Finite Group Databasebeta

← #7506779

Subgroups — #7506779

Order 512.

1813 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic, complemented
2#35530C211yesmaximal, characteristic
3#31960C214yesmaximal, complemented
4#31955C214yesmaximal, complemented
5#31936C214yesmaximal, complemented
6#31931C214yesmaximal, complemented
7C2×#2095C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23627C211yesmaximal, characteristic, complemented
10#23611C211yesmaximal, characteristic, complemented
11#23488C211yesmaximal, characteristic
12#23394C211yesmaximal, characteristic
13#21237C211yesmaximal, characteristic, complemented
14#21048C211yesmaximal, characteristic, complemented
15#20930C211yesmaximal, characteristic, complemented
16#20913C211yesmaximal, characteristic, complemented
17#20847C211yesmaximal, characteristic, complemented
18#20764C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×#537C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38#2502C2212yes—
39#2501C2212yes—
40#2500C2212yes—
41#2499C2212yes—
42#2498—24no—
43#2448—22no—
44#2446—22no—
45#2443—24no—
46#2423—22no—
47#2423C2214yes—
48#2422—22no—
49#2421C2214yes—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2311C2211yescharacteristic
54#2297C2211yescharacteristic
55#2297C2211yescharacteristic
56C2×#391C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#384C2211yescharacteristic
59#2095C2214yes—
60#2095C2214yes—
61#2085C2214yes—
62#2083C2214yes—
63#2074C2212yes—
64#2073C2212yes—
65#2072C2212yes—
66#2071C2212yes—
67#2068C2214yes—
68#2067C2214yes—
69#2036C2214yes—
70#2033C2214yes—
71#2001C2214yes—
72#2000C2214yes—
73#1973C2212yes—
74#1971C2212yes—
75#1969C2212yes—
76#1968C2212yes—
77#1924C2214yes—
78#1923C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1873C2214yes—
82#1872C2214yes—
83#1854C2214yes—
84#1852C2214yes—
85#1836C2214yes—
86#1834C2214yes—
87#1805C2214yes—
88#1803C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×#41C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—42no—
103C22×Q16C2312yes—
104C22×SD16C2312yes—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#537—42no—
110#537—42no—
111#536—24no—
112#536—42no—
113C2×C4⋊Q8—21no—
114C2×#124—21no—
115C2×#123—22no—
116C2×#123—24no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2314yes—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—22no—
122C2×#121C2312yes—
123#520—42no—
124#520—42no—
125#519—24no—
126#519—42no—
127C2×C4×Q8—21no—
128C2×C4×Q8—22no—
129C2×C4×Q8C2314yes—
130C2×C4×D4—21no—
131C2×C4×D4—22no—
132C2×#117—22no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—22no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C42C2311yescharacteristic
144#428—42no—
145C2×#107—42no—
146C2×#107—42no—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#107C2314yes—
150C2×#106C2312yes—
151#424—42no—
152C2×C4⋊C8C2314yes—
153#419—42no—
154#417—42no—
155#416—42no—
156#416—42no—
157C2×#103—42no—
158C2×#103—42no—
159C2×#103C2312yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#102C2312yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169#408—24no—
170#408—42no—
171C2×C22⋊C8C2314yes—
172C2×C22⋊C8C2314yes—
173C4×M16—24no—
174C2×#97C2312yes—
175C2×C4×C8C2312yes—
176#400—24no—
177#399—24no—
178#399—24no—
179#398—24no—
180#398—24no—
181#397—24no—
182#397—24no—
183#394—24no—
184#391—21no—
185#391—22no—
186#391D412yes—
187#391C2312yes—
188#391C2311yescharacteristic
189#391C2311yescharacteristic
190#391C2312yes—
191#391C2312yes—
192#390—21no—
193#390D412yes—
194#390C2312yes—
195#390C2311yescharacteristic
196#390C2311yescharacteristic
197#389—22no—
198#389—22no—
199#389C2314yes—
200#388—24no—
201#388—24no—
202#388—24no—
203#387—24no—
204#387—24no—
205#387—24no—
206#386—22no—
207#386—22no—
208#386—22no—
209#386D414yes—
210#385—22no—
211#385—24no—
212#384—22no—
213#384C2312yes—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#382C2312yes—
218#382C2312yes—
219#380—22no—
220#380—22no—
221#380—22no—
222#380C2314yes—
223C4×C4⋊C4—22no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4C2312yes—
226C4×C22⋊C4C2312yes—
227C2×#95—22no—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8C2×D411yescharacteristic
258C22×Q8#4114yes—
259C22×Q8C2411yescharacteristic
260C22×Q8C2412yes—
261C22×D4—21no—
262C22×D4—22no—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×D412yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275#137—416no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16#4114yes—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—44no—
299#124—44no—
300#123—48no—
301#123—88no—
302C22⋊Q8—28no—
303C22⋊Q8—44no—
304C22⋊Q8—42no—
305C22⋊Q8—42no—
306C22⋊Q8—44no—
307#121—42no—
308#121—42no—
309#121—48no—
310#121—44no—
311C4×Q8—28no—
312C4×Q8—44no—
313C4×Q8—48no—
314C4×Q8—44no—
315C4×D4—48no—
316C4×D4—44no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—48no—
326#117—42no—
327#117—42no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—42no—
348C2×C4⋊C4—42no—
349C2×C4⋊C4—84no—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4C2×D412yes—
370C2×C22⋊C4C2×D414yes—
371C2×C22⋊C4C2×D414yes—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—24no—
375C2×C42—21no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383#107—44no—
384#107—44no—
385#107—42no—
386#106—42no—
387C4⋊C8—44no—
388#103—44no—
389#103—44no—
390#103—42no—
391#103—42no—
392#103—42no—
393#103—42no—
394#103—42no—
395#102—42no—
396#102—42no—
397#102—42no—
398C22⋊C8—28no—
399C22⋊C8—44no—
400#97—24no—
401C4×C8—24no—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95C2×D414yes—
413#95C2×D414yes—
414C24—21no—
415C24C22×D412yes—
416C24C22×D411yescharacteristic
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423#41—44no—
424#41—44no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—22no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—44no—
446C2×Q8C2×#4114yes—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—44no—
463C2×D4—84no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—44no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—84no—
495C22×C4—82no—
496C22×C4C22×D412yes—
497C22×C4C22×D414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C22×C4C2511yescharacteristic
508Q16—48no—
509SD16—48no—
510M16—28no—
511M16—44no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—44no—
524C2×C8—44no—
525C2×C8C2×#4114yes—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—44no—
544C4⋊C4—48no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—82no—
551C4⋊C4—84no—
552C4⋊C4—84no—
553C4⋊C4—84no—
554C22⋊C4—24no—
555C22⋊C4—28no—
556C22⋊C4—44no—
557C22⋊C4—48no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—42no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—44no—
566C22⋊C4—84no—
567C22⋊C4—84no—
568C42—24no—
569C42—21no—
570C42—21no—
571C42—22no—
572C42—22no—
573C42—21no—
574C42—21no—
575C42—48no—
576C42—44no—
577C42—42no—
578C42—42no—
579C42—42no—
580C23—22no—
581C23—22no—
582C23—24no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—24no—
587C23—42no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—42no—
592C23#53512yes—
593C23#54612yes—
594C23#54712yes—
595C23C23×D411yescharacteristic
596Q8—24no—
597Q8—24no—
598Q8—42no—
599Q8—48no—
600Q8—44no—
601Q8—42no—
602Q8—41no—
603Q8—41no—
604Q8—41no—
605Q8—41no—
606D4—48no—
607D4—41no—
608D4—41no—
609D4—41no—
610D4—44no—
611D4—41no—
612D4—84no—
613D4—164no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—48no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—162no—
655C2×C4—162no—
656C2×C4—162no—
657C2×C4—164no—
658C2×C4#53512yes—
659C2×C4#53914yes—
660C2×C4#54014yes—
661C2×C4#54612yes—
662C2×C4#54712yes—
663C2×C4#55114yes—
664C2×C4#56214yes—
665C2×C4#57814yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×H(5,𝔽2)11yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×H(5,𝔽2)11yescharacteristic
671C2×C4C2×2−1+411yescharacteristic
672C8—24no—
673C8—42no—
674C8—42no—
675C8—42no—
676C22—24no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—82no—
686C22—82no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#270614yes—
691C22C2×#53511yescharacteristic
692C22C2×#54611yescharacteristic
693C22C2×#54711yescharacteristic
694C22C2×#57812yes—
695C22#358512yes—
696C4—24no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—81no—
704C4—82no—
705C4—81no—
706C4—82no—
707C4—82no—
708C4—81no—
709C4#270614yes—
710C4C2×#53911yescharacteristic
711C4C2×#54011yescharacteristic
712C4C2×#55111yescharacteristic
713C4C2×#56211yescharacteristic
714C2—24no—
715C2—82no—
716C2—161no—
717C2C2×#270611yescharacteristic
718C2#6082111yescharacteristic
719C2#6082711yescharacteristic
720C2#6120312yes—
721C2#6127312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35673C2semidirect
3. #31960C2semidirect
4. #31955C2semidirect
5. #31936C2semidirect
6. #31931C2semidirect
7. C2×#2095C2semidirect
8. #23660C2semidirect
9. #23627C2semidirect
10. #23611C2semidirect
13. #21237C2semidirect
14. #21048C2semidirect
15. #20930C2semidirect
16. #20913C2semidirect
17. #20847C2semidirect
18. #20764C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356732
#355302
#319602
#319552
#319362
#319312
C2×#20952
#236602
#236272
#236112
#234882
#233942
#212372
#210482
#209302
#209132
#208472
#207642
#206512