Finite Group Databasebeta

← #6792513

Subgroups — #6792513

Order 512.

1873 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36116C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#32905C211yesmaximal, characteristic, complemented
4#31645C214yesmaximal, complemented
5#31529C214yesmaximal, complemented
6#29913C212yesmaximal, complemented
7#28398C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#25902C211yesmaximal, characteristic, complemented
10#25319C212yesmaximal, complemented
11#24812C212yesmaximal, complemented
12#24501C212yesmaximal, complemented
13#24299C211yesmaximal, characteristic, complemented
14#23936C212yesmaximal
15#22599C211yesmaximal, characteristic, complemented
16#21861C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#487C2212yescomplemented
22C2×#476C2212yescomplemented
23C2×#461C2212yes—
24C2×#450C2212yes—
25C2×#447C2211yescharacteristic
26C2×C4×D8C2212yes—
27C2×#431C2212yes—
28#2857C2211yescharacteristic
29#2855C2212yes—
30#2563—21no—
31#2563—22no—
32#2563—21no—
33#2563C2214yes—
34#2563C2212yes—
35#2563C2212yes—
36#2552C2211yescharacteristic, complemented
37#2547C2211yescharacteristic, complemented
38#2496—21no—
39#2486—21no—
40#2478—22no—
41#2453C2212yescomplemented
42#2447C2212yescomplemented
43#2419—22no—
44C2×#393—22no—
45#2365C2211yescharacteristic, complemented
46D4×C4⋊C4—21no—
47D4×C4⋊C4C2211yescharacteristic, complemented
48#2344—21no—
49C4×#121C2211yescharacteristic
50C2×#390C2212yes—
51C2×#382C2212yes—
52C2×#380—21no—
53C2×#380—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yescomplemented
56#2097C2214yescomplemented
57#2085C2214yescomplemented
58#2074C2214yescomplemented
59#2065C2214yes—
60#2058C2212yescomplemented
61#2055C2214yes—
62#2025C2214yes—
63#2024C2214yescomplemented
64#2023C2212yescomplemented
65#2012C2214yes—
66#1991C2214yescomplemented
67#1990C2212yescomplemented
68#1987C2212yescomplemented
69#1968C2214yes—
70#1958C2214yescomplemented
71#1939C2212yescomplemented
72#1934C2214yes—
73#1916C2214yescomplemented
74#1880C2214yes—
75#1879C2214yes—
76#1879C2212yes—
77#1877C2214yes—
78#1874C2211yescharacteristic
79#1874C2212yes—
80#1872C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1862C2212yes—
84#1853C2212yes—
85#1852C2211yescharacteristic
86#1817C2212yes—
87#1805C2214yescomplemented
88#1804C2214yescomplemented
89#1802C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2314yes—
99C22×D8C2312yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106#529—28no—
107#529—44no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×C22⋊Q8C2312yes—
113C2×#121C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×D4—22no—
116C2×C4×D4—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—44no—
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×#117—22no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#487—44no—
137#476—44no—
138#461—44no—
139#450—44no—
140#447—24no—
141C4×D8—44no—
142#431—44no—
143C2×#107C2312yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2314yes—
161#400—22no—
162#398—22no—
163#398—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#393—24no—
168#393—24no—
169#393—24no—
170#393—24no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#390—22no—
175#390—22no—
176#390C2314yes—
177#390C2314yes—
178#390C2311yescharacteristic
179#389—21no—
180#389—22no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2311yescharacteristic
185#389C2312yes—
186#389C2312yes—
187#388—21no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—21no—
193#385—22no—
194#385—22no—
195#385—22no—
196#384—21no—
197#384C2311yescharacteristic
198#382—22no—
199#382—22no—
200#382C2314yes—
201#382C2314yes—
202#381—22no—
203#381—22no—
204#380—21no—
205#380—21no—
206#380—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—24no—
211#380—24no—
212#380—21no—
213#380—21no—
214#380—21no—
215#380C2311yescharacteristic
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C2×#95—22no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2412yes—
246C22×D4—21no—
247C22×D4—24no—
248C22×D4—21no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—42no—
252C22×D4C2411yescharacteristic
253C22×D4C2412yes—
254C23×C4—22no—
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—44no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D412yes—
267C23×C4C2411yescharacteristic
268C2×SD16—28no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×D8—24no—
272C2×D8—24no—
273C2×D8—42no—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C4⋊Q8—24no—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283#126—48no—
284#125—24no—
285#125—42no—
286#124—24no—
287#124—42no—
288#123—48no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291#121—22no—
292#121#4114yes—
293C4×Q8—24no—
294C4×Q8—42no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—42no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—48no—
305C4×D4—44no—
306#117—48no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—42no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—84no—
357C2×C22⋊C4—84no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369C2×C42C2412yes—
370C2×C42C2411yescharacteristic
371#107—42no—
372#106—44no—
373C4⋊C8—24no—
374C4⋊C8—44no—
375#103—44no—
376#103—44no—
377#103—44no—
378#102—24no—
379#102—24no—
380#102—42no—
381#102—42no—
382#102—44no—
383#102—44no—
384C22⋊C8—22no—
385C22⋊C8—22no—
386C22⋊C8—24no—
387C4×C8—24no—
388C4×C8—44no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95C2×D412yes—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408C24—24no—
409C24—21no—
410C24—22no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24C22×D411yescharacteristic
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—816no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—48no—
440C2×D4—48no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4C2×#4112yes—
446C2×D42−1+414yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—82no—
485C22×C4—816no—
486C22×C4#13514yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C2×#4112yes—
497C22×C4H(5,𝔽2)12yes—
498C22×C4C2511yescharacteristic
499SD16—88no—
500D8—48no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4C2×#4112yes—
541C22⋊C4—24no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—81no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—22no—
557C42—22no—
558C42—24no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—44no—
566C42—42no—
567C42H(5,𝔽2)14yes—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—22no—
573C23—24no—
574C23—48no—
575C23—48no—
576C23—48no—
577C23—42no—
578C23—41no—
579C23—42no—
580C23—41no—
581C23—42no—
582C23—41no—
583C23—41no—
584C23—42no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—41no—
589C23—84no—
590C23—84no—
591C23—84no—
592C23—84no—
593C23#54612yes—
594C23C2×#13512yes—
595C23#57712yes—
596C23C23×D411yescharacteristic
597Q8—44no—
598Q8—44no—
599Q8—88no—
600Q8—88no—
601D4—24no—
602D4—24no—
603D4—24no—
604D4—48no—
605D4—48no—
606D4—42no—
607D4—42no—
608D4—42no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—44no—
635C2×C4—48no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—88no—
657C2×C4—84no—
658C2×C4—88no—
659C2×C4—84no—
660C2×C4—88no—
661C2×C4—84no—
662C2×C4—88no—
663C2×C4#53512yes—
664C2×C4#53512yes—
665C2×C4#54612yes—
666C2×C4C2×#13512yes—
667C2×C4C2×#13512yes—
668C2×C4#57512yes—
669C2×C4C22×#4111yescharacteristic
670C2×C4C2×H(5,𝔽2)12yes—
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+412yes—
673C8—42no—
674C8—42no—
675C8—44no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—44no—
687C22—42no—
688C22—816no—
689C22—82no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—82no—
694C22—81no—
695C22—82no—
696C22—82no—
697C22#330314yes—
698C22C2×#53511yescharacteristic
699C22C2×#54611yescharacteristic
700C22#348611yescharacteristic
701C22C22×#13511yescharacteristic
702C22C2×#57511yescharacteristic
703C22C2×#57711yescharacteristic
704C22#358511yescharacteristic
705C4—22no—
706C4—22no—
707C4—22no—
708C4—41no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—81no—
715C4—82no—
716C4—81no—
717C4—84no—
718C4—82no—
719C4—82no—
720C4#321114yes—
721C4#352012yes—
722C4#353612yes—
723C2—24no—
724C2—82no—
725C2—84no—
726C2#3890511yescharacteristic
727C2C2×#321111yescharacteristic
728C2#6081211yescharacteristic
729C2C2×#330311yescharacteristic
730C2#6116611yescharacteristic
731C2#6181911yescharacteristic
732C2#6184111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#447C22
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. #36116C2semidirect
2. C2×#2563C2semidirect
3. #32905C2semidirect
4. #31645C2semidirect
5. #31529C2semidirect
6. #29913C2semidirect
7. #28398C2semidirect
8. #27814C2semidirect
9. #25902C2semidirect
10. #25319C2semidirect
11. #24812C2semidirect
12. #24501C2semidirect
13. #24299C2semidirect
15. #22599C2semidirect
16. #21861C2semidirect
21. C2×#487C22semidirect
22. C2×#476C22semidirect
36. #2552C22semidirect
37. #2547C22semidirect
41. #2453C22semidirect
42. #2447C22semidirect
45. #2365C22semidirect
47. D4×C4⋊C4C22semidirect
55. #2101C22semidirect
56. #2097C22semidirect
57. #2085C22semidirect
58. #2074C22semidirect
60. #2058C22semidirect
63. #2024C22semidirect
64. #2023C22semidirect
66. #1991C22semidirect
67. #1990C22semidirect
68. #1987C22semidirect
70. #1958C22semidirect
71. #1939C22semidirect
73. #1916C22semidirect
87. #1805C22semidirect
88. #1804C22semidirect
92. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#361162
C2×#25632
#329052
#316452
#315292
#299132
#283982
#278142
#259022
#253192
#248122
#245012
#242992
#239362
#225992
#218612
C4×#4472