Finite Group Databasebeta

← #6802506

Subgroups — #6802506

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36586C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3#32905C211yesmaximal, characteristic, complemented
4#32171C214yesmaximal, complemented
5#30738C214yesmaximal, complemented
6#29719C212yesmaximal, complemented
7#28459C212yesmaximal, complemented
8#27748C212yesmaximal, complemented
9#25724C212yesmaximal
10#25722C211yesmaximal, characteristic, complemented
11#25365C212yesmaximal, complemented
12#24627C212yesmaximal, complemented
13#24217C212yesmaximal, complemented
14#24209C211yesmaximal, characteristic, complemented
15#22625C211yesmaximal, characteristic, complemented
16#22044C212yesmaximal, complemented
17C4×#447C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#495C2214yes—
22C2×#494C2212yescomplemented
23C2×#464C2214yes—
24C2×#461C2212yes—
25C2×#461C2212yescomplemented
26C2×#451C2212yes—
27C2×#451C2212yes—
28C2×#448C2212yes—
29C2×#450C2212yes—
30C2×#447C2211yescharacteristic
31C2×C4×D8C2212yes—
32#2857C2211yescharacteristic
33#2850C2212yes—
34#2596—22no—
35#2571C2211yescharacteristic, complemented
36#2547—22no—
37#2547—21no—
38#2547—21no—
39#2547C2214yes—
40#2547C2212yes—
41#2547C2212yes—
42#2547C2211yescharacteristic, complemented
43#2514C2212yescomplemented
44#2510—22no—
45#2494—21no—
46#2487—21no—
47#2448C2212yescomplemented
48C2×#397—22no—
49#2365C2211yescharacteristic, complemented
50#2354C2211yescharacteristic, complemented
51#2352—21no—
52#2347—21no—
53C4×C22⋊Q8C2212yes—
54C4×#121C2211yescharacteristic
55C2×#391C2212yes—
56C2×#380—21no—
57C2×#380—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2074C2214yescomplemented
60#2056C2214yescomplemented
61#2055C2214yescomplemented
62#2043C2212yescomplemented
63#2042C2214yes—
64#2016C2214yescomplemented
65#2015C2214yes—
66#2015C2214yes—
67#2015C2214yescomplemented
68#2013C2212yescomplemented
69#2012C2214yes—
70#1983C2214yes—
71#1981C2212yescomplemented
72#1969C2214yescomplemented
73#1958C2214yes—
74#1956C2212yescomplemented
75#1939C2212yescomplemented
76#1933C2214yescomplemented
77#1928C2214yescomplemented
78#1916C2214yescomplemented
79#1879C2212yes—
80#1878C2214yescomplemented
81#1877C2214yes—
82#1874C2211yescharacteristic
83#1872C2212yes—
84#1862C2212yes—
85#1852C2211yescharacteristic
86#1802C2212yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1786C2211yescharacteristic
91C8×C4⋊C4C2212yescomplemented
92C4×#102C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×Q16C2314yes—
99C22×Q16C2314yes—
100C22×SD16C2314yes—
101C22×SD16C2314yes—
102C22×D8C2312yes—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#124—22no—
110C2×#124—22no—
111C2×#124—48no—
112C2×#124C2314yes—
113C2×#124C2314yes—
114C2×#124C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—44no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C22⋊C4—21no—
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#495—48no—
137#494—44no—
138#464—48no—
139#461—44no—
140#461—44no—
141#451—28no—
142#451—44no—
143#450—44no—
144#448—44no—
145#447—24no—
146C4×D8—44no—
147C2×#107C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2311yescharacteristic
163C2×C22⋊C8C2311yescharacteristic
164C2×C4×C8C2312yes—
165C2×C4×C8C2314yes—
166#401—22no—
167#400—22no—
168#400—22no—
169#399—22no—
170#397—22no—
171#397—24no—
172#397—24no—
173#397—24no—
174#397—24no—
175#397—22no—
176#391—22no—
177#391—22no—
178#391C2314yes—
179#391C2312yes—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#390C2312yes—
184#390C2311yescharacteristic
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389C2312yes—
190#389C2311yescharacteristic
191#388—22no—
192#388—22no—
193#388—22no—
194#388—21no—
195#388—22no—
196#388—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#386—22no—
201#386—22no—
202#386—21no—
203#385—22no—
204#384C2311yescharacteristic
205#383—22no—
206#382—21no—
207#382C2312yes—
208#380—21no—
209#380—21no—
210#380—21no—
211#380—21no—
212#380—21no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380—22no—
217#380—24no—
218#380—24no—
219#380C2311yescharacteristic
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D412yes—
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—21no—
232C2×#95—22no—
233C2×#95—22no—
234C2×#95D412yes—
235#340C2314yes—
236#340C2312yes—
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×Q8C2412yes—
249C22×Q8C2414yes—
250C22×D4—24no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—42no—
256C22×D4C2411yescharacteristic
257C22×D4C2412yes—
258C23×C4—22no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—44no—
264C23×C4—44no—
265C23×C4—44no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×Q16—28no—
272C2×Q16—44no—
273C2×Q16—44no—
274C2×Q16—44no—
275C2×Q16—44no—
276C2×Q16—44no—
277C2×SD16—28no—
278C2×SD16—44no—
279C2×SD16—44no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16—44no—
283C2×D8—24no—
284C2×D8—24no—
285C2×D8—42no—
286C22×C8C2414yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295#124—28no—
296#124—28no—
297#124—44no—
298#124—44no—
299#124—44no—
300#124—48no—
301#124—48no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304#121—22no—
305#121—22no—
306C4×Q8—44no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—42no—
311C4×D4—42no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—44no—
315#117—44no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—42no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—44no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—44no—
362C2×C22⋊C4—48no—
363C2×C22⋊C4—48no—
364C2×C42—24no—
365C2×C42—22no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42—44no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377#107—42no—
378C4⋊C8—24no—
379C4⋊C8—44no—
380#103—28no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385#103—44no—
386#102—24no—
387#102—24no—
388#102—42no—
389#102—42no—
390#102—44no—
391C22⋊C8—24no—
392C22⋊C8—22no—
393C22⋊C8#4114yes—
394C4×C8—24no—
395C4×C8—44no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—24no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—21no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—21no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95C2×D412yes—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417C24—24no—
418C24—22no—
419C24—21no—
420C24—42no—
421C24—42no—
422C24—42no—
423C24—42no—
424C24—42no—
425C24—42no—
426C24C22×D411yescharacteristic
427C2×Q8—24no—
428C2×Q8—22no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—24no—
436C2×Q8—24no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—44no—
440C2×Q8—88no—
441C2×Q8—88no—
442C2×D4—24no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—48no—
453C2×D4—48no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4H(5,𝔽2)14yes—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—44no—
487C22×C4—42no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—816no—
496C22×C4#13514yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C4C2511yescharacteristic
509Q16—88no—
510Q16—88no—
511SD16—88no—
512SD16—88no—
513D8—48no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—24no—
522C2×C8C2×#4114yes—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—21no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—81no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—48no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—42no—
558C22⋊C4—44no—
559C22⋊C4—42no—
560C22⋊C4—42no—
561C22⋊C4—42no—
562C22⋊C4—81no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—22no—
567C42—22no—
568C42—24no—
569C42—24no—
570C42—22no—
571C42—42no—
572C42—42no—
573C42—44no—
574C42—42no—
575C42—82no—
576C42H(5,𝔽2)14yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—24no—
582C23—48no—
583C23—48no—
584C23—48no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23—84no—
602C23#55012yes—
603C23C2×#13512yes—
604C23#57812yes—
605C23C23×D411yescharacteristic
606Q8—44no—
607Q8—44no—
608Q8—44no—
609Q8—44no—
610Q8—44no—
611Q8—44no—
612Q8—88no—
613D4—24no—
614D4—24no—
615D4—48no—
616D4—42no—
617D4—42no—
618D4—42no—
619D4—42no—
620D4—84no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—42no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—88no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4—84no—
671C2×C4—88no—
672C2×C4—84no—
673C2×C4—88no—
674C2×C4#53512yes—
675C2×C4#53512yes—
676C2×C4#55012yes—
677C2×C4C2×#13512yes—
678C2×C4C2×#13512yes—
679C2×C4#57512yes—
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×H(5,𝔽2)12yes—
682C2×C4C2×H(5,𝔽2)12yes—
683C2×C4C2×H(5,𝔽2)11yescharacteristic
684C8—24no—
685C8—44no—
686C8—42no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—816no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—81no—
702C22—81no—
703C22—81no—
704C22—81no—
705C22—81no—
706C22—81no—
707C22—81no—
708C22#339314yes—
709C22C2×#53511yescharacteristic
710C22C2×#55011yescharacteristic
711C22#348411yescharacteristic
712C22C22×#13511yescharacteristic
713C22C2×#57511yescharacteristic
714C22C2×#57811yescharacteristic
715C22#358611yescharacteristic
716C4—22no—
717C4—22no—
718C4—22no—
719C4—42no—
720C4—42no—
721C4—41no—
722C4—42no—
723C4—42no—
724C4—81no—
725C4—82no—
726C4—81no—
727C4—81no—
728C4—82no—
729C4—84no—
730C4—82no—
731C4#321114yes—
732C4#351912yes—
733C4#353712yes—
734C2—24no—
735C2—84no—
736C2—161no—
737C2#3913211yescharacteristic
738C2C2×#321111yescharacteristic
739C2#6081411yescharacteristic
740C2C2×#339311yescharacteristic
741C2#6140311yescharacteristic
742C2#6156511yescharacteristic
743C2#6159511yescharacteristic
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. #36586C2semidirect
2. C2×#2547C2semidirect
3. #32905C2semidirect
4. #32171C2semidirect
5. #30738C2semidirect
6. #29719C2semidirect
7. #28459C2semidirect
8. #27748C2semidirect
10. #25722C2semidirect
11. #25365C2semidirect
12. #24627C2semidirect
13. #24217C2semidirect
14. #24209C2semidirect
15. #22625C2semidirect
16. #22044C2semidirect
22. C2×#494C22semidirect
25. C2×#461C22semidirect
35. #2571C22semidirect
42. #2547C22semidirect
43. #2514C22semidirect
47. #2448C22semidirect
49. #2365C22semidirect
50. #2354C22semidirect
59. #2074C22semidirect
60. #2056C22semidirect
61. #2055C22semidirect
62. #2043C22semidirect
64. #2016C22semidirect
67. #2015C22semidirect
68. #2013C22semidirect
71. #1981C22semidirect
72. #1969C22semidirect
74. #1956C22semidirect
75. #1939C22semidirect
76. #1933C22semidirect
77. #1928C22semidirect
78. #1916C22semidirect
80. #1878C22semidirect
91. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#365862
C2×#25472
#329052
#321712
#307382
#297192
#284592
#277482
#257242
#257222
#253652
#246272
#242172
#242092
#226252
#220442
C4×#4472