Finite Group Databasebeta

← #482098

Subgroups — #482098

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35728C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17353C211yesmaximal, characteristic, complemented
4#11876C214yesmaximal, complemented
5#11873C214yesmaximal, complemented
6#11693C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#546—21no—
9C2×D42C411yescharacteristic, complemented
10C2×#520C2211yescharacteristic
11#2982—21no—
12C2×#410—21no—
13#2619C2211yescharacteristic
14#2618C412yescomplemented
15#2548C411yescharacteristic, complemented
16#2543C2212yes—
17#2536C411yescharacteristic, complemented
18#2426—21no—
19#2424—22no—
20#2416—21no—
21#2411—22no—
22#2408C2212yes—
23#2392C2211yescharacteristic
24C2×#394C412yescomplemented
25#2349C411yescharacteristic, complemented
26D4×C22⋊C4—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yescomplemented
30#1740C2212yescomplemented
31#1460—21no—
32#1460C2214yescomplemented
33#1458—22no—
34#1458C2212yescomplemented
35#1344C2214yescomplemented
36#1343C2214yes—
37#1343C2218yescomplemented
38C22×#41D411yescharacteristic
39C23×D4—21no—
40C23×D4C2×C412yes—
41#546—44no—
42#546—44no—
43D42—44no—
44C2×#127C2×C411yescharacteristic
45C2×#124—21no—
46C2×#123—22no—
47C2×C22⋊Q8—21no—
48C2×#121—22no—
49C2×#121—21no—
50C2×#121—22no—
51C2×#121—21no—
52C2×#121C2×C412yes—
53C2×#120—22no—
54C2×#120—22no—
55#520—44no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2312yes—
65C2×#117—22no—
66C2×#117D412yes—
67C2×#117C2311yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C8×Q8—44no—
73#431—24no—
74#431—22no—
75#410—44no—
76C2×#99—24no—
77#398C2×C411yescharacteristic
78#398C2×C412yes—
79#397C2311yescharacteristic
80#396—22no—
81#396—22no—
82#396—21no—
83#394—22no—
84#394—22no—
85#394—21no—
86#394—21no—
87#394C2×C412yes—
88#394C2×C412yes—
89#394C2×C411yescharacteristic
90#394C2×C412yes—
91#394C2×C412yes—
92#392—21no—
93#392C2312yes—
94#392C2311yescharacteristic
95#390—21no—
96#390—22no—
97#390—21no—
98#390—22no—
99#389—21no—
100#389—44no—
101#388—22no—
102#388—22no—
103#386—22no—
104#386—22no—
105#386—22no—
106#385—22no—
107#382C2×C411yescharacteristic
108#380—21no—
109#380—24no—
110#380C2×C412yes—
111#379—22no—
112C4×C4⋊C4C2×C411yescharacteristic
113C4×C22⋊C4—21no—
114C2×#95C2×C412yes—
115#342—22no—
116#342C2314yes—
117#324—22no—
118#323—24no—
119#323C2314yes—
120C25—22no—
121C2×#41—42no—
122C2×#41—42no—
123C2×#41D814yes—
124C2×#41SD1614yes—
125C22×Q8C22⋊C411yescharacteristic
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—84no—
137C22×D4C22⋊C411yescharacteristic
138C22×D4C22×C411yescharacteristic
139C22×D4C22×C412yes—
140C22×D4C2×D412yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148#127—42no—
149#124—44no—
150#123—48no—
151C22⋊Q8—44no—
152#121—44no—
153#121—44no—
154#121—42no—
155#121—42no—
156#121—44no—
157#121—44no—
158#121—84no—
159#120—48no—
160#120—44no—
161#120—44no—
162C4×Q8—24no—
163C4×Q8—42no—
164C4×D4—24no—
165C4×D4—42no—
166C4×D4—44no—
167C4×D4—42no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—42no—
171C4×D4—42no—
172#117—42no—
173#117—44no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4C22⋊C412yes—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—48no—
211C2×C22⋊C4—84no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C2×D411yescharacteristic
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2411yescharacteristic
216C2×C42—21no—
217C2×C42—42no—
218C2×C42C22⋊C412yes—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C4⋊C8—44no—
223C4⋊C8—42no—
224#99—48no—
225C22⋊C8—24no—
226C4×C8—44no—
227C4×C8—42no—
228#95—22no—
229#95—21no—
230#95—22no—
231#95—21no—
232#95—21no—
233#95—44no—
234#95C22×C411yescharacteristic
235#95C22×C412yes—
236#95C22×C411yescharacteristic
237C24—21no—
238C24—22no—
239C24—44no—
240C24—44no—
241C24—44no—
242C24—44no—
243C24—82no—
244C24—82no—
245C24—82no—
246C24C2×C22⋊C412yes—
247C24C22×D411yescharacteristic
248#41—48no—
249#41—48no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8#10214yes—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—44no—
262C2×D4—42no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—44no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—81no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—816no—
281C2×D4—88no—
282C2×D4—88no—
283C2×D4#10214yes—
284C2×D4C2×D814yes—
285C2×D4C2×SD1614yes—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—44no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4#10218yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×D812yes—
316C22×C4C2×SD1612yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C2×C8—44no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—84no—
331C4⋊C4—81no—
332C4⋊C4—82no—
333C4⋊C4—164no—
334C4⋊C4#9914yes—
335C22⋊C4—24no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—42no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—81no—
354C22⋊C4—82no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—82no—
358C42—24no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—42no—
363C42—84no—
364C23—22no—
365C23—21no—
366C23—22no—
367C23—21no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—88no—
387C23—88no—
388C23—88no—
389C23—88no—
390C23—164no—
391C23—164no—
392C23—164no—
393C23C2×#10214yes—
394C23C22×C22⋊C411yescharacteristic
395C23C22×D811yescharacteristic
396C23C22×SD1611yescharacteristic
397Q8—28no—
398D4—28no—
399D4—44no—
400D4—44no—
401D4—42no—
402D4—42no—
403D4—88no—
404D4—84no—
405D4—88no—
406D4—82no—
407D4—82no—
408D4—84no—
409D4—162no—
410D4—162no—
411D4—162no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—162no—
449C2×C4—164no—
450C2×C4C2×#9914yes—
451C2×C4C2×#9912yes—
452C2×C4C2×#10212yes—
453C2×C4C2×#10214yes—
454C2×C4C2×#10212yes—
455C2×C4#42112yes—
456C8—84no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—24no—
462C22—44no—
463C22—41no—
464C22—41no—
465C22—44no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—42no—
470C22—41no—
471C22—42no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—81no—
481C22—81no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—1616no—
486C22C22×#9911yescharacteristic
487C22C22×#10211yescharacteristic
488C22C2×#42111yescharacteristic
489C4—24no—
490C4—22no—
491C4—24no—
492C4—41no—
493C4—42no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—81no—
498C4—82no—
499C4—81no—
500C4—81no—
501C4—164no—
502C4#151914yes—
503C2—21no—
504C2—21no—
505C2—41no—
506C2—44no—
507C2—41no—
508C2—82no—
509C2—162no—
510C2C2×#151911yescharacteristic
511C2#3334811yescharacteristic
512C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35728C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17353C2semidirect
4. #11876C2semidirect
5. #11873C2semidirect
6. #11693C2semidirect
7. #11691C2semidirect
9. C2×D42C4semidirect
14. #2618C4semidirect
15. #2548C4semidirect
17. #2536C4semidirect
24. C2×#394C4semidirect
25. #2349C4semidirect
29. #1799C22semidirect
30. #1740C22semidirect
32. #1460C22semidirect
34. #1458C22semidirect
35. #1344C22semidirect
37. #1343C22semidirect

Maximal subgroups

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

subgroupindex
#357282
#206342
#173532
#118762
#118732
#116932
#116912