4⋊C4—24no—
55C4×C4⋊C4D418yes—
56#376—24no—
57C2×#95C2×C411yescharacteristic
58C2×#95C2×C411yescharacteristic
59C2×#95C2×C412yes—
60C2×#95C2×C412yes—
61C2×#95C2×C412yes—
62C2×#95C2×C412yes—
63C2×#95C2×C412yes—
64C2×#95C2311yescharacteristic
65C2×#95C2311yescharacteristic
66C2×#95C2312yes—
67#336—24no—
68#336—28no—
69#336—24no—
70C22×Q8—24no—
71C22×Q8—24no—
72C22×Q8—24no—
73C22×Q8—82no—
74C22×Q8—82no—
75C23×C4C22×C411yescharacteristic
76C23×C4C22×C411yescharacteristic
77C23×C4C22×C411yescharacteristic
78C23×C4C22×C412yes—
79C23×C4C22×C412yes—
80C23×C4C22×C412yes—
81C23×C4C22×C411yescharacteristic
82C23×C4C22×C412yes—
83C23×C4C22×C411yescharacteristic
84C23×C4C22×C411yescharacteristic
85C23×C4C2411yescharacteristic
86C22×C8—28no—
87C4⋊Q8—232no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
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⋊C4C22⋊C414yes—
103C2×C4⋊C4C22⋊C414yes—
104C2×C4⋊C4M1614yes—
105C2×C42—22no—
106C2×C42—22no—
107C2×C42—24no—
108C2×C42—22no—
109C2×C42—24no—
110C2×C42—42no—
111C2×C42—82no—
112C2×C42—82no—
113C2×C42M1614yes—
114C2×C42C2×D414yes—
115C2×C42C2×D414yes—
116C2×C42C2×D414yes—
117C4⋊C8—84no—
118C22⋊C8—216no—
119#97—84no—
120#95—22no—
121#95—22no—
122#95—24no—
123#95—24no—
124#95—24no—
125#95—24no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—22no—
130#95C22⋊C418yes—
131C24C23×C411yescharacteristic
132C2×Q8—216no—
133C2×Q8—84no—
134C2×Q8—84no—
135C2×Q8—88no—
136C2×Q8—88no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4C2×C22⋊C414yes—
176C22×C4C2×C22⋊C412yes—
177C22×C4C2×C22⋊C414yes—
178C22×C4C2×C22⋊C412yes—
179C22×C4C2×M1612yes—
180C22×C4C2×M1612yes—
181C22×C4C2×M1612yes—
182C22×C4C2×M1612yes—
183C22×C4C2×M1614yes—
184C22×C4C22×D412yes—
185C22×C42−1+414yes—
186C2×C8—48no—
187C2×C8—48no—
188C2×C8—48no—
189C4⋊C4—216no—
190C4⋊C4—28no—
191C4⋊C4—28no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—84no—
195C4⋊C4—84no—
196C42—28no—
197C42—28no—
198C42—44no—
199C42—44no—
200C42—44no—
201C42—84no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23C22×C22⋊C411yescharacteristic
207C23#51912yes—
208C23C22×M1611yescharacteristic
209C23C22×M1611yescharacteristic
210C23#56812yes—
211Q8—816no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—44no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—44no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—84no—
256C2×C4—82no—
257C2×C4—84no—
258C2×C4—82no—
259C2×C4#40714yes—
260C2×C4#53614yes—
261C8—88no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—22no—
270C22#231011yescharacteristic
271C22C2×#40711yescharacteristic
272C22#287911yescharacteristic
273C22#288311yescharacteristic
274C22#288411yescharacteristic
275C22#288811yescharacteristic
276C22#288911yescharacteristic
277C22#298312yes—
278C22#298312yes—
279C4—24no—
280C4—24no—
281C4—42no—
282C4—44no—
283C4—44no—
284C4—42no—
285C4—42no—
286C4—44no—
287C4—81no—
288C4—82no—
289C4—81no—
290C2—22no—
291C2—22no—
292C2#1736911yescharacteristic
293C2#1737111yescharacteristic
294C2#1740711yescharacteristic
295C2#1740911yescharacteristic
296C2#2182211yescharacteristic
297C2#2182811yescharacteristic
298C2#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×#23312
#219162
#98822
#98812
#87012
#86992