Finite Group Databasebeta

← #6802525

Subgroups — #6802525

Order 512.

1919 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#36599C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3#32905C211yesmaximal, characteristic
4#32187C214yesmaximal, complemented
5#30754C214yesmaximal, complemented
6#29715C212yesmaximal, complemented
7#28471C212yesmaximal, complemented
8#27756C212yesmaximal, complemented
9#25730C212yesmaximal
10#25722C211yesmaximal, characteristic, complemented
11#25365C212yesmaximal, complemented
12#24629C212yesmaximal, complemented
13#24217C212yesmaximal
14#24212C211yesmaximal, characteristic, complemented
15#22629C211yesmaximal, characteristic, complemented
16#22050C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#497C2214yes—
22C2×#494C2212yescomplemented
23C2×#464C2214yes—
24C2×#462C2212yescomplemented
25C2×#461C2212yes—
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×#451C2212yes—
29C2×#450C2211yescharacteristic
30C2×#450C2212yes—
31C2×C4×SD16C2212yes—
32#2857C2212yes—
33#2857C2211yescharacteristic
34#2585—22no—
35#2569—22no—
36#2552C2211yescharacteristic, complemented
37#2547—22no—
38#2547—21no—
39#2547—21no—
40#2547C2212yes—
41#2547C2214yes—
42#2547C2212yes—
43#2546C2211yescharacteristic, complemented
44#2511C2212yescomplemented
45#2509—21no—
46#2446C2212yescomplemented
47#2442—21no—
48C2×#397—22no—
49#2370—21no—
50#2365C2211yescharacteristic, complemented
51#2355C2211yescharacteristic, complemented
52#2345—21no—
53C4×C22⋊Q8C2212yes—
54C4×C22⋊Q8C2211yescharacteristic
55C2×#391C2212yes—
56C2×#380—21no—
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2072C2214yescomplemented
60#2056C2214yescomplemented
61#2056C2214yescomplemented
62#2043C2212yescomplemented
63#2042C2214yes—
64#2016C2214yescomplemented
65#2016C2212yescomplemented
66#2016C2214yescomplemented
67#2015C2214yes—
68#2015C2214yes—
69#2015C2214yes—
70#1983C2214yes—
71#1981C2212yescomplemented
72#1971C2214yescomplemented
73#1959C2214yes—
74#1956C2212yescomplemented
75#1939C2212yescomplemented
76#1935C2214yescomplemented
77#1927C2214yescomplemented
78#1917C2214yescomplemented
79#1878C2214yescomplemented
80#1877C2212yes—
81#1877C2214yes—
82#1872C2211yescharacteristic
83#1872C2212yes—
84#1864C2212yes—
85#1853C2211yescharacteristic
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1786C2211yescharacteristic
91#1770C2212yescomplemented
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×Q16C2314yes—
101C22×SD16C2312yes—
102C22×SD16C2314yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#124—22no—
112C2×#124—22no—
113C2×#124—48no—
114C2×#124C2314yes—
115C2×#124C2314yes—
116C2×C22⋊Q8C2311yescharacteristic
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—44no—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—22no—
132C22×C22⋊C4—22no—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#497—48no—
136#494—44no—
137#464—48no—
138#462—44no—
139#461—44no—
140#451—28no—
141#451—44no—
142#451—44no—
143#450—24no—
144#450—44no—
145C4×SD16—44no—
146C2×#106C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2314yes—
159C2×#102C2312yes—
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2311yescharacteristic
162C2×C22⋊C8C2311yescharacteristic
163C2×#97C2314yes—
164C2×C4×C8C2312yes—
165#400—22no—
166#400—22no—
167#400—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#397—24no—
172#397—24no—
173#397—24no—
174#397—24no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#391C2314yes—
180#391C2314yes—
181#391C2312yes—
182#391C2311yescharacteristic
183#391C2312yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—21no—
192#389C2311yescharacteristic
193#389C2312yes—
194#387—22no—
195#387—22no—
196#387—21no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#385—21no—
203#385—22no—
204#384C2311yescharacteristic
205#384C2312yes—
206#382—21no—
207#380—21no—
208#380—22no—
209#380—22no—
210#380—21no—
211#380—22no—
212#380—22no—
213#380—22no—
214#380—24no—
215#380—22no—
216#380—24no—
217#380C2311yescharacteristic
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2311yescharacteristic
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2312yes—
236#340C2314yes—
237#340C2312yes—
238#336C2314yes—
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C25—21no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—44no—
247C22×Q8—44no—
248C22×Q8C2411yescharacteristic
249C22×Q8C2412yes—
250C22×Q8C2414yes—
251C22×D4—24no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—42no—
256C22×D4C2412yes—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4—44no—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×Q16—28no—
270C2×Q16—44no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×Q16—44no—
278C2×SD16—24no—
279C2×SD16—28no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×SD16—44no—
283C2×SD16—44no—
284C22×C8C2414yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—24no—
288C4⋊Q8—24no—
289C4⋊Q8—24no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294#125—44no—
295#124—28no—
296#124—28no—
297#124—44no—
298#124—44no—
299#124—48no—
300#124—48no—
301C22⋊Q8—22no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—22no—
305C4×Q8—24no—
306C4×Q8—44no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—44no—
312C4×D4—44no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—42no—
334C2×C4⋊C4—42no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—48no—
359C2×C22⋊C4—48no—
360C2×C42—24no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374#106—42no—
375C4⋊C8—24no—
376C4⋊C8—44no—
377#103—28no—
378#103—42no—
379#103—44no—
380#103—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#102—24no—
385#102—24no—
386#102—42no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8—22no—
390C22⋊C8#4114yes—
391#97—44no—
392C4×C8—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—21no—
398#95—21no—
399#95—22no—
400#95—22no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—21no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95C2×D412yes—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416C24—24no—
417C24—22no—
418C24—42no—
419C24—42no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24C22×D411yescharacteristic
425C2×Q8—24no—
426C2×Q8—21no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—22no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—88no—
443C2×Q8—88no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—48no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4H(5,𝔽2)14yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—42no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—816no—
489C22×C4#13714yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4112yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C4C2511yescharacteristic
502Q16—88no—
503Q16—88no—
504Q16—88no—
505SD16—48no—
506SD16—88no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—24no—
515C2×C8C2×#4114yes—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—81no—
544C4⋊C4—81no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—44no—
558C42—22no—
559C42—22no—
560C42—24no—
561C42—24no—
562C42—22no—
563C42—42no—
564C42—42no—
565C42—42no—
566C42—42no—
567C42—82no—
568C42—82no—
569C42H(5,𝔽2)14yes—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—24no—
575C23—48no—
576C23—48no—
577C23—48no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—84no—
585C23—84no—
586C23—84no—
587C23—84no—
588C23#55012yes—
589C23C2×#13712yes—
590C23#57812yes—
591C23C23×D411yescharacteristic
592Q8—42no—
593Q8—44no—
594Q8—44no—
595Q8—44no—
596Q8—44no—
597Q8—42no—
598Q8—44no—
599Q8—44no—
600Q8—88no—
601D4—24no—
602D4—24no—
603D4—48no—
604D4—42no—
605D4—42no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4#53512yes—
662C2×C4#53512yes—
663C2×C4#55012yes—
664C2×C4C2×#13712yes—
665C2×C4C2×#13712yes—
666C2×C4#57512yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)12yes—
669C2×C4C2×H(5,𝔽2)12yes—
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C8—24no—
672C8—44no—
673C8—42no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—816no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#339814yes—
689C22C2×#53511yescharacteristic
690C22C2×#55011yescharacteristic
691C22#348411yescharacteristic
692C22C22×#13711yescharacteristic
693C22C2×#57511yescharacteristic
694C22C2×#57811yescharacteristic
695C22#358611yescharacteristic
696C4—22no—
697C4—22no—
698C4—22no—
699C4—41no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—82no—
705C4—81no—
706C4—81no—
707C4—82no—
708C4—81no—
709C4—84no—
710C4—81no—
711C4—82no—
712C4#321514yes—
713C4#351912yes—
714C4#353712yes—
715C2—24no—
716C2—84no—
717C2#3913211yescharacteristic
718C2C2×#321511yescharacteristic
719C2#6081411yescharacteristic
720C2C2×#339811yescharacteristic
721C2#6140311yescharacteristic
722C2#6158611yescharacteristic
723C2#6159511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36599C2semidirect
4. #32187C2semidirect
5. #30754C2semidirect
6. #29715C2semidirect
7. #28471C2semidirect
8. #27756C2semidirect
10. #25722C2semidirect
11. #25365C2semidirect
12. #24629C2semidirect
14. #24212C2semidirect
15. #22629C2semidirect
16. #22050C2semidirect
22. C2×#494C22semidirect
24. C2×#462C22semidirect
36. #2552C22semidirect
43. #2546C22semidirect
44. #2511C22semidirect
46. #2446C22semidirect
50. #2365C22semidirect
51. #2355C22semidirect
59. #2072C22semidirect
60. #2056C22semidirect
61. #2056C22semidirect
62. #2043C22semidirect
64. #2016C22semidirect
65. #2016C22semidirect
66. #2016C22semidirect
71. #1981C22semidirect
72. #1971C22semidirect
74. #1956C22semidirect
75. #1939C22semidirect
76. #1935C22semidirect
77. #1927C22semidirect
78. #1917C22semidirect
79. #1878C22semidirect
91. #1770C22semidirect

Maximal subgroups

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

subgroupindex
#365992
C2×#25472
#329052
#321872
#307542
#297152
#284712
#277562
#257302
#257222
#253652
#246292
#242172
#242122
#226292
#220502
C4×#4502