Finite Group Databasebeta

← #6500763

Subgroups — #6500763

Order 512.

2538 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35512C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27595C212yesmaximal, complemented
4#27591C212yesmaximal, complemented
5#24878C214yesmaximal, complemented
6#24877C214yesmaximal, complemented
7#24864C214yesmaximal, complemented
8#24861C214yesmaximal, complemented
9#24007C212yesmaximal, complemented
10#24002C212yesmaximal, complemented
11#21947C212yesmaximal, complemented
12#21744C211yesmaximal, characteristic, complemented
13#20602C212yesmaximal, complemented
14C22×H(5,𝔽2)C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C2×#525C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C2×#484C2212yescomplemented
19C2×#481C2212yescomplemented
20C2×#480C2214yescomplemented
21C2×#417C2212yescomplemented
22C2×#418C2212yescomplemented
23C2×#416C2212yescomplemented
24C22×#102C2212yescomplemented
25C2×#408C2212yes—
26#2408—22no—
27#2408C2214yes—
28#2407—22no—
29#2407C2214yes—
30#2405—22no—
31#2403—22no—
32#2376—22no—
33#2358—22no—
34#2352C2214yescomplemented
35D4×C4⋊C4C2214yescomplemented
36#2329—22no—
37#2325—22no—
38#2314C2212yescomplemented
39#2311C2211yescharacteristic
40#2311C2212yes—
41#2311C2211yescharacteristic
42#2309C2211yescharacteristic
43C2×#380C2212yescomplemented
44#2095C2212yescomplemented
45#2093C2212yescomplemented
46#2092C2214yescomplemented
47#2074C2212yescomplemented
48#2073C2212yescomplemented
49#2071C2214yescomplemented
50#2035C2214yescomplemented
51#2033C2214yescomplemented
52#2033C2214yescomplemented
53#2033C2214yescomplemented
54#1928C2214yescomplemented
55#1926C2214yescomplemented
56#1925C2214yescomplemented
57#1925C2214yescomplemented
58#1874C2212yescomplemented
59#1874C2214yescomplemented
60#1872C2212yescomplemented
61#1870C2214yescomplemented
62#1869C2214yescomplemented
63#1854C2214yescomplemented
64#1854C2214yescomplemented
65#1852C2214yescomplemented
66#1852C2214yescomplemented
67#1818C2212yescomplemented
68#1809C2214yescomplemented
69#1808C2214yescomplemented
70#1788C2214yes—
71#1788C2214yescomplemented
72#1786C2214yes—
73#1786C2214yescomplemented
74#1779C2212yescomplemented
75C2×H(5,𝔽2)—28no—
76C2×H(5,𝔽2)—44no—
77C2×H(5,𝔽2)—42no—
78C2×H(5,𝔽2)—42no—
79C22×#41—21no—
80C22×#41C2311yescharacteristic
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C23×D4—21no—
84C23×D4—22no—
85C23×D4—21no—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C22×M16C2312yes—
90C23×C8C2312yes—
91#525—28no—
92#525—42no—
93#525—42no—
94C2×#123—21no—
95C2×#123D412yes—
96C2×C22⋊Q8D412yes—
97C2×C22⋊Q8C2312yes—
98C2×#121—22no—
99C2×#121—44no—
100C2×#121D412yes—
101C2×#121C2314yes—
102C2×#121C2312yes—
103C2×#120—21no—
104C2×#120—24no—
105C2×#120D412yes—
106C2×C4×D4—22no—
107C2×C4×D4—24no—
108C2×C4×D4C2314yes—
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—22no—
118#484—44no—
119#481—44no—
120#480—48no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125#418—44no—
126#417—44no—
127#416—44no—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102—44no—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137#408—28no—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143#394—24no—
144#394—24no—
145#392—24no—
146#392—24no—
147#391—22no—
148#391—21no—
149#391C2312yes—
150#390—22no—
151#390—21no—
152#390—22no—
153#390C2314yes—
154#390C2312yes—
155#389—24no—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#386—22no—
161#386—22no—
162#386—22no—
163#386—24no—
164#386—22no—
165#386—24no—
166#385—24no—
167#385—24no—
168#385—24no—
169#384C2312yes—
170#382C2312yes—
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176#380C2314yes—
177#380C2314yes—
178#379—22no—
179#379—24no—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—24no—
182C2×#95—22no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C25—21no—
191C25—22no—
192C25C2×D411yescharacteristic
193H(5,𝔽2)—432no—
194H(5,𝔽2)—432no—
195C2×#41—22no—
196C2×#41—22no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—24no—
202C2×#41—42no—
203C2×#41—41no—
204C2×#41—41no—
205C2×#41—42no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—44no—
209C2×#41—44no—
210C2×#41—44no—
211C2×#41#4118yes—
212C22×Q8C2×D411yescharacteristic
213C22×Q8C2411yescharacteristic
214C22×D4—24no—
215C22×D4—24no—
216C22×D4—21no—
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—24no—
221C22×D4—21no—
222C22×D4—22no—
223C22×D4—24no—
224C22×D4—24no—
225C22×D4—24no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—41no—
229C22×D4—41no—
230C22×D4—44no—
231C22×D4—48no—
232C22×D4—44no—
233C22×D4—48no—
234C22×D4—48no—
235C22×D4—44no—
236C22×D4—44no—
237C22×D4—84no—
238C22×D4C2×D414yes—
239C22×D4C2×D412yes—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—24no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—42no—
251C23×C4—41no—
252C23×C4—41no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C2×M16—24no—
259C2×M16—42no—
260C2×M16—42no—
261C22×C8—24no—
262C22×C8—42no—
263C22×C8—42no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266#123—28no—
267#123—44no—
268C22⋊Q8—28no—
269C22⋊Q8—24no—
270C22⋊Q8—42no—
271#121—28no—
272#121—28no—
273#121—24no—
274#121—44no—
275#121—42no—
276#121—48no—
277#120—28no—
278#120—44no—
279#120—88no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—48no—
283C4×D4—88no—
284#117—24no—
285#117—41no—
286#117—41no—
287#117—42no—
288#117—42no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D414yes—
303C2×C4⋊C4C2414yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—84no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D414yes—
323C2×C22⋊C4C2×D414yes—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—24no—
327C2×C42C2412yes—
328#107—44no—
329#107—44no—
330#107—44no—
331#106—44no—
332#103—28no—
333#103—44no—
334#102—28no—
335#102—28no—
336#102—28no—
337#102—44no—
338#102—44no—
339#102—44no—
340C22⋊C8—28no—
341C22⋊C8—28no—
342C22⋊C8—44no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352C24—24no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—41no—
358C24—42no—
359C24—41no—
360C24—44no—
361C24—44no—
362C24—42no—
363C24—44no—
364C24—42no—
365C24—82no—
366C24—82no—
367C24—84no—
368C24—84no—
369C24C22×D411yescharacteristic
370C24C22×D412yes—
371C24C22×D412yes—
372#41—216no—
373#41—48no—
374#41—48no—
375#41—48no—
376#41—48no—
377#41—48no—
378#41—48no—
379#41—48no—
380#41—432no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8#12314yes—
387C2×Q8H(5,𝔽2)12yes—
388C2×D4—216no—
389C2×D4—24no—
390C2×D4—24no—
391C2×D4—28no—
392C2×D4—28no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—41no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—42no—
409C2×D4—44no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—42no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—432no—
418C2×D4—464no—
419C2×D4—42no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—48no—
423C2×D4—44no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—82no—
428C2×D4—84no—
429C2×D4—816no—
430C2×D4#12318yes—
431C2×D4#12314yes—
432C2×D4H(5,𝔽2)12yes—
433C2×D4H(5,𝔽2)14yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—44no—
454C22×C4—41no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—41no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—84no—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2511yescharacteristic
482M16—84no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—84no—
490C4⋊C4—24no—
491C4⋊C4—28no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—44no—
500C4⋊C4—42no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C4—84no—
505C22⋊C4—24no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—84no—
517C22⋊C4—82no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—42no—
522C42—82no—
523C42—84no—
524C23—22no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—22no—
532C23—21no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—44no—
541C23—48no—
542C23—416no—
543C23—48no—
544C23—44no—
545C23—42no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—42no—
551C23—44no—
552C23—82no—
553C23—84no—
554C23—82no—
555C23—84no—
556C23—84no—
557C23—168no—
558C23—164no—
559C23—168no—
560C23—164no—
561C23—164no—
562C23—164no—
563C23C2×#12312yes—
564C23C2×#12314yes—
565C23#57614yes—
566C23C23×D411yescharacteristic
567Q8—24no—
568Q8—44no—
569Q8—42no—
570D4—24no—
571D4—24no—
572D4—24no—
573D4—44no—
574D4—48no—
575D4—48no—
576D4—416no—
577D4—416no—
578D4—48no—
579D4—416no—
580D4—42no—
581D4—42no—
582D4—42no—
583D4—88no—
584D4—816no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—84no—
620C2×C4—82no—
621C2×C4—84no—
622C2×C4—164no—
623C2×C4—164no—
624C2×C4—164no—
625C2×C4C2×#12314yes—
626C2×C4C2×#12312yes—
627C2×C4C22×#4112yes—
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×H(5,𝔽2)12yes—
631C8—44no—
632C8—44no—
633C22—22no—
634C22—24no—
635C22—24no—
636C22—28no—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—44no—
645C22—41no—
646C22—42no—
647C22—42no—
648C22—41no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—88no—
656C22—84no—
657C22—82no—
658C22—84no—
659C22—81no—
660C22—81no—
661C22—168no—
662C22—1616no—
663C22C22×#12311yescharacteristic
664C22#348411yescharacteristic
665C22#348411yescharacteristic
666C22C2×#57612yes—
667C22#358612yes—
668C4—22no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—84no—
674C4—82no—
675C4—82no—
676C4—84no—
677C4#349012yes—
678C4#349012yes—
679C2—44no—
680C2—42no—
681C2—44no—
682C2—82no—
683C2—84no—
684C2#3535011yescharacteristic
685C2#6052112yes—
686C2#6162412yes—
687C2#6163011yescharacteristic
688C2#6164911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35512C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34360C2semidirect
3. #27595C2semidirect
4. #27591C2semidirect
5. #24878C2semidirect
6. #24877C2semidirect
7. #24864C2semidirect
8. #24861C2semidirect
9. #24007C2semidirect
10. #24002C2semidirect
11. #21947C2semidirect
12. #21744C2semidirect
13. #20602C2semidirect
18. C2×#484C22semidirect
19. C2×#481C22semidirect
20. C2×#480C22semidirect
21. C2×#417C22semidirect
22. C2×#418C22semidirect
23. C2×#416C22semidirect
24. C22×#102C22semidirect
34. #2352C22semidirect
35. D4×C4⋊C4C22semidirect
38. #2314C22semidirect
43. C2×#380C22semidirect
44. #2095C22semidirect
45. #2093C22semidirect
46. #2092C22semidirect
47. #2074C22semidirect
48. #2073C22semidirect
49. #2071C22semidirect
50. #2035C22semidirect
51. #2033C22semidirect
52. #2033C22semidirect
53. #2033C22semidirect
54. #1928C22semidirect
55. #1926C22semidirect
56. #1925C22semidirect
57. #1925C22semidirect
58. #1874C22semidirect
59. #1874C22semidirect
60. #1872C22semidirect
61. #1870C22semidirect
62. #1869C22semidirect
63. #1854C22semidirect
64. #1854C22semidirect
65. #1852C22semidirect
66. #1852C22semidirect
67. #1818C22semidirect
68. #1809C22semidirect
69. #1808C22semidirect
71. #1788C22semidirect
73. #1786C22semidirect
74. #1779C22semidirect

Maximal subgroups

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

subgroupindex
#355122
#343602
#275952
#275912
#248782
#248772
#248642
#248612
#240072
#240022
#219472
#217442
#206022