Finite Group Databasebeta

← #482982

Subgroups — #482982

Order 512.

752 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#62247C211yesmaximal, characteristic
2#33488C211yesmaximal, characteristic, complemented
3#22717C211yesmaximal, characteristic, complemented
4#11895C211yesmaximal, characteristic, complemented
5#11894C211yesmaximal, characteristic, complemented
6#11893C211yesmaximal, characteristic, complemented
7#11892C211yesmaximal, characteristic, complemented
8#11866C211yesmaximal, characteristic, complemented
9#11861C211yesmaximal, characteristic, complemented
10#11750C211yesmaximal, characteristic, complemented
11#11749C211yesmaximal, characteristic, complemented
12#11712C211yesmaximal, characteristic, complemented
13#11710C211yesmaximal, characteristic, complemented
14#11709C211yesmaximal, characteristic, complemented
15C2×#1349C211yesmaximal, characteristic, complemented
16#3505—21no—
17#3339C411yescharacteristic
18#3339C411yescharacteristic
19#3336C411yescharacteristic
20#3336C411yescharacteristic
21#3301—21no—
22#3301—21no—
23#3296—21no—
24#3296—21no—
25#3285C411yescharacteristic
26#3285C411yescharacteristic
27#3280C411yescharacteristic
28#3280C411yescharacteristic
29#3250—21no—
30#3191—21no—
31#3187—21no—
32#3168C2211yescharacteristic
33#3168C2211yescharacteristic
34#3050C2211yescharacteristic
35#3050C2211yescharacteristic
36C2×#467C2211yescharacteristic
37C2×#466C2211yescharacteristic
38D4×M16—21no—
39C2×C8×D4C2211yescharacteristic
40#2169—21no—
41#2168—21no—
42#2168—21no—
43D4×C16—21no—
44#2163C2211yescharacteristic, complemented
45#2162C2211yescharacteristic, complemented
46#2112C2211yescharacteristic
47#2112C2211yescharacteristic, complemented
48C2×C22⋊C16C2211yescharacteristic
49C2×C22⋊C16C2211yescharacteristic, complemented
50#1950—21no—
51#1855—21no—
52#1855—21no—
53#1837C2211yescharacteristic, complemented
54#1810C2211yescharacteristic
55#1810C2211yescharacteristic, complemented
56C4×#108C2211yescharacteristic, complemented
57C2×#341C2211yescharacteristic
58C2×#341C2211yescharacteristic, complemented
59#1350C2211yescharacteristic, complemented
60#1350C2211yescharacteristic, complemented
61#1350C2211yescharacteristic, complemented
62#1350C2211yescharacteristic, complemented
63#1350C2211yescharacteristic, complemented
64#1350C2211yescharacteristic, complemented
65#1350C2211yescharacteristic, complemented
66#1350C2211yescharacteristic, complemented
67#1349C2211yescharacteristic
68#1349C2211yescharacteristic
69#1349C2211yescharacteristic
70#1349C2211yescharacteristic
71#1349C2211yescharacteristic
72#1349C2211yescharacteristic
73#1349C2211yescharacteristic
74#1349C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C2×#135C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C22×D8C2×C411yescharacteristic
79C22×M16—21no—
80C22×M16—21no—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83#553—21no—
84#553—21no—
85#553—41no—
86#553—41no—
87#552—21no—
88#552—21no—
89#551—41no—
90#551—41no—
91#548—21no—
92#548—21no—
93#546—21no—
94#546—21no—
95#539—41no—
96#538—41no—
97#534—41no—
98#529—21no—
99#529—21no—
100#525—21no—
101#525—21no—
102C2×C22⋊Q8—21no—
103C2×#121—21no—
104#518—41no—
105C2×C4×D4D411yescharacteristic
106C2×M32—21no—
107C2×M32—21no—
108C22×C16—21no—
109C22×C16—21no—
110#499C2×C411yescharacteristic
111#499C2×C411yescharacteristic
112#495C2×C411yescharacteristic
113#495C2×C411yescharacteristic
114#488—21no—
115#488—21no—
116#485—21no—
117#485—21no—
118#483—21no—
119#483—21no—
120#482—21no—
121#482—21no—
122#481—21no—
123#481—21no—
124#479—21no—
125#479—21no—
126#478—21no—
127#478—21no—
128#467—21no—
129#467—21no—
130#467C2×C411yescharacteristic
131#467C2×C411yescharacteristic
132#467C2×C411yescharacteristic
133#467C2×C411yescharacteristic
134#467C2311yescharacteristic
135#467C2311yescharacteristic
136#466—21no—
137#466—21no—
138#466C2×C411yescharacteristic
139#466C2×C411yescharacteristic
140#466C2×C411yescharacteristic
141#466C2×C411yescharacteristic
142#466C2311yescharacteristic
143#466C2311yescharacteristic
144#465—21no—
145#465—21no—
146#465—21no—
147#465—21no—
148#465—21no—
149#465—21no—
150#464—21no—
151#464—21no—
152#463—21no—
153#463—21no—
154#452—21no—
155#452—21no—
156#450—21no—
157#450—21no—
158#449—21no—
159#449—21no—
160#448—21no—
161#448—21no—
162C4×Q16C2×C411yescharacteristic
163C4×Q16C2×C411yescharacteristic
164C4×SD16—21no—
165C4×SD16—21no—
166C4×D8C2×C411yescharacteristic
167C4×D8C2×C411yescharacteristic
168#436—41no—
169#435—41no—
170#435—41no—
171C8×D4—41no—
172C8×D4C2×C411yescharacteristic
173C8×D4C2×C411yescharacteristic
174C8×D4C2×C411yescharacteristic
175C8×D4C2×C411yescharacteristic
176C8×D4D411yescharacteristic
177C8×D4D411yescharacteristic
178C8×D4D411yescharacteristic
179C8×D4D411yescharacteristic
180C2×#108—21no—
181C2×#108—21no—
182C2×#108—21no—
183#427C2×C411yescharacteristic
184#427C2×C411yescharacteristic
185C2×#107C2×C411yescharacteristic
186C2×#107C2×C411yescharacteristic
187#423—21no—
188C2×C4⋊C8D411yescharacteristic
189#416—21no—
190#416—21no—
191C2×#103—21no—
192C2×#102—21no—
193#407—21no—
194#407—21no—
195C2×C22⋊C8D411yescharacteristic
196C2×C22⋊C8D411yescharacteristic
197C4×M16—21no—
198C2×C4×C8C2311yescharacteristic
199C4⋊C16—21no—
200C4⋊C16—21no—
201C22⋊C16—21no—
202C22⋊C16—21no—
203C22⋊C16—21no—
204C22⋊C16—21no—
205C22⋊C16C2311yescharacteristic
206C22⋊C16C2311yescharacteristic
207C22⋊C16C2311yescharacteristic
208C22⋊C16C2311yescharacteristic
209#346—21no—
210C4×C16—21no—
211#341—21no—
212#341—21no—
213#341—21no—
214#341C2311yescharacteristic
215#341C2311yescharacteristic
216#341C2311yescharacteristic
217#341C2311yescharacteristic
218C2×#41—21no—
219C2×#41—21no—
220C22×Q8—21no—
221C22×D4—21no—
222C22×D4—41no—
223C22×D4SD1611yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226#135—81no—
227#135—81no—
228C2×Q16—21no—
229C2×Q16—81no—
230C2×Q16C22×C411yescharacteristic
231C2×Q16C22×C411yescharacteristic
232C2×SD16—21no—
233C2×SD16—21no—
234C2×SD16—21no—
235C2×SD16—21no—
236C2×D8—21no—
237C2×D8—81no—
238C2×D8C22×C411yescharacteristic
239C2×D8C22×C411yescharacteristic
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—41no—
245C2×M16—41no—
246C2×M16—42no—
247C2×M16—42no—
248C2×M16—81no—
249C2×M16—81no—
250C22×C8—41no—
251C22×C8—41no—
252C22×C8—81no—
253C22×C8—81no—
254C22×C8C22×C411yescharacteristic
255C22×C8C22×C411yescharacteristic
256C22×C8C22×C411yescharacteristic
257C22×C8C22×C411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2411yescharacteristic
266#126—41no—
267#126—41no—
268#126—41no—
269#126—41no—
270#126—41no—
271#126—41no—
272#125—21no—
273#125—21no—
274#125—41no—
275#125—41no—
276#124—21no—
277#124—21no—
278#124—41no—
279#124—41no—
280#123—41no—
281#123—41no—
282#123—41no—
283#123—41no—
284#123—41no—
285#123—41no—
286#123—41no—
287#123—41no—
288#123—41no—
289#123—41no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—21no—
296C22⋊Q8—41no—
297C22⋊Q8—41no—
298C22⋊Q8—41no—
299C22⋊Q8—41no—
300#121—21no—
301#121—21no—
302#121—21no—
303#121—21no—
304#121—21no—
305#121—21no—
306#121—41no—
307#121—41no—
308#121—41no—
309#121—41no—
310#121—41no—
311#121—41no—
312#121—41no—
313#121—41no—
314#120—41no—
315#120—41no—
316#120—41no—
317#120—41no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×D4—21no—
321C4×D4—21no—
322C4×D4—41no—
323C4×D4—41no—
324C4×D4—41no—
325C4×D4—41no—
326C4×D4—41no—
327C4×D4—41no—
328C4×D4—41no—
329C4×D4—41no—
330C4×D4—41no—
331C4×D4—41no—
332C4×D4C22⋊C411yescharacteristic
333C4×D4C22⋊C411yescharacteristic
334C4×D4C22⋊C411yescharacteristic
335C4×D4C22⋊C411yescharacteristic
336C4×D4D811yescharacteristic
337C4×D4D811yescharacteristic
338C4×D4D811yescharacteristic
339C4×D4D811yescharacteristic
340#117—21no—
341#117—21no—
342#117—41no—
343#117—41no—
344#117—41no—
345#117—41no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4SD1611yescharacteristic
350C2×C22⋊C4—41no—
351C2×C22⋊C4—41no—
352C2×C22⋊C4—41no—
353C2×C22⋊C4—41no—
354C2×C22⋊C4SD1611yescharacteristic
355C2×C22⋊C4SD1611yescharacteristic
356C2×C42C2×D411yescharacteristic
357M32—81no—
358M32—81no—
359C2×C16—21no—
360C2×C16—21no—
361C2×C16—21no—
362C2×C16—21no—
363C2×C16—81no—
364C2×C16—81no—
365#108—42no—
366#108—81no—
367#108—81no—
368#107—21no—
369#107—21no—
370#107C22×C411yescharacteristic
371#107C22×C411yescharacteristic
372#107C22×C411yescharacteristic
373#107C22×C411yescharacteristic
374#106—21no—
375#106—21no—
376#106—21no—
377#106—21no—
378C4⋊C8—41no—
379C4⋊C8—41no—
380C4⋊C8C22⋊C411yescharacteristic
381C4⋊C8C22⋊C411yescharacteristic
382C4⋊C8C22⋊C411yescharacteristic
383C4⋊C8C22⋊C411yescharacteristic
384#103—21no—
385#103—21no—
386#103—21no—
387#103—21no—
388#103—21no—
389#103—21no—
390#103—21no—
391#103—21no—
392#102—21no—
393#102—21no—
394#102—21no—
395#102—21no—
396#102—21no—
397#102—21no—
398#102—21no—
399#102—21no—
400C22⋊C8—41no—
401C22⋊C8—41no—
402C22⋊C8—41no—
403C22⋊C8—41no—
404C22⋊C8C22⋊C411yescharacteristic
405C22⋊C8C22⋊C411yescharacteristic
406C22⋊C8C22⋊C411yescharacteristic
407C22⋊C8C22⋊C411yescharacteristic
408C22⋊C8C22⋊C411yescharacteristic
409C22⋊C8C22⋊C411yescharacteristic
410C22⋊C8C22⋊C411yescharacteristic
411C22⋊C8C22⋊C411yescharacteristic
412#97—41no—
413C4×C8—41no—
414C4×C8C22×C411yescharacteristic
415C4×C8C22×C411yescharacteristic
416C4×C8C2×D411yescharacteristic
417C4×C8C2×D411yescharacteristic
418C24—41no—
419C24C2×SD1611yescharacteristic
420C24C2×SD1611yescharacteristic
421#41—81no—
422#41—81no—
423#41—81no—
424#41—81no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—41no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—42no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4—81no—
443C2×D4—81no—
444C2×D4—84no—
445C2×D4#10211yescharacteristic
446C2×D4#10211yescharacteristic
447C2×D4#10211yescharacteristic
448C2×D4#10211yescharacteristic
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×D811yescharacteristic
465C22×C4C2×D811yescharacteristic
466C22×C4C2×D811yescharacteristic
467C22×C4C2×D811yescharacteristic
468C22×C4C2×SD1611yescharacteristic
469C22×C4C2×SD1611yescharacteristic
470C22×C4C2×SD1611yescharacteristic
471C22×C4C2×SD1611yescharacteristic
472C22×C4C22×D411yescharacteristic
473Q16—41no—
474Q16—41no—
475Q16—81no—
476SD16—81no—
477SD16—81no—
478SD16—81no—
479SD16—81no—
480D8—41no—
481D8—41no—
482D8—81no—
483M16—22no—
484M16—41no—
485M16—41no—
486M16—41no—
487M16—41no—
488M16—41no—
489M16—41no—
490M16—41no—
491M16—82no—
492M16—82no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—82no—
510C2×C8—82no—
511C2×C8C2×C22⋊C411yescharacteristic
512C2×C8C2×C22⋊C411yescharacteristic
513C2×C8C2×C22⋊C411yescharacteristic
514C2×C8C2×C22⋊C411yescharacteristic
515C2×C8C2×C22⋊C411yescharacteristic
516C2×C8C2×C22⋊C411yescharacteristic
517C2×C8C23×C411yescharacteristic
518C2×C8C22×D411yescharacteristic
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—41no—
534C4⋊C4—41no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4#10211yescharacteristic
538C4⋊C4#10211yescharacteristic
539C4⋊C4#10211yescharacteristic
540C4⋊C4#10211yescharacteristic
541C22⋊C4—41no—
542C22⋊C4—41no—
543C22⋊C4—41no—
544C22⋊C4—41no—
545C22⋊C4—41no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—41no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—41no—
553C22⋊C4—41no—
554C22⋊C4—41no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4#10211yescharacteristic
562C22⋊C4#10211yescharacteristic
563C22⋊C4#10211yescharacteristic
564C22⋊C4#10211yescharacteristic
565C22⋊C4#10211yescharacteristic
566C22⋊C4#10211yescharacteristic
567C22⋊C4#10211yescharacteristic
568C22⋊C4#10211yescharacteristic
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42C2×C22⋊C411yescharacteristic
574C42C2×C22⋊C411yescharacteristic
575C42C2×D811yescharacteristic
576C42C2×D811yescharacteristic
577C16—41no—
578C16—41no—
579C16—41no—
580C16—41no—
581C23—22no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—82no—
590C23—161no—
591C23—161no—
592C23C2×#10211yescharacteristic
593C23C2×#10211yescharacteristic
594C23C2×#10211yescharacteristic
595C23C2×#10211yescharacteristic
596C23C22×SD1611yescharacteristic
597Q8—41no—
598Q8—41no—
599Q8—41no—
600Q8—41no—
601Q8—161no—
602D4—22no—
603D4—22no—
604D4—41no—
605D4—41no—
606D4—41no—
607D4—41no—
608D4—41no—
609D4—41no—
610D4—82no—
611D4—82no—
612D4—82no—
613D4—82no—
614D4—161no—
615D4—161no—
616D4—161no—
617D4—161no—
618D4—161no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—82no—
644C2×C4—161no—
645C2×C4—161no—
646C2×C4—161no—
647C2×C4—161no—
648C2×C4—161no—
649C2×C4—161no—
650C2×C4C2×#10211yescharacteristic
651C2×C4C2×#10211yescharacteristic
652C2×C4C2×#10211yescharacteristic
653C2×C4C2×#10211yescharacteristic
654C2×C4C2×#10211yescharacteristic
655C2×C4C2×#10211yescharacteristic
656C2×C4C2×#10211yescharacteristic
657C2×C4C2×#10211yescharacteristic
658C2×C4C22×C22⋊C411yescharacteristic
659C2×C4C22×D811yescharacteristic
660C8—21no—
661C8—21no—
662C8—21no—
663C8—21no—
664C8—21no—
665C8—21no—
666C8—21no—
667C8—21no—
668C8—42no—
669C8—81no—
670C8—81no—
671C8—81no—
672C22—21no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—82no—
687C22—82no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—322no—
695C22#214812yes—
696C22C22×#10211yescharacteristic
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—21no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4#288311yescharacteristic
713C4#290111yescharacteristic
714C2—22no—
715C2—41no—
716C2—41no—
717C2—41no—
718C2—41no—
719C2—161no—
720C2—161no—
721C2#1761311yescharacteristic
722C2#3338511yescharacteristic
723C2C2×#214811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radical#62247C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33488C2semidirect
3. #22717C2semidirect
4. #11895C2semidirect
5. #11894C2semidirect
6. #11893C2semidirect
7. #11892C2semidirect
8. #11866C2semidirect
9. #11861C2semidirect
10. #11750C2semidirect
11. #11749C2semidirect
12. #11712C2semidirect
13. #11710C2semidirect
14. #11709C2semidirect
15. C2×#1349C2semidirect
44. #2163C22semidirect
45. #2162C22semidirect
47. #2112C22semidirect
49. C2×C22⋊C16C22semidirect
53. #1837C22semidirect
55. #1810C22semidirect
56. C4×#108C22semidirect
58. C2×#341C22semidirect
59. #1350C22semidirect
60. #1350C22semidirect
61. #1350C22semidirect
62. #1350C22semidirect
63. #1350C22semidirect
64. #1350C22semidirect
65. #1350C22semidirect
66. #1350C22semidirect

Maximal subgroups

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

subgroupindex
#622472
#334882
#227172
#118952
#118942
#118932
#118922
#118662
#118612
#117502
#117492
#117122
#117102
#117092
C2×#13492