Finite Group Databasebeta

← #163722

Subgroups — #163722

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35110C211yesmaximal, characteristic
2#22787C211yesmaximal, characteristic
3#22002C211yesmaximal, characteristic
4#7725C211yesmaximal, characteristic, complemented
5#7721C211yesmaximal, characteristic, complemented
6#7711C211yesmaximal, characteristic, complemented
7#7705C211yesmaximal, characteristic, complemented
8#7701C211yesmaximal, characteristic, complemented
9#7701C211yesmaximal, characteristic, complemented
10#7691C211yesmaximal, characteristic, complemented
11#7681C211yesmaximal, characteristic, complemented
12#7650C211yesmaximal, characteristic, complemented
13#7644C211yesmaximal, characteristic, complemented
14#7619C211yesmaximal, characteristic, complemented
15#7611C211yesmaximal, characteristic, complemented
16#2857C2211yescharacteristic
17#2855C2211yescharacteristic
18#2855C2211yescharacteristic
19#2851C2211yescharacteristic
20#2655—21no—
21#2638—21no—
22#2565C411yescharacteristic
23#2562C411yescharacteristic
24#2550C411yescharacteristic
25#2536C411yescharacteristic
26#2530—21no—
27#2527—21no—
28#2520—21no—
29#2496C411yescharacteristic
30#2494C411yescharacteristic
31#2492C411yescharacteristic
32#2489C411yescharacteristic
33#2468—21no—
34#2334—21no—
35#2332—21no—
36C4×#125C2211yescharacteristic
37#2295C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1924—21no—
40#1924—21no—
41#1917—21no—
42#1917—21no—
43#1870—21no—
44#1869—21no—
45#1851—21no—
46#1839C2211yescharacteristic
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1752—21no—
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1745C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#126—21no—
79C2×#125C2×C411yescharacteristic
80C2×#125C2×C411yescharacteristic
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102#401—21no—
103#400—21no—
104#400—21no—
105#400—21no—
106#400—21no—
107#398—21no—
108#398—21no—
109#398—21no—
110#398—21no—
111#397—21no—
112#397—21no—
113#397—21no—
114#395—21no—
115#394—21no—
116#394—21no—
117#393—21no—
118#391—21no—
119#391—21no—
120#391—21no—
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2311yescharacteristic
125#389—21no—
126#389—21no—
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2311yescharacteristic
131#389C2311yescharacteristic
132#389C2311yescharacteristic
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#384C2×C411yescharacteristic
159#382—21no—
160#382C2×C411yescharacteristic
161#382C2×C411yescharacteristic
162#381—21no—
163#381—21no—
164#380—21no—
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169#380C2×C411yescharacteristic
170#380C2×C411yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186#376—21no—
187C43C2311yescharacteristic
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8—21no—
207C22×Q8C22×C411yescharacteristic
208C22×D4C22⋊C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C2×M16—42no—
216C2×M16—81no—
217C2×M16—81no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—81no—
229C4⋊Q8—81no—
230#126—44no—
231#125—42no—
232#125—81no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—81no—
236C22⋊Q8—81no—
237C4×Q8—42no—
238C4×D4—42no—
239C4×D4—41no—
240C4×D4—41no—
241C4×D4—42no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—81no—
287C2×C42—81no—
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2411yescharacteristic
301C4⋊C8—81no—
302C4⋊C8—81no—
303C4⋊C8—81no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—42no—
307C22⋊C8—81no—
308C22⋊C8—81no—
309C22⋊C8—81no—
310#97—42no—
311#97—81no—
312#97—81no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346C24C2×C22⋊C411yescharacteristic
347C24C2×C22⋊C411yescharacteristic
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400M16—84no—
401C2×C8—22no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—84no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C22⋊C4—22no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—42no—
485C42—42no—
486C42—81no—
487C42—81no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23#42112yes—
502C23C22×C22⋊C411yescharacteristic
503Q8—82no—
504Q8—162no—
505D4—44no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4#56812yes—
577C2×C4#57511yescharacteristic
578C2×C4#57511yescharacteristic
579C2×C4#57711yescharacteristic
580C2×C4#57811yescharacteristic
581C8—42no—
582C8—81no—
583C8—81no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22#287911yescharacteristic
604C22#288311yescharacteristic
605C22#288911yescharacteristic
606C22#289611yescharacteristic
607C22#289811yescharacteristic
608C22C2×#42111yescharacteristic
609C22#290111yescharacteristic
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—42no—
634C2—81no—
635C2#1740711yescharacteristic
636C2#1749411yescharacteristic
637C2#1750611yescharacteristic
638C2#1759811yescharacteristic
639C2#1762311yescharacteristic
640C2#1768011yescharacteristic
641C2#1776711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7725C2semidirect
5. #7721C2semidirect
6. #7711C2semidirect
7. #7705C2semidirect
8. #7701C2semidirect
9. #7701C2semidirect
10. #7691C2semidirect
11. #7681C2semidirect
12. #7650C2semidirect
13. #7644C2semidirect
14. #7619C2semidirect
15. #7611C2semidirect

Maximal subgroups

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

subgroupindex
#351102
#227872
#220022
#77252
#77212
#77112
#77052
#77012
#76912
#76812
#76502
#76442
#76192
#76112