Finite Group Databasebeta

← #482003

Subgroups — #482003

Order 512.

868 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36011C211yesmaximal, characteristic
2#20725C211yesmaximal, characteristic, complemented
3#17372C211yesmaximal, characteristic, complemented
4#11824C212yesmaximal, complemented
5#11822C212yesmaximal, complemented
6#11813C212yesmaximal, complemented
7#11798C212yesmaximal, complemented
8#11729C212yesmaximal, complemented
9#11727C212yesmaximal, complemented
10C2×#555C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12#2983—21no—
13#2635C2211yescharacteristic
14#2635C2211yescharacteristic
15#2628C2211yescharacteristic
16#2591C411yescharacteristic, complemented
17#2591C411yescharacteristic, complemented
18#2568C411yescharacteristic, complemented
19#2536C411yescharacteristic, complemented
20#2484C411yescharacteristic, complemented
21#2484C411yescharacteristic, complemented
22#2474—21no—
23#2474—21no—
24#2459—21no—
25#2459—21no—
26#2420—21no—
27#2407—21no—
28#2395C2211yescharacteristic
29#2395C2211yescharacteristic
30#2392C2211yescharacteristic
31#2362—21no—
32#2357C411yescharacteristic, complemented
33#2352—21no—
34#2313—21no—
35#1820C2212yes—
36#1799—21no—
37#1799C2212yes—
38#1782—21no—
39C4×#99—21no—
40C4×#99C2212yes—
41#1466—21no—
42#1466C2212yes—
43#1464—21no—
44#1464C2212yes—
45#1460C2212yes—
46#1458—21no—
47#1352C2214yes—
48#1352C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41D411yescharacteristic
52#555—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124—21no—
56C2×C22⋊Q8—21no—
57C2×#121—21no—
58C2×#121C2×C411yescharacteristic
59C2×#121C2×C411yescharacteristic
60#520—44no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80#446—44no—
81#432—22no—
82#432—22no—
83#431—22no—
84C2×#99—22no—
85C2×#99—22no—
86#400C2×C411yescharacteristic
87#400C2×C411yescharacteristic
88#400C2×C411yescharacteristic
89#400C2×C411yescharacteristic
90#400C2×C411yescharacteristic
91#400C2×C411yescharacteristic
92#399C2311yescharacteristic
93#399C2311yescharacteristic
94#398C2×C411yescharacteristic
95#398C2×C411yescharacteristic
96#398C2×C411yescharacteristic
97#397—21no—
98#397—21no—
99#397C2311yescharacteristic
100#396—21no—
101#395—21no—
102#394—21no—
103#394—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#394C2×C411yescharacteristic
108#394C2×C411yescharacteristic
109#394C2×C411yescharacteristic
110#393—21no—
111#392—21no—
112#392—21no—
113#392C2311yescharacteristic
114#391—21no—
115#391—21no—
116#391—21no—
117#390—21no—
118#389—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—44no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#385—21no—
141#385—21no—
142#382—21no—
143#382C2×C411yescharacteristic
144#381—21no—
145#381—21no—
146#381—22no—
147#381C2×C411yescharacteristic
148#381C2×C411yescharacteristic
149#380—21no—
150#380—21no—
151#380—22no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4C2×C411yescharacteristic
161C4×C22⋊C4C2×C411yescharacteristic
162#342—22no—
163#342C2314yes—
164#324—22no—
165#323—22no—
166#323—22no—
167#323C2314yes—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41—42no—
171C22×Q8—21no—
172C22×Q8C22⋊C411yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22×C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C23×C4—21no—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C4⋊Q8—42no—
191#124—44no—
192#124—44no—
193C22⋊Q8—44no—
194#121—44no—
195#121—42no—
196#121—42no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×D4—24no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—44no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—44no—
211#117—42no—
212#117—42no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4C22⋊C411yescharacteristic
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2411yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—42no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279#99—44no—
280#99—44no—
281C22⋊C8—24no—
282#97—42no—
283#97—42no—
284C4×C8—42no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—42no—
305#95—42no—
306#95—44no—
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310#95C22×C411yescharacteristic
311C24—21no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C22×D411yescharacteristic
315#41—416no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—81no—
321C2×Q8—81no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—81no—
340C2×D4—81no—
341C2×D4—81no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C2×C8—44no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—162no—
408C4⋊C4#9914yes—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—81no—
446C42—81no—
447C42—162no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C22×C22⋊C411yescharacteristic
470C23#57512yes—
471Q8—44no—
472Q8—162no—
473D4—44no—
474D4—44no—
475D4—44no—
476D4—82no—
477D4—84no—
478D4—162no—
479D4—162no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—161no—
537C2×C4—161no—
538C2×C4C2×#9912yes—
539C2×C4C2×#9912yes—
540C2×C4C2×#9912yes—
541C2×C4#42112yes—
542C8—84no—
543C22—21no—
544C22—21no—
545C22—22no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22C22×#9911yescharacteristic
568C22#289611yescharacteristic
569C22C2×#42111yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—164no—
590C2—21no—
591C2—21no—
592C2—42no—
593C2—81no—
594C2—81no—
595C2—161no—
596C2#1755911yescharacteristic
597C2#3335011yescharacteristic
598C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36011C2
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. #20725C2semidirect
3. #17372C2semidirect
4. #11824C2semidirect
5. #11822C2semidirect
6. #11813C2semidirect
7. #11798C2semidirect
8. #11729C2semidirect
9. #11727C2semidirect
10. C2×#555C4semidirect
16. #2591C4semidirect
17. #2591C4semidirect
18. #2568C4semidirect
19. #2536C4semidirect
20. #2484C4semidirect
21. #2484C4semidirect
32. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#360112
#207252
#173722
#118242
#118222
#118132
#117982
#117292
#117272