Finite Group Databasebeta

← #6461909

Subgroups — #6461909

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35338C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic, complemented
3#29570C214yesmaximal, complemented
4#29566C214yesmaximal, complemented
5#29562C214yesmaximal, complemented
6#29556C214yesmaximal, complemented
7#23551C211yesmaximal, characteristic, complemented
8#23506C211yesmaximal, characteristic, complemented
9#23454C211yesmaximal, characteristic, complemented
10C2×#2012C211yesmaximal, characteristic
11#21233C211yesmaximal, characteristic
12#21231C211yesmaximal, characteristic, complemented
13#21091C211yesmaximal, characteristic, complemented
14#21090C211yesmaximal, characteristic, complemented
15#21066C211yesmaximal, characteristic, complemented
16#21065C211yesmaximal, characteristic, complemented
17#20934C211yesmaximal, characteristic
18#20933C211yesmaximal, characteristic, complemented
19#20368C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C23×D8C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#518C2211yescharacteristic
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#418C2211yescharacteristic
32C2×#418C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×#405C2211yescharacteristic
36C2×#405C2211yescharacteristic
37#2861C2214yescomplemented
38#2640C2214yescomplemented
39#2592—22no—
40#2592—22no—
41#2561—24no—
42#2357—22no—
43#2354—22no—
44#2342—24no—
45#2319C2214yescomplemented
46#2317C2211yescharacteristic
47#2310C2211yescharacteristic
48#2310C2211yescharacteristic
49#2309C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2296C2211yescharacteristic
53#2293C2214yescomplemented
54C2×#390C2211yescharacteristic
55#2274C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2059C2212yescomplemented
58#2059C2212yescomplemented
59#2058C2212yescomplemented
60#2057C2212yescomplemented
61#2042C2214yescomplemented
62#2042C2214yescomplemented
63#2025C2214yescomplemented
64#2023C2214yescomplemented
65#2012C2212yes—
66#2012C2212yes—
67#2012C2212yes—
68#2012C2212yes—
69#1973C2214yescomplemented
70#1972C2214yes—
71#1970C2214yes—
72#1968C2214yescomplemented
73#1936C2214yescomplemented
74#1935C2214yescomplemented
75#1934C2214yescomplemented
76#1933C2214yescomplemented
77#1928C2214yescomplemented
78#1927C2214yescomplemented
79#1926C2214yescomplemented
80#1925C2214yescomplemented
81#1879C2214yes—
82#1879C2214yes—
83#1877C2214yescomplemented
84#1877C2214yescomplemented
85#1746C2212yescomplemented
86#1746C2212yescomplemented
87#1745C2212yescomplemented
88#1744C2212yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1728C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C23×Q8C2311yescharacteristic
98C23×D4C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C22×SD16—24no—
101C22×SD16—24no—
102C22×SD16—42no—
103C22×SD16C2312yes—
104C22×SD16C2312yes—
105C22×D8—24no—
106C22×D8—24no—
107C22×D8—42no—
108C22×D8C2312yes—
109C22×D8C2312yes—
110C22×M16C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C23×C8C2311yescharacteristic
114#533—42no—
115#533—42no—
116C2×C4⋊Q8C2312yes—
117C2×#125C2312yes—
118C2×#124—22no—
119C2×#124—42no—
120C2×#124—42no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—22no—
124#518—24no—
125#518—24no—
126C2×C4×Q8—22no—
127C2×C4×D4—22no—
128C2×C4×D4C2314yes—
129C2×#117—24no—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—22no—
138C22×C22⋊C4—22no—
139C22×C22⋊C4—22no—
140C22×C42C2311yescharacteristic
141#427—42no—
142#427—42no—
143C2×#107C2312yes—
144C2×#107C2312yes—
145C2×#106C2312yes—
146C2×#106C2312yes—
147#419—24no—
148#419—42no—
149#418—24no—
150#418—42no—
151#418—42no—
152#418—42no—
153C2×#103—24no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102—24no—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165#405—24no—
166#405—24no—
167C2×#97C2312yes—
168C2×#97C2312yes—
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#397—24no—
172#397—24no—
173#393—24no—
174#393—24no—
175#391—22no—
176#391—21no—
177#391—21no—
178#391C2314yes—
179#391C2314yes—
180#390—22no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—22no—
186#390C2314yes—
187#390C2314yes—
188#390C2314yes—
189#390C2312yes—
190#390C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2314yes—
194#388—24no—
195#388—24no—
196#388—24no—
197#388—24no—
198#388—24no—
199#388—24no—
200#388—24no—
201#388—24no—
202#385—22no—
203#385—22no—
204#384—21no—
205#382—21no—
206#381—22no—
207#381—22no—
208#381—24no—
209#380—24no—
210#379—22no—
211#379—22no—
212#379—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4D412yes—
216C4×C4⋊C4D412yes—
217C4×C4⋊C4C2314yes—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—24no—
220C4×C22⋊C4D414yes—
221#376C2312yes—
222C43C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2311yescharacteristic
233#336C2312yes—
234#336C2312yes—
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238#336C2312yes—
239C25—21no—
240C25—21no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8C2412yes—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4#4114yes—
264C22×D4C2412yes—
265C22×D4C2412yes—
266C23×C4—24no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—24no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—42no—
276C2×SD16—42no—
277C2×SD16—416no—
278C2×D8—24no—
279C2×D8—24no—
280C2×D8—24no—
281C2×D8—24no—
282C2×D8—42no—
283C2×D8—42no—
284C2×D8—416no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8#4114yes—
296C22×C8#4114yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2411yescharacteristic
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C4⋊Q8—42no—
304C4⋊Q8—42no—
305#125—42no—
306#125—42no—
307#124—42no—
308#124—42no—
309#124—42no—
310#124—42no—
311#124—48no—
312C22⋊Q8—48no—
313C4×Q8—48no—
314C4×D4—216no—
315C4×D4—28no—
316C4×D4—28no—
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—22no—
326#117—22no—
327#117—22no—
328#117—88no—
329#117#4114yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C22⋊C4—216no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—48no—
364C2×C22⋊C4—84no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42C2×D412yes—
375C2×C42C2×D414yes—
376C2×C42C2×D414yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—42no—
382#107—42no—
383#106—42no—
384#106—42no—
385#103—24no—
386#103—24no—
387#103—28no—
388#103—44no—
389#102—24no—
390#102—24no—
391#102—28no—
392#102—44no—
393#102—44no—
394#102—44no—
395#97—24no—
396#97—24no—
397C4×C8—24no—
398C4×C8—24no—
399#95—22no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—22no—
410C24—24no—
411C24—22no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—82no—
418C24—82no—
419C24—82no—
420C24—82no—
421C24C22×D411yescharacteristic
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—44no—
433C2×Q8—44no—
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—28no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—44no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4C2×#4114yes—
464C22×C4—22no—
465C22×C4—216no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—82no—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C4C2511yescharacteristic
494SD16—48no—
495SD16—48no—
496D8—48no—
497D8—48no—
498M16—44no—
499M16—44no—
500C2×C8—28no—
501C2×C8—28no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C2×#4112yes—
517C2×C8C2×#4112yes—
518C2×C8C2×#4112yes—
519C2×C8C2×#4112yes—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4C2×#4114yes—
540C22⋊C4—216no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—42no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—84no—
557C42—24no—
558C42—24no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—44no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—42no—
572C42—84no—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—44no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—48no—
581C23—44no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—88no—
587C23—88no—
588C23—84no—
589C23—162no—
590C23—162no—
591C23—162no—
592C23—162no—
593C23—164no—
594C23—164no—
595C23#52512yes—
596C23#53212yes—
597C23#53512yes—
598C23C23×D411yescharacteristic
599Q8—42no—
600Q8—42no—
601Q8—42no—
602Q8—42no—
603Q8—88no—
604Q8—84no—
605D4—24no—
606D4—24no—
607D4—48no—
608D4—42no—
609D4—42no—
610D4—42no—
611D4—42no—
612D4—42no—
613D4—42no—
614D4—84no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—84no—
646C2×C4—816no—
647C2×C4—84no—
648C2×C4—162no—
649C2×C4#52512yes—
650C2×C4#53212yes—
651C2×C4#53512yes—
652C2×C4#55114yes—
653C2×C4#55114yes—
654C2×C4#55514yes—
655C2×C4#55514yes—
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4C2×H(5,𝔽2)11yescharacteristic
661C2×C4C2×H(5,𝔽2)11yescharacteristic
662C8—22no—
663C8—22no—
664C8—22no—
665C8—22no—
666C8—42no—
667C8—42no—
668C22—22no—
669C22—22no—
670C22—24no—
671C22—24no—
672C22—44no—
673C22—48no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—84no—
681C22—84no—
682C22—168no—
683C22—168no—
684C22#261014yes—
685C22C2×#52511yescharacteristic
686C22C2×#53211yescharacteristic
687C22C2×#53511yescharacteristic
688C22#358412yes—
689C22#358512yes—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—44no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4#261014yes—
702C4C2×#55111yescharacteristic
703C4C2×#55111yescharacteristic
704C4C2×#55511yescharacteristic
705C4C2×#55511yescharacteristic
706C2—24no—
707C2—84no—
708C2—162no—
709C2C2×#261011yescharacteristic
710C2#6059212yes—
711C2#6071311yescharacteristic
712C2#6071911yescharacteristic
713C2#6081512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35338C2semidirect
2. #34255C2semidirect
3. #29570C2semidirect
4. #29566C2semidirect
5. #29562C2semidirect
6. #29556C2semidirect
7. #23551C2semidirect
8. #23506C2semidirect
9. #23454C2semidirect
12. #21231C2semidirect
13. #21091C2semidirect
14. #21090C2semidirect
15. #21066C2semidirect
16. #21065C2semidirect
18. #20933C2semidirect
19. #20368C2semidirect
37. #2861C22semidirect
38. #2640C22semidirect
45. #2319C22semidirect
53. #2293C22semidirect
57. #2059C22semidirect
58. #2059C22semidirect
59. #2058C22semidirect
60. #2057C22semidirect
61. #2042C22semidirect
62. #2042C22semidirect
63. #2025C22semidirect
64. #2023C22semidirect
69. #1973C22semidirect
72. #1968C22semidirect
73. #1936C22semidirect
74. #1935C22semidirect
75. #1934C22semidirect
76. #1933C22semidirect
77. #1928C22semidirect
78. #1927C22semidirect
79. #1926C22semidirect
80. #1925C22semidirect
83. #1877C22semidirect
84. #1877C22semidirect
85. #1746C22semidirect
86. #1746C22semidirect
87. #1745C22semidirect
88. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#353382
#342552
#295702
#295662
#295622
#295562
#235512
#235062
#234542
C2×#20122
#212332
#212312
#210912
#210902
#210662
#210652
#209342
#209332
#203682