Finite Group Databasebeta

← #493035

Subgroups — #493035

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2C2×#2126C212yesmaximal
3#20393C212yesmaximal, complemented
4#12627C214yesmaximal, complemented
5C4×#353C214yesmaximal, complemented
6#12583C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C2×#520C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10C2×#410C2214yes—
11#2361—24no—
12#2354—22no—
13D4×C4⋊C4—22no—
14#2326C412yescomplemented
15#2326C2212yes—
16#2311C411yescharacteristic, complemented
17#2311C2211yescharacteristic
18C4×C4⋊Q8C412yescomplemented
19C4×#127C2212yes—
20C2×#384C411yescharacteristic, complemented
21#2274—22no—
22C4×#117C411yescharacteristic, complemented
23#2126C4116yescomplemented
24C2×#354C2214yes—
25C2×#353C2214yes—
26C4×#99—24no—
27C4×#99C2218yes—
28#1740—22no—
29#1393C2218yes—
30C22×#41C2×C411yescharacteristic
31C22×#41C2311yescharacteristic
32#532—24no—
33#532C2×C418yes—
34C2×C4⋊Q8C2×C411yescharacteristic
35C2×#127C2311yescharacteristic
36C2×#124C2×C411yescharacteristic
37C2×#124C2311yescharacteristic
38C2×#121—22no—
39#520—44no—
40#520—44no—
41C2×C4×Q8C2×C412yes—
42C2×C4×Q8C2×C412yes—
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4—22no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D414yes—
48C2×C4×D4C2312yes—
49C2×#117—21no—
50C2×#117—24no—
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117D412yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4D412yes—
59C22×C42C2×C411yescharacteristic
60#410—28no—
61#410C2×C4116yescomplemented
62C2×#99—24no—
63C2×#99C2318yes—
64#391C2×C412yes—
65#391C2×C412yes—
66#390—24no—
67#390C2×C412yes—
68#390C2312yes—
69#389—24no—
70#388—24no—
71#388—24no—
72#386—22no—
73#386—22no—
74#385—24no—
75#385—24no—
76#384—22no—
77#384C2×C412yes—
78#384C2×C412yes—
79#380—22no—
80#380—22no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4—48no—
85C4×C22⋊C4—22no—
86C4×C22⋊C4—28no—
87#376C2×C412yes—
88C43C2×C412yes—
89#354C2×C4116yescomplemented
90#353C2×C4116yescomplemented
91#342—28no—
92C2×#41—22no—
93C2×#41—42no—
94C2×#41—42no—
95C2×#41—42no—
96C2×#41—42no—
97C2×#41C4214yescomplemented
98C2×#41C22×C414yes—
99C2×#41#4114yes—
100C22×Q8C22×C411yescharacteristic
101C22×Q8C22×C411yescharacteristic
102C22×D4—21no—
103C22×D4—22no—
104C22×D4C22×C411yescharacteristic
105C22×D4C2×D412yes—
106C22×D4C2411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C22⋊C412yes—
112C23×C4C22×C411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D414yes—
115C4⋊Q8—22no—
116C4⋊Q8C4214yescomplemented
117#127—22no—
118#127C22×C414yes—
119#124—22no—
120#124—22no—
121#124C4214yescomplemented
122#124C22×C414yes—
123#121—216no—
124C4×Q8—44no—
125C4×Q8—44no—
126C4×D4—216no—
127C4×D4—44no—
128C4×D4—44no—
129C4×D4—48no—
130C4×D4—48no—
131C4×D4—88no—
132#117—24no—
133#117—216no—
134#117—44no—
135#117—44no—
136#117—48no—
137#117—42no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C414yes—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C411yescharacteristic
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C2×D414yes—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—44no—
171C2×C42—48no—
172C2×C42C4214yescomplemented
173C2×C42C22⋊C414yes—
174C2×C42C22×C412yes—
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C412yes—
178#99—216no—
179#99—28no—
180#99C22×C4116yes—
181#95—22no—
182#95—22no—
183#95—24no—
184#95—24no—
185#95—48no—
186C24—21no—
187C24C2×C22⋊C411yescharacteristic
188C24C22×D412yes—
189#41—44no—
190#41—44no—
191#41—48no—
192#41—48no—
193#41—48no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8C2×C4212yes—
200C2×Q8C2×C4212yes—
201C2×Q8#11712yes—
202C2×D4—28no—
203C2×D4—24no—
204C2×D4—21no—
205C2×D4—42no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4C2×C4212yes—
216C2×D4C4×D418yes—
217C2×D4C23×C412yes—
218C2×D4C2×#4112yes—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—84no—
240C22×C4—82no—
241C22×C4—84no—
242C22×C4—82no—
243C22×C4C2×C4212yes—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C414yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4#11712yes—
250C22×C4C4×D418yes—
251C22×C4C23×C411yescharacteristic
252C4⋊C4—28no—
253C4⋊C4—24no—
254C4⋊C4—28no—
255C4⋊C4—22no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—48no—
259C4⋊C4—416no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C4⋊C4#9914yes—
265C22⋊C4—28no—
266C22⋊C4—28no—
267C22⋊C4—24no—
268C22⋊C4—216no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—48no—
272C22⋊C4—42no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C42—21no—
276C42—21no—
277C42—24no—
278C42—24no—
279C42—44no—
280C42—42no—
281C42—44no—
282C42—44no—
283C42—44no—
284C42—416no—
285C42—168no—
286C42C2×C4212yes—
287C42C2×C4212yes—
288C23—21no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—41no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23C4×C22⋊C412yes—
302C23C22×C22⋊C411yescharacteristic
303C23C2×C4×D414yes—
304Q8—42no—
305Q8—42no—
306Q8—42no—
307Q8—42no—
308D4—48no—
309D4—42no—
310D4—42no—
311D4—42no—
312D4—416no—
313D4—42no—
314D4—48no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—28no—
323C2×C4—28no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—48no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—416no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—88no—
346C2×C4—88no—
347C2×C4—88no—
348C2×C4—164no—
349C2×C4—162no—
350C2×C4—162no—
351C2×C4C4×C22⋊C414yes—
352C2×C4C4×C22⋊C412yes—
353C2×C4C2×#9914yes—
354C2×C4C2×#9912yes—
355C2×C4#41012yes—
356C2×C4C22×C4211yescharacteristic
357C2×C4C2×#11711yescharacteristic
358C22—21no—
359C22—21no—
360C22—24no—
361C22—22no—
362C22—42no—
363C22—44no—
364C22—42no—
365C22—44no—
366C22—42no—
367C22—41no—
368C22—42no—
369C22—41no—
370C22—42no—
371C22—82no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22C2×C4×C22⋊C411yescharacteristic
376C22C22×#9911yescharacteristic
377C22C2×#41011yescharacteristic
378C4—24no—
379C4—24no—
380C4—21no—
381C4—24no—
382C4—21no—
383C4—42no—
384C4—44no—
385C4—42no—
386C4—42no—
387C4—82no—
388C4—816no—
389C4—82no—
390C4C4×#9914yes—
391C2—21no—
392C2—21no—
393C2—44no—
394C2—42no—
395C2—82no—
396C2C2×C4×#9911yescharacteristic
397C2#3335911yescharacteristic
398C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×D4C24
prime radicalC2×#532C22
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
3. #20393C2semidirect
4. #12627C2semidirect
5. C4×#353C2semidirect
6. #12583C2semidirect
8. C2×#520C4semidirect
14. #2326C4semidirect
16. #2311C4semidirect
18. C4×C4⋊Q8C4semidirect
20. C2×#384C4semidirect
22. C4×#117C4semidirect
23. #2126C4semidirect
61. #410C2×C4semidirect
89. #354C2×C4semidirect
90. #353C2×C4semidirect
97. C2×#41C42semidirect
116. C4⋊Q8C42semidirect
121. #124C42semidirect
172. C2×C42C42semidirect

Maximal subgroups

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

subgroupindex
#342072
C2×#21262
#203932
#126272
C4×#3532
#125832