Finite Group Databasebeta

← #480319

Subgroups — #480319

Order 512.

705 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60126C211yesmaximal, characteristic
2#33427C212yesmaximal, complemented
3#11524C214yesmaximal, complemented
4#11523C214yesmaximal, complemented
5C2×#1328C212yesmaximal, complemented
6#11438C212yesmaximal
7#3428C411yescharacteristic
8#2925—22no—
9#2925—22no—
10#2924—24no—
11#2923C412yes—
12#2923C2212yes—
13#2922C412yes—
14#2922C2212yes—
15#2905C411yescharacteristic
16C2×#424C2211yescharacteristic
17#2879C2211yescharacteristic
18C2×#408C411yescharacteristic
19#2873C2211yescharacteristic
20C2×#405C411yescharacteristic
21#2166—24no—
22#2165C2218yescomplemented
23#2153—22no—
24C2×#349C2214yes—
25#2110—22no—
26#1328C2214yescomplemented
27#1328C2214yescomplemented
28#1321C2218yes—
29C22×#41C2×C411yescharacteristic
30C22×M16C2×C411yescharacteristic
31C22×M16C2×C411yescharacteristic
32C22×M16C2311yescharacteristic
33C23×C8C2×C411yescharacteristic
34#520—22no—
35#520—22no—
36#519—21no—
37#518—21no—
38#518—22no—
39C4×#41—21no—
40C4×#41—21no—
41C4×#41C2×C412yes—
42C4×#41C2×C412yes—
43C2×#117—22no—
44C2×#117C2×C411yescharacteristic
45C2×#117C2×C411yescharacteristic
46C2×M32—24no—
47#433—24no—
48#433—24no—
49#433—24no—
50#433—24no—
51#432—24no—
52#432—24no—
53#431—24no—
54#431—24no—
55#424—22no—
56#424—22no—
57#424C2×C411yescharacteristic
58#424C2×C411yescharacteristic
59#424C2×C412yes—
60#424C2×C412yes—
61#424C2311yescharacteristic
62#424C2311yescharacteristic
63#423—22no—
64C2×C4⋊C8—22no—
65#408—22no—
66#408—22no—
67#408C2×C412yes—
68#408C2×C412yes—
69#408C2×C412yes—
70#408C2312yes—
71#407—22no—
72C2×C22⋊C8—22no—
73#405—22no—
74#405—21no—
75#405—21no—
76#405D412yes—
77#405D414yes—
78#405D412yes—
79C4×M16C2×C412yes—
80C4×M16C2312yes—
81C2×#97C2×C412yes—
82C2×C4×C8C2×C412yes—
83#364—28no—
84#349—24no—
85#349C2318yes—
86#347—28no—
87C2×#41—21no—
88C2×#41—21no—
89C2×#41—42no—
90C2×#41—42no—
91C2×#41M1612yes—
92C2×#41C22×C412yes—
93C22×Q8M1611yescharacteristic
94C22×D4—21no—
95C22×D4M1611yescharacteristic
96C23×C4—21no—
97C23×C4C22×C411yescharacteristic
98C2×M16—21no—
99C2×M16—21no—
100C2×M16—42no—
101C2×M16—42no—
102C2×M16—42no—
103C2×M16C22⋊C412yes—
104C2×M16C22×C412yes—
105C2×M16C22×C411yescharacteristic
106C2×M16C2×D412yes—
107C2×M16C2411yescharacteristic
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—42no—
111C22×C8C22⋊C412yes—
112C22×C8C22×C411yescharacteristic
113C22×C8C22×C411yescharacteristic
114C22×C8C22×C412yes—
115C22×C8C2×D412yes—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×D4—24no—
121C4×D4—24no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—24no—
127C4×D4—24no—
128#117—24no—
129#117—24no—
130#117—21no—
131#117—21no—
132#117—21no—
133#117—21no—
134#117—22no—
135#117—22no—
136#117C22⋊C412yes—
137#117C22⋊C411yescharacteristic
138#117C22⋊C411yescharacteristic
139#117M1612yes—
140#117C22×C411yescharacteristic
141#117C22×C411yescharacteristic
142C2×C4⋊C4—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4M1612yes—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42C22×C412yes—
152M32—28no—
153M32—44no—
154C2×C16—44no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—24no—
158C4⋊C8—24no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C22⋊C8—24no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—24no—
165#97—22no—
166#97—22no—
167#97C22⋊C414yes—
168#97C2×D414yes—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8C22⋊C414yes—
172C4×C8C2×D414yes—
173C24—21no—
174C24C2×M1611yescharacteristic
175#41—44no—
176#41—44no—
177#41—42no—
178#41—42no—
179#41—42no—
180#41—42no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×Q8C2×M1612yes—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—42no—
188C2×D4—44no—
189C2×D4C2×M1612yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4C2×C22⋊C411yescharacteristic
202C22×C4C2×C22⋊C411yescharacteristic
203C22×C4C2×M1611yescharacteristic
204C22×C4C2×M1611yescharacteristic
205C22×C4C2×M1611yescharacteristic
206C22×C4C23×C411yescharacteristic
207M16—24no—
208M16—41no—
209M16—42no—
210M16—42no—
211M16—41no—
212M16—42no—
213M16—42no—
214C2×C8—24no—
215C2×C8—21no—
216C2×C8—22no—
217C2×C8—21no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8C2×C22⋊C412yes—
225C2×C8C2×C22⋊C414yes—
226C2×C8C22×D412yes—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—21no—
240C22⋊C4—21no—
241C22⋊C4C2×M1612yes—
242C22⋊C4C2×M1612yes—
243C42—24no—
244C42—24no—
245C42—21no—
246C42—21no—
247C42—22no—
248C42—22no—
249C42C2×C22⋊C412yes—
250C42C2×C22⋊C412yes—
251C16—48no—
252C23—22no—
253C23—22no—
254C23—42no—
255C23—81no—
256C23—82no—
257C23#40711yescharacteristic
258C23#40711yescharacteristic
259C23C22×M1611yescharacteristic
260Q8—42no—
261Q8—42no—
262D4—42no—
263D4—44no—
264D4—44no—
265D4—42no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4#40711yescharacteristic
287C2×C4#40711yescharacteristic
288C2×C4#41012yes—
289C2×C4#41312yes—
290C2×C4C22×C22⋊C411yescharacteristic
291C2×C4C22×M1611yescharacteristic
292C8—22no—
293C8—22no—
294C8—24no—
295C8—41no—
296C8—41no—
297C8—42no—
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—44no—
304C22—42no—
305C22—82no—
306C22—162no—
307C22#146412yes—
308C22C2×#40711yescharacteristic
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—42no—
314C4—42no—
315C4—42no—
316C4#146412yes—
317C4C2×#41011yescharacteristic
318C4C2×#41311yescharacteristic
319C2—21no—
320C2—22no—
321C2—41no—
322C2—41no—
323C2—82no—
324C2C2×#146411yescharacteristic
subgroupG/H
centerC4C2×#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC2C2×#1464
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ #117 ◁ #2873 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33427C2semidirect
3. #11524C2semidirect
4. #11523C2semidirect
5. C2×#1328C2semidirect
22. #2165C22semidirect
26. #1328C22semidirect
27. #1328C22semidirect

Maximal subgroups

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

subgroupindex
#601262
#334272
#115242
#115232
C2×#13282
#114382