Finite Group Databasebeta

← #6653019

Subgroups — #6653019

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic, complemented
2#34963C211yesmaximal, characteristic
3#32905C212yesmaximal, complemented
4#32615C214yesmaximal, complemented
5#32029C212yesmaximal, complemented
6#31682C212yesmaximal, complemented
7#31065C212yesmaximal, complemented
8#30500C212yesmaximal, complemented
9#30403C214yesmaximal, complemented
10#29926C212yesmaximal, complemented
11#27627C212yesmaximal, complemented
12#24751C212yesmaximal, complemented
13#24659C212yesmaximal, complemented
14#24209C212yesmaximal, complemented
15#21851C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C2×C4×#41C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#465C2214yes—
20C2×#461C2212yes—
21C2×#461C2212yes—
22C2×#448C2214yes—
23C2×#450C2214yes—
24C2×#432C2211yescharacteristic, complemented
25C22×C4⋊C8C2211yescharacteristic, complemented
26#2857C2212yes—
27#2857C2212yescomplemented
28#2628—21no—
29#2628—21no—
30#2598—21no—
31#2598—21no—
32#2580C2212yes—
33#2568C2212yescomplemented
34#2567—21no—
35#2563—21no—
36#2542C2212yes—
37#2535C2212yes—
38#2487—21no—
39#2487—21no—
40#2487C2211yescharacteristic
41#2487C2211yescharacteristic
42#2482—21no—
43#2409C2212yes—
44#2409C2212yescomplemented
45#2408C2211yescharacteristic
46#2408C2211yescharacteristic
47#2404—21no—
48#2398—21no—
49#2397—21no—
50#2317—21no—
51#2311—21no—
52C4×#121C2212yes—
53C4×#121C2212yes—
54C4×#120—21no—
55C4×#120—21no—
56#2285C2211yescharacteristic, complemented
57#2283C2211yescharacteristic
58#2097C2214yescomplemented
59#2077C2214yescomplemented
60#2074C2214yescomplemented
61#2066C2214yescomplemented
62#2055C2214yes—
63#2043C2214yescomplemented
64#2036C2214yes—
65#2036C2214yescomplemented
66#2035C2214yes—
67#2033C2214yes—
68#2013C2214yes—
69#2000C2214yes—
70#1985C2212yescomplemented
71#1981C2212yescomplemented
72#1981C2212yescomplemented
73#1973C2214yes—
74#1955C2212yes—
75#1948C2212yescomplemented
76#1947C2212yescomplemented
77#1935C2214yescomplemented
78#1926C2214yes—
79#1921C2214yescomplemented
80#1892C2214yescomplemented
81#1877C2214yes—
82#1877C2214yes—
83#1872C2214yes—
84#1869C2212yes—
85#1869C2212yescomplemented
86#1802C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2212yescomplemented
90#1787C2214yescomplemented
91C22×#41D411yescharacteristic
92C23×Q8C2312yes—
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C22×SD16C2314yes—
96C22×SD16C2314yes—
97C23×C8C2312yes—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8—42no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2314yes—
106C2×#121C2312yes—
107C2×#121C2312yes—
108C2×#120—21no—
109C2×#120—21no—
110C4×#41—42no—
111C4×#41—42no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—42no—
119C2×C4×D4—42no—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C2×#117D411yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#465—48no—
140#461—44no—
141#461—44no—
142#450—48no—
143#448—48no—
144#432—42no—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×C4⋊C8—42no—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C22⋊C8C2312yes—
162C2×#97C2312yes—
163#400—22no—
164#400—22no—
165#400—21no—
166#400—21no—
167#399—22no—
168#399—21no—
169#399—21no—
170#399—22no—
171#399—22no—
172#397—22no—
173#395—22no—
174#395—22no—
175#394—21no—
176#394—21no—
177#393—22no—
178#393—22no—
179#393—22no—
180#392—22no—
181#392—21no—
182#392—21no—
183#391—22no—
184#391C2314yes—
185#391C2314yes—
186#390—22no—
187#390C2312yes—
188#390C2312yes—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#386—22no—
198#386—22no—
199#386—21no—
200#386—21no—
201#386—22no—
202#386—22no—
203#386—22no—
204#386—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#385—22no—
209#385—21no—
210#385—21no—
211#384—22no—
212#384C2312yes—
213#383—21no—
214#382—22no—
215#382—21no—
216#382C2312yes—
217#381—21no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380—22no—
222#380—21no—
223#380C2312yes—
224#380C2312yes—
225#380C2312yes—
226#380C2311yescharacteristic
227#380C2311yescharacteristic
228#379—21no—
229#379—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2312yes—
233C4×C22⋊C4—21no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236#376—21no—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C25—21no—
244C2×#41—24no—
245C2×#41—42no—
246C2×#41—42no—
247C22×Q8—22no—
248C22×Q8—42no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—44no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2414yes—
254C22×D4—24no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—41no—
259C22×D4—41no—
260C22×D4—42no—
261C22×D4C2×D411yescharacteristic
262C22×D4C2×D411yescharacteristic
263C22×D4C2×D411yescharacteristic
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—44no—
270C23×C4—84no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×SD16—28no—
281C2×SD16—28no—
282C2×SD16—44no—
283C2×SD16—44no—
284C2×SD16—44no—
285C2×SD16—44no—
286C22×C8—42no—
287C22×C8—44no—
288C22×C8C2414yes—
289C22×C8C2414yes—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—42no—
293C4⋊Q8—42no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C22⋊Q8—44no—
299C22⋊Q8—44no—
300#121—24no—
301#121—24no—
302#121—42no—
303#121—42no—
304#120—44no—
305#120—44no—
306C4×Q8—42no—
307C4×Q8—41no—
308C4×Q8—41no—
309C4×D4—24no—
310C4×D4—24no—
311C4×D4—42no—
312C4×D4—42no—
313C4×D4—42no—
314C4×D4—42no—
315C4×D4—42no—
316C4×D4—41no—
317C4×D4—41no—
318C4×D4—41no—
319C4×D4—41no—
320C4×D4—41no—
321C4×D4—41no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—84no—
325C4×D4—84no—
326#117—42no—
327#117—42no—
328#117—42no—
329#117—44no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—41no—
342C2×C4⋊C4—41no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—44no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2414yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—44no—
396#106—44no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—42no—
400#103—44no—
401#103—44no—
402#103—44no—
403#103—44no—
404#103—44no—
405#103—44no—
406#102—28no—
407#102—44no—
408C22⋊C8—42no—
409C22⋊C8—44no—
410C22⋊C8—44no—
411#97—42no—
412#95—21no—
413#95—21no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427C24—22no—
428C24—44no—
429C24—42no—
430C24—42no—
431C24—42no—
432C24—82no—
433C24C22×D411yescharacteristic
434C24C22×D411yescharacteristic
435C24C22×D411yescharacteristic
436#41—416no—
437C2×Q8—22no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—44no—
447C2×Q8—84no—
448C2×Q8—84no—
449C2×Q8—84no—
450C2×Q8—88no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—41no—
461C2×D4—41no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—44no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—48no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—82no—
474C2×D4—82no—
475C2×D4#13714yes—
476C2×D4H(5,𝔽2)14yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—24no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—41no—
508C22×C4—41no—
509C22×C4—44no—
510C22×C4—42no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—82no—
516C22×C4—82no—
517C22×C4—82no—
518C22×C4—84no—
519C22×C4—816no—
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4C22×D411yescharacteristic
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D411yescharacteristic
528C22×C4C22×D411yescharacteristic
529C22×C4C22×D411yescharacteristic
530C22×C4C22×D411yescharacteristic
531C22×C4C2511yescharacteristic
532SD16—88no—
533SD16—88no—
534C2×C8—24no—
535C2×C8—24no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—24no—
539C2×C8—24no—
540C2×C8—84no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—24no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—84no—
565C4⋊C4—84no—
566C22⋊C4—41no—
567C22⋊C4—41no—
568C22⋊C4—41no—
569C22⋊C4—41no—
570C22⋊C4—41no—
571C22⋊C4—41no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—42no—
577C22⋊C4—42no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—84no—
586C22⋊C4—84no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—21no—
594C42—21no—
595C42—21no—
596C42—21no—
597C42—21no—
598C42—21no—
599C42—44no—
600C42—44no—
601C42H(5,𝔽2)12yes—
602C42H(5,𝔽2)12yes—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—22no—
607C23—22no—
608C23—22no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—22no—
613C23—22no—
614C23—22no—
615C23—24no—
616C23—48no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—88no—
624C23—84no—
625C23—88no—
626C23—84no—
627C23C2×#13712yes—
628C23C2×#13712yes—
629C23#57512yes—
630C23C23×D411yescharacteristic
631Q8—44no—
632Q8—44no—
633Q8—44no—
634Q8—44no—
635Q8—44no—
636Q8—88no—
637D4—24no—
638D4—24no—
639D4—44no—
640D4—44no—
641D4—44no—
642D4—48no—
643D4—42no—
644D4—42no—
645D4—88no—
646D4—88no—
647C2×C4—28no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—41no—
676C2×C4—41no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—44no—
699C2×C4—88no—
700C2×C4—84no—
701C2×C4—88no—
702C2×C4—88no—
703C2×C4#53412yes—
704C2×C4#53412yes—
705C2×C4C2×#13712yes—
706C2×C4#57714yes—
707C2×C4C2×H(5,𝔽2)11yescharacteristic
708C2×C4C2×H(5,𝔽2)12yes—
709C2×C4C2×H(5,𝔽2)11yescharacteristic
710C2×C4#58512yes—
711C8—44no—
712C8—44no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—41no—
726C22—41no—
727C22—41no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—816no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22C2×#53411yescharacteristic
737C22#348812yes—
738C22C22×#13711yescharacteristic
739C22C2×#57511yescharacteristic
740C22C2×#57712yes—
741C4—22no—
742C4—22no—
743C4—41no—
744C4—41no—
745C4—41no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—41no—
751C4—84no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—84no—
756C4#320014yes—
757C4#353711yescharacteristic
758C4#354011yescharacteristic
759C4#354212yes—
760C2—42no—
761C2—42no—
762C2—42no—
763C2—84no—
764C2#3726911yescharacteristic
765C2C2×#320011yescharacteristic
766C2#6075111yescharacteristic
767C2#6197812yes—
768C2#6198012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34963C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35279C2semidirect
3. #32905C2semidirect
4. #32615C2semidirect
5. #32029C2semidirect
6. #31682C2semidirect
7. #31065C2semidirect
8. #30500C2semidirect
9. #30403C2semidirect
10. #29926C2semidirect
11. #27627C2semidirect
12. #24751C2semidirect
13. #24659C2semidirect
14. #24209C2semidirect
15. #21851C2semidirect
16. C22×C4⋊Q8C22semidirect
24. C2×#432C22semidirect
25. C22×C4⋊C8C22semidirect
27. #2857C22semidirect
33. #2568C22semidirect
44. #2409C22semidirect
56. #2285C22semidirect
58. #2097C22semidirect
59. #2077C22semidirect
60. #2074C22semidirect
61. #2066C22semidirect
63. #2043C22semidirect
65. #2036C22semidirect
70. #1985C22semidirect
71. #1981C22semidirect
72. #1981C22semidirect
75. #1948C22semidirect
76. #1947C22semidirect
77. #1935C22semidirect
79. #1921C22semidirect
80. #1892C22semidirect
85. #1869C22semidirect
89. #1797C22semidirect
90. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#352792
#349632
#329052
#326152
#320292
#316822
#310652
#305002
#304032
#299262
#276272
#247512
#246592
#242092
#218512