Finite Group Databasebeta

← #6355045

Subgroups — #6355045

Order 512.

1676 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60962C211yesmaximal, characteristic
2C2×#3226C211yesmaximal, characteristic, complemented
3#60800C211yesmaximal, characteristic, complemented
4#60237C211yesmaximal, characteristic, complemented
5#19507C214yesmaximal, complemented
6#19505C214yesmaximal, complemented
7#19459C214yesmaximal, complemented
8#19457C214yesmaximal, complemented
9#19235C214yesmaximal, complemented
10#19231C214yesmaximal, complemented
11#17810C211yesmaximal, characteristic, complemented
12C2×#1556C211yesmaximal, characteristic, complemented
13#17713C211yesmaximal, characteristic, complemented
14#3444C2211yescharacteristic
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17#3246C2212yes—
18#3245C2212yes—
19#3239—21no—
20#3235C2212yes—
21#3226—22no—
22#3226—22no—
23#3226C2214yes—
24#3226C2214yes—
25#3218—22no—
26#3218—22no—
27#3217—21no—
28#3214C2212yes—
29#3209C2212yes—
30#3209C2212yescomplemented
31#3206C2212yes—
32#3206C2212yescomplemented
33#3188—22no—
34#3180—22no—
35C2×#500—21no—
36#3152C2212yescomplemented
37#3139C2212yescomplemented
38C2×#486—21no—
39#3115—22no—
40#3110C2212yes—
41#3109C2212yes—
42C2×#480—22no—
43C2×#474C2214yes—
44#3039—22no—
45#3004—22no—
46#2981C2214yes—
47#2980—22no—
48#2974C2214yes—
49#2973—22no—
50#2935C2214yes—
51#2932—22no—
52#2920C2211yescharacteristic
53C2×#432C2211yescharacteristic
54C2×#432C2211yescharacteristic
55C2×#431C2211yescharacteristic
56#2880—22no—
57#1703C2218yescomplemented
58#1699C2218yescomplemented
59#1689C2218yescomplemented
60#1685C2218yescomplemented
61#1683C2218yescomplemented
62#1679C2218yescomplemented
63#1669C2218yescomplemented
64#1665C2218yescomplemented
65#1573C2214yes—
66#1572C2214yescomplemented
67#1565C2214yescomplemented
68#1556C2214yes—
69#1556C2214yescomplemented
70#1554C2212yescomplemented
71#1553C2212yescomplemented
72#1552C2212yes—
73#1551C2212yes—
74#1542C2212yescomplemented
75#1540C2212yescomplemented
76C2×#332C2214yescomplemented
77C22×#41—21no—
78C23×Q8—21no—
79C2×#137—24no—
80C2×#131—24no—
81#533C2311yescharacteristic
82#533C2311yescharacteristic
83#533C2311yescharacteristic
84#532—21no—
85#532—21no—
86#532—21no—
87#532—21no—
88#532D412yes—
89#532D412yes—
90#532C2311yescharacteristic
91#532C2311yescharacteristic
92#532C2311yescharacteristic
93#532C2312yes—
94#532C2312yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×#127D411yescharacteristic
109#529—21no—
110#529—21no—
111#529—21no—
112#529—21no—
113#529—21no—
114#529—21no—
115C2×#124—21no—
116#525—21no—
117#525—21no—
118#525—21no—
119C2×#121—21no—
120C2×#121D412yes—
121C4×#41—21no—
122C4×#41—21no—
123C4×#41—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C42C2311yescharacteristic
129#501—22no—
130#500—24no—
131#500—24no—
132#497—24no—
133#496—22no—
134#494C2314yes—
135#492C2314yes—
136#491—24no—
137#490—22no—
138#489—22no—
139#487C2314yes—
140#487C2314yes—
141#486—24no—
142#486—24no—
143#484—22no—
144#482—24no—
145#481—22no—
146#480—28no—
147#480—28no—
148#476C2314yes—
149#475—22no—
150#474—22no—
151#474—24no—
152#474—24no—
153#474C2314yes—
154#474C2318yes—
155#474C2318yes—
156#466—24no—
157#465—24no—
158#462—24no—
159#462—24no—
160#461—24no—
161#461—24no—
162#461C2314yes—
163#460—22no—
164#459—22no—
165#459C2314yes—
166#451—24no—
167#451—24no—
168#450—22no—
169#449—24no—
170#448—24no—
171#448—24no—
172#447—22no—
173#442—24no—
174#440—24no—
175#436C2318yes—
176#435—28no—
177#435—28no—
178#435—28no—
179C8×D4C2318yes—
180#432—21no—
181#432—21no—
182#432—21no—
183#432—21no—
184#432D412yes—
185#432D412yes—
186#432C2312yes—
187#432C2312yes—
188#432C2312yes—
189#432C2312yes—
190#431—21no—
191#431—21no—
192#431C2312yes—
193#431C2312yes—
194#431C2312yes—
195#431C2312yes—
196C2×#107—24no—
197#423D412yes—
198C2×C4⋊C8C2312yes—
199C2×C4⋊C8C2312yes—
200C2×C4⋊C8C2312yes—
201#419—24no—
202C2×#102—24no—
203#408—24no—
204#408—28no—
205#407—24no—
206#407D412yes—
207C2×C22⋊C8—21no—
208C2×C22⋊C8—21no—
209C2×C22⋊C8—21no—
210C2×C22⋊C8—24no—
211C4×M16C2312yes—
212C2×#97—21no—
213C2×#97—21no—
214C2×C4×C8—21no—
215#335C2318yes—
216#334C2318yes—
217#333C2314yes—
218#332C2318yes—
219#332C2318yes—
220#332C2314yes—
221#332C2314yes—
222#331C2314yes—
223C2×#41—22no—
224C2×#41—22no—
225C2×#41—22no—
226C2×#41—22no—
227C2×#41—21no—
228C2×#41—21no—
229C2×#41—21no—
230C2×#41—84no—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—41no—
236C22×Q8—41no—
237C22×Q8—41no—
238C22×Q8—84no—
239C22×D4—22no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C2×D412yes—
243C23×C4—22no—
244C23×C4—41no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4#4111yescharacteristic
248#137—88no—
249C2×Q16—28no—
250C2×SD16—24no—
251C2×SD16—28no—
252C2×D8—24no—
253#131—88no—
254C2×M16—28no—
255C2×M16C2×D414yes—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—28no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—22no—
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—41no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275C4⋊Q8—41no—
276C4⋊Q8—41no—
277C4⋊Q8C2×D412yes—
278C4⋊Q8C2×D412yes—
279C4⋊Q8C2414yes—
280C4⋊Q8C2411yescharacteristic
281C4⋊Q8C2411yescharacteristic
282C4⋊Q8C2411yescharacteristic
283#127—21no—
284#127—21no—
285#127C2×D412yes—
286#127C2×D412yes—
287#127C2411yescharacteristic
288#126—41no—
289#126—41no—
290#125—21no—
291#125—21no—
292#125—21no—
293#124—22no—
294#124—22no—
295#124—21no—
296#124—21no—
297#124—22no—
298#124—22no—
299#124—21no—
300#123—41no—
301#123—41no—
302#123—41no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309#121—22no—
310#121—22no—
311#121—22no—
312#121—22no—
313#121—22no—
314#121—22no—
315#121—21no—
316#121—21no—
317#121—21no—
318#121—21no—
319#121—21no—
320#121—21no—
321#121C2×D414yes—
322#121C2×D414yes—
323#120—41no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—22no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—41no—
343C4×D4—41no—
344C4×D4—41no—
345C4×D4—41no—
346C4×D4—41no—
347C4×D4—41no—
348C4×D4C2×D414yes—
349C4×D4C2414yes—
350#117—21no—
351#117—21no—
352#117—21no—
353#117—41no—
354#117—41no—
355#117—41no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—41no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4C2×D412yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C42—41no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42#4111yescharacteristic
380C2×C42#4111yescharacteristic
381C2×C42#4111yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#107—24no—
388#106—24no—
389C4⋊C8—22no—
390C4⋊C8—22no—
391C4⋊C8—22no—
392C4⋊C8C2×D414yes—
393C4⋊C8C2×D414yes—
394C4⋊C8C2414yes—
395C4⋊C8C2414yes—
396C4⋊C8C2414yes—
397#103—28no—
398#103—28no—
399#103—24no—
400#102—28no—
401#102—28no—
402#102—24no—
403#102—24no—
404#102—24no—
405C22⋊C8—22no—
406C22⋊C8—22no—
407C22⋊C8—22no—
408C22⋊C8—22no—
409C22⋊C8—22no—
410C22⋊C8—22no—
411C22⋊C8—28no—
412C22⋊C8—28no—
413C22⋊C8C2×D414yes—
414C22⋊C8C2×D414yes—
415#97—22no—
416#97—22no—
417#97—22no—
418#97—22no—
419#97C2×D414yes—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8C2414yes—
423C24—22no—
424C24#12111yescharacteristic
425#41—48no—
426#41—44no—
427#41—84no—
428#41—84no—
429#41—82no—
430#41—84no—
431#41—82no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—21no—
435C2×Q8—24no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—48no—
442C2×Q8—84no—
443C2×Q8—84no—
444C2×Q8—82no—
445C2×Q8—84no—
446C2×Q8—84no—
447C2×Q8—84no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—44no—
459C2×D4—41no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—82no—
463C2×D4—82no—
464C2×D4—84no—
465C2×D4C22×D414yes—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4#12111yescharacteristic
481C22×C4#12111yescharacteristic
482C22×C4#12111yescharacteristic
483C22×C4#12111yescharacteristic
484C22×C4#12111yescharacteristic
485C22×C4#12111yescharacteristic
486C22×C4#12111yescharacteristic
487C22×C4#12111yescharacteristic
488C22×C4#12111yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4111yescharacteristic
496C22×C4C2×#4111yescharacteristic
497C22×C4C2×#4111yescharacteristic
498Q16—88no—
499SD16—88no—
500SD16—84no—
501D8—84no—
502M16—28no—
503M16—88no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—44no—
508C2×C8—44no—
509C2×C8—88no—
510C2×C8—82no—
511C2×C8C22×D418yes—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—24no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4C22×D414yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—41no—
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42#12111yescharacteristic
553C42#12111yescharacteristic
554C42#12111yescharacteristic
555C42#12111yescharacteristic
556C42#12111yescharacteristic
557C42#12111yescharacteristic
558C42C22×D411yescharacteristic
559C42C22×D411yescharacteristic
560C42C22×D411yescharacteristic
561C42C22×D411yescharacteristic
562C42C22×D411yescharacteristic
563C42C22×D411yescharacteristic
564C42C2×#4111yescharacteristic
565C42C2×#4111yescharacteristic
566C42C2×#4111yescharacteristic
567C42C2511yescharacteristic
568C23—22no—
569C23—22no—
570C23—24no—
571C23—41no—
572C23—44no—
573C23—41no—
574C23—82no—
575C23—82no—
576C23C2×#12111yescharacteristic
577C23C2×#12111yescharacteristic
578C23C2×#12111yescharacteristic
579Q8—44no—
580Q8—42no—
581Q8—48no—
582Q8—42no—
583Q8—82no—
584Q8—81no—
585Q8—84no—
586Q8—84no—
587Q8—84no—
588D4—24no—
589D4—24no—
590D4—48no—
591D4—42no—
592D4—44no—
593D4—42no—
594D4—82no—
595D4—82no—
596D4—82no—
597D4—82no—
598D4—81no—
599D4—84no—
600D4—82no—
601D4—82no—
602D4—82no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—48no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—84no—
627C2×C4—82no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—82no—
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×#12111yescharacteristic
636C2×C4C2×#12111yescharacteristic
637C2×C4C2×#12111yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4C2×#12111yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12111yescharacteristic
643C2×C4#57614yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C8—28no—
648C8—44no—
649C8—44no—
650C8—44no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—164no—
666C22#304112yes—
667C22#306912yes—
668C22C22×#12111yescharacteristic
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—44no—
679C4—84no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4#304112yes—
684C4C2×#57612yes—
685C4#358612yes—
686C2—22no—
687C2—22no—
688C2—41no—
689C2—84no—
690C2—161no—
691C2C2×#304111yescharacteristic
692C2#6045511yescharacteristic
693C2C2×#306911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60962C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3226C2semidirect
3. #60800C2semidirect
4. #60237C2semidirect
5. #19507C2semidirect
6. #19505C2semidirect
7. #19459C2semidirect
8. #19457C2semidirect
9. #19235C2semidirect
10. #19231C2semidirect
11. #17810C2semidirect
12. C2×#1556C2semidirect
13. #17713C2semidirect
30. #3209C22semidirect
32. #3206C22semidirect
36. #3152C22semidirect
37. #3139C22semidirect
57. #1703C22semidirect
58. #1699C22semidirect
59. #1689C22semidirect
60. #1685C22semidirect
61. #1683C22semidirect
62. #1679C22semidirect
63. #1669C22semidirect
64. #1665C22semidirect
66. #1572C22semidirect
67. #1565C22semidirect
69. #1556C22semidirect
70. #1554C22semidirect
71. #1553C22semidirect
74. #1542C22semidirect
75. #1540C22semidirect
76. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
#609622
C2×#32262
#608002
#602372
#195072
#195052
#194592
#194572
#192352
#192312
#178102
C2×#15562
#177132