Finite Group Databasebeta

← #493713

Subgroups — #493713

Order 512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2C2×#2126C212yesmaximal
3#20634C212yesmaximal, complemented
4#12800C214yesmaximal, complemented
5#12797C214yesmaximal, 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#2316—22no—
17#2311C411yescharacteristic, complemented
18#2311C2211yescharacteristic
19C4×C4⋊Q8C412yescomplemented
20C4×#127C2212yes—
21C2×#384C411yescharacteristic, complemented
22C4×#117C411yescharacteristic, complemented
23#2126—28no—
24C2×#354C2214yes—
25C2×#353C2214yes—
26#1799—24no—
27#1799C2218yes—
28#1740—22no—
29#1393C2218yes—
30C22×#41C2×C411yescharacteristic
31C22×#41C2311yescharacteristic
32#532—24no—
33#532D414yes—
34#532Q814yes—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×#127C2311yescharacteristic
37C2×#124C2×C411yescharacteristic
38C2×#124C2311yescharacteristic
39C2×#121—22no—
40#520—44no—
41#520—44no—
42C2×C4×Q8C2×C412yes—
43C2×C4×Q8C2×C412yes—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4—22no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D414yes—
49C2×C4×D4C2312yes—
50C2×#117—21no—
51C2×#117—24no—
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C411yescharacteristic
54C2×#117D412yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C42C2×C411yescharacteristic
61#410—28no—
62#410—28no—
63C2×#99—24no—
64C2×#99C2318yes—
65#391C2×C412yes—
66#391C2×C412yes—
67#390—24no—
68#390C2×C412yes—
69#390C2312yes—
70#389—24no—
71#389—48no—
72#388—24no—
73#388—24no—
74#386—22no—
75#386—22no—
76#385—24no—
77#385—24no—
78#384—22no—
79#384C2×C412yes—
80#384C2×C412yes—
81#380—22no—
82#380—22no—
83#380—28no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—24no—
87C4×C22⋊C4—22no—
88#376C2×C412yes—
89C43C2×C412yes—
90#354—28no—
91#353—28no—
92#342—28no—
93C2×#41—22no—
94C2×#41—42no—
95C2×#41—42no—
96C2×#41—42no—
97C2×#41—42no—
98C2×#41C4⋊C414yescomplemented
99C2×#41C2×D412yes—
100C2×#41C2×Q812yes—
101C2×#41#4114yes—
102C22×Q8C22×C411yescharacteristic
103C22×Q8C22×C411yescharacteristic
104C22×D4—21no—
105C22×D4—22no—
106C22×D4C22×C411yescharacteristic
107C22×D4C2×D412yes—
108C22×D4C2411yescharacteristic
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4C22⋊C412yes—
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D414yes—
117C4⋊Q8—22no—
118C4⋊Q8C4⋊C414yescomplemented
119#127—22no—
120#127C2×D412yes—
121#127C2×Q812yes—
122#124—22no—
123#124—22no—
124#124C4⋊C414yescomplemented
125#124C2×D412yes—
126#124C2×Q812yes—
127#121—28no—
128#121—28no—
129C4×Q8—44no—
130C4×Q8—44no—
131C4×D4—28no—
132C4×D4—28no—
133C4×D4—44no—
134C4×D4—44no—
135C4×D4—48no—
136C4×D4—48no—
137C4×D4—88no—
138#117—24no—
139#117—44no—
140#117—44no—
141#117—42no—
142#117—48no—
143#117—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—84no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22⋊C411yescharacteristic
160C2×C4⋊C4C22⋊C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4C22⋊C412yes—
171C2×C22⋊C4C2×D414yes—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—44no—
178C2×C42C22⋊C414yes—
179C2×C42C4⋊C414yescomplemented
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C412yes—
184#99—28no—
185#99—28no—
186#99—48no—
187#95—22no—
188#95—22no—
189#95—24no—
190#95—24no—
191#95—48no—
192C24—21no—
193C24C2×C22⋊C411yescharacteristic
194C24C22×D412yes—
195#41—44no—
196#41—44no—
197#41—48no—
198#41—48no—
199#41—48no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8C2×C4⋊C412yes—
206C2×Q8C2×C4⋊C412yes—
207C2×Q8#11712yes—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—21no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—42no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—44no—
219C2×D4—44no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4C2×C4⋊C412yes—
223C2×D4#12114yes—
224C2×D4C22⋊Q814yes—
225C2×D4C22×D411yescharacteristic
226C2×D4C22×Q811yescharacteristic
227C2×D4C2×#4112yes—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—48no—
248C22×C4—84no—
249C22×C4—82no—
250C22×C4—84no—
251C22×C4—82no—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C4⋊C412yes—
258C22×C4#11712yes—
259C22×C4#12114yes—
260C22×C4C22⋊Q814yes—
261C22×C4C23×C411yescharacteristic
262C4⋊C4—28no—
263C4⋊C4—24no—
264C4⋊C4—28no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—88no—
273C4⋊C4—84no—
274C4⋊C4—168no—
275C4⋊C4#9914yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—48no—
284C22⋊C4—48no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C42—21no—
288C42—21no—
289C42—24no—
290C42—24no—
291C42—44no—
292C42—42no—
293C42—44no—
294C42—44no—
295C42—44no—
296C42—88no—
297C42C2×C4⋊C412yes—
298C42C2×C4⋊C412yes—
299C23—21no—
300C23—22no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—41no—
305C23—41no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—41no—
311C23—41no—
312C23#38012yes—
313C23C22×C22⋊C411yescharacteristic
314C23C2×#12112yes—
315C23C2×C22⋊Q812yes—
316Q8—42no—
317Q8—42no—
318Q8—42no—
319Q8—42no—
320D4—44no—
321D4—44no—
322D4—41no—
323D4—41no—
324D4—42no—
325D4—42no—
326D4—48no—
327D4—48no—
328D4—42no—
329D4—48no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—28no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—48no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—48no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—88no—
360C2×C4—88no—
361C2×C4—88no—
362C2×C4—88no—
363C2×C4—164no—
364C2×C4—162no—
365C2×C4—162no—
366C2×C4#38014yes—
367C2×C4#38012yes—
368C2×C4C2×#9914yes—
369C2×C4C2×#9912yes—
370C2×C4#41012yes—
371C2×C4C22×C4⋊C411yescharacteristic
372C2×C4C2×#11711yescharacteristic
373C22—21no—
374C22—21no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—44no—
383C22—42no—
384C22—41no—
385C22—42no—
386C22—41no—
387C22—42no—
388C22—82no—
389C22—82no—
390C22—81no—
391C22—81no—
392C22C2×#38011yescharacteristic
393C22C22×#9911yescharacteristic
394C22C2×#41011yescharacteristic
395C4—24no—
396C4—24no—
397C4—21no—
398C4—21no—
399C4—24no—
400C4—42no—
401C4—44no—
402C4—42no—
403C4—42no—
404C4—82no—
405C4—82no—
406C4—168no—
407C4#179914yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—42no—
412C2—42no—
413C2—82no—
414C2C2×#179911yescharacteristic
415C2#3335911yescharacteristic
416C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC2×#532C22
LCSG ▷ C22×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. #20634C2semidirect
4. #12800C2semidirect
5. #12797C2semidirect
6. #12583C2semidirect
8. C2×#520C4semidirect
14. #2326C4semidirect
17. #2311C4semidirect
19. C4×C4⋊Q8C4semidirect
21. C2×#384C4semidirect
22. C4×#117C4semidirect
98. C2×#41C4⋊C4semidirect
118. C4⋊Q8C4⋊C4semidirect
124. #124C4⋊C4semidirect
179. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342072
C2×#21262
#206342
#128002
#127972
#125832