Finite Group Databasebeta

← #493012

Subgroups — #493012

Order 512.

1275 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2C2×#2129C212yesmaximal
3#20393C212yesmaximal
4#12627C214yesmaximal, complemented
5C4×#354C214yesmaximal, complemented
6#12581C212yesmaximal
7C2×#536C2211yescharacteristic
8C2×#520C2211yescharacteristic
9C2×#519C411yescharacteristic, complemented
10C2×#410C2214yes—
11#2362—22no—
12#2360—22no—
13#2359—24no—
14#2327C412yescomplemented
15#2326C2212yes—
16#2317C411yescharacteristic, complemented
17C4×C4⋊Q8C412yescomplemented
18C4×#124C2212yes—
19#2297C2211yescharacteristic
20C2×#391C411yescharacteristic, complemented
21#2274—22no—
22C4×#117C411yescharacteristic, complemented
23#2129C4116yescomplemented
24C2×#354C2218yes—
25C4×#99—24no—
26C4×#99C2218yes—
27#1740—22no—
28#1393C2214yes—
29#1393C2214yes—
30C22×#41C2311yescharacteristic
31C23×Q8C2×C411yescharacteristic
32#536—24no—
33#536C2×C418yes—
34C2×C4⋊Q8C2×C412yes—
35C2×#124C2312yes—
36C2×#123—22no—
37C2×C22⋊Q8—22no—
38#520—44no—
39#519—44no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×D4D414yes—
46C2×C4×D4C2312yes—
47C2×#117—22no—
48C2×#117—24no—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C42C2×C411yescharacteristic
56#410—28no—
57#410C2×C4116yes—
58C2×#99—24no—
59C2×#99C2318yes—
60#391—22no—
61#391C2×C411yescharacteristic
62#391C2×C412yes—
63#391C2×C412yes—
64#391C2×C411yescharacteristic
65#390C2311yescharacteristic
66#390C2311yescharacteristic
67#389—24no—
68#389—22no—
69#388—24no—
70#388—24no—
71#387—24no—
72#387—24no—
73#386—22no—
74#385—22no—
75#385—24no—
76#384C2×C412yes—
77#382C2×C412yes—
78C4×C4⋊C4—21no—
79C4×C4⋊C4—21no—
80C4×C4⋊C4—24no—
81C4×C4⋊C4—48no—
82C4×C22⋊C4—24no—
83C4×C22⋊C4—22no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—28no—
86#376C2×C412yes—
87C2×#95—22no—
88C43C2×C412yes—
89#354C2×C4132yescomplemented
90#342—28no—
91C2×#41—42no—
92C2×#41—42no—
93C2×#41C22×C414yes—
94C2×#41#4114yes—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8C4214yescomplemented
100C22×Q8C22×C412yes—
101C22×Q8C22×C411yescharacteristic
102C22×D4C2×D412yes—
103C22×D4C2411yescharacteristic
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C22⋊C412yes—
107C23×C4C22×C411yescharacteristic
108C23×C4C2×D412yes—
109C23×C4C2×D414yes—
110C4⋊Q8—24no—
111C4⋊Q8C4218yescomplemented
112#124—24no—
113#124C22×C418yes—
114#123—216no—
115C22⋊Q8—216no—
116C4×Q8—44no—
117C4×Q8—44no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—44no—
121C4×D4—44no—
122C4×D4—48no—
123#117—22no—
124#117—24no—
125#117—216no—
126#117—48no—
127#117—44no—
128#117—48no—
129#117—42no—
130#117C4214yescomplemented
131C2×C4⋊C4—21no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4C22⋊C412yes—
140C2×C4⋊C4C22⋊C414yes—
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22×C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C412yes—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4C22⋊C412yes—
153C2×C22⋊C4C2×D414yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—21no—
157C2×C42—22no—
158C2×C42—21no—
159C2×C42—22no—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42—48no—
163C2×C42C22⋊C414yes—
164C2×C42C22×C412yes—
165C2×C42C22×C412yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168#99—216no—
169#99—28no—
170#99C22×C4116yes—
171#95—24no—
172#95—24no—
173#95—22no—
174#95—22no—
175#95—22no—
176#95—24no—
177#95—22no—
178#95—48no—
179C24C2×C22⋊C411yescharacteristic
180C24C22×D412yes—
181#41—48no—
182#41—44no—
183#41—44no—
184C2×Q8—22no—
185C2×Q8—24no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—48no—
189C2×Q8—48no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8C2×C4214yes—
195C2×Q8C2×C4212yes—
196C2×Q8#11712yes—
197C2×D4—42no—
198C2×D4—41no—
199C2×D4—42no—
200C2×D4—41no—
201C2×D4C4×D418yes—
202C2×D4C23×C412yes—
203C2×D4C2×#4112yes—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—48no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—84no—
224C22×C4—82no—
225C22×C4C2×C4212yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4#11712yes—
232C22×C4C4×D418yes—
233C22×C4C23×C411yescharacteristic
234C4⋊C4—24no—
235C4⋊C4—216no—
236C4⋊C4—24no—
237C4⋊C4—28no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C4⋊C4—416no—
244C4⋊C4—44no—
245C4⋊C4—84no—
246C4⋊C4#9914yes—
247C22⋊C4—216no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—216no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—48no—
254C22⋊C4—42no—
255C22⋊C4—82no—
256C42—22no—
257C42—24no—
258C42—44no—
259C42—41no—
260C42—41no—
261C42—44no—
262C42—44no—
263C42—416no—
264C42—44no—
265C42—42no—
266C42—42no—
267C42—82no—
268C42—168no—
269C42C2×C4214yes—
270C23—21no—
271C23—22no—
272C23—21no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—42no—
278C23C4×C22⋊C412yes—
279C23C22×C22⋊C411yescharacteristic
280C23C2×C4×D414yes—
281Q8—42no—
282Q8—42no—
283Q8—44no—
284Q8—44no—
285Q8—48no—
286D4—48no—
287D4—42no—
288D4—42no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—28no—
294C2×C4—28no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—48no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—416no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—88no—
321C2×C4—84no—
322C2×C4—88no—
323C2×C4—88no—
324C2×C4—162no—
325C2×C4C4×C22⋊C414yes—
326C2×C4C4×C22⋊C412yes—
327C2×C4C2×#9914yes—
328C2×C4C2×#9912yes—
329C2×C4#41012yes—
330C2×C4C22×C4211yescharacteristic
331C2×C4C2×#11711yescharacteristic
332C22—21no—
333C22—21no—
334C22—24no—
335C22—22no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—82no—
344C22C2×C4×C22⋊C411yescharacteristic
345C22C22×#9911yescharacteristic
346C22C2×#41011yescharacteristic
347C4—24no—
348C4—21no—
349C4—24no—
350C4—21no—
351C4—44no—
352C4—42no—
353C4—42no—
354C4—44no—
355C4—44no—
356C4—81no—
357C4—816no—
358C4—81no—
359C4—81no—
360C4C4×#9914yes—
361C2—21no—
362C2—21no—
363C2—44no—
364C2—42no—
365C2C2×C4×#9911yescharacteristic
366C2#3336011yescharacteristic
367C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12627C2semidirect
5. C4×#354C2semidirect
9. C2×#519C4semidirect
14. #2327C4semidirect
16. #2317C4semidirect
17. C4×C4⋊Q8C4semidirect
20. C2×#391C4semidirect
22. C4×#117C4semidirect
23. #2129C4semidirect
89. #354C2×C4semidirect
99. C22×Q8C42semidirect
111. C4⋊Q8C42semidirect
130. #117C42semidirect

Maximal subgroups

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

subgroupindex
#342762
C2×#21292
#203932
#126272
C4×#3542
#125812