Finite Group Databasebeta

← #6461896

Subgroups — #6461896

Order 512.

1921 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35264C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic, complemented
3#29571C214yesmaximal, complemented
4#29569C214yesmaximal, complemented
5#29557C214yesmaximal, complemented
6#29555C214yesmaximal, complemented
7#23556C211yesmaximal, characteristic, complemented
8#23509C211yesmaximal, characteristic
9#23457C211yesmaximal, characteristic, complemented
10#23423C211yesmaximal, characteristic
11#21237C211yesmaximal, characteristic, complemented
12C2×#1968C211yesmaximal, characteristic, complemented
13#21095C211yesmaximal, characteristic, complemented
14#21083C211yesmaximal, characteristic, complemented
15#21068C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic, complemented
17#20936C211yesmaximal, characteristic, complemented
18#20930C211yesmaximal, characteristic, complemented
19#20366C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C2×#535C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#418C2211yescharacteristic
30C2×#418C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C2×C4×M16C2211yescharacteristic
36C22×C4×C8C2211yescharacteristic
37#2861C2212yes—
38#2858C2212yes—
39#2641C2212yes—
40#2640C2212yes—
41#2491—22no—
42#2490—22no—
43#2472—24no—
44#2356—22no—
45#2355—22no—
46#2343—24no—
47#2320C2214yescomplemented
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2310C2211yescharacteristic
51#2309C2211yescharacteristic
52#2297C2211yescharacteristic
53#2296C2211yescharacteristic
54#2295C2214yescomplemented
55C2×#391C2211yescharacteristic
56C2×#390C2211yescharacteristic
57#2274C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#2059C2214yescomplemented
60#2058C2214yescomplemented
61#2042C2214yes—
62#2041C2214yes—
63#2026C2214yescomplemented
64#2024C2214yescomplemented
65#2015C2214yes—
66#2014C2214yes—
67#1973C2212yescomplemented
68#1971C2212yescomplemented
69#1969C2212yescomplemented
70#1968C2214yes—
71#1968C2214yes—
72#1968C2212yescomplemented
73#1936C2214yes—
74#1934C2214yes—
75#1934C2212yescomplemented
76#1934C2212yescomplemented
77#1933C2212yescomplemented
78#1933C2212yescomplemented
79#1927C2214yescomplemented
80#1926C2214yes—
81#1926C2214yescomplemented
82#1925C2214yes—
83#1880C2214yes—
84#1880C2214yes—
85#1880C2214yescomplemented
86#1878C2214yescomplemented
87#1746C2214yescomplemented
88#1744C2214yescomplemented
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1729C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—42no—
103C22×Q16C2312yes—
104C22×SD16—24no—
105C22×SD16—24no—
106C22×SD16—42no—
107C22×SD16C2314yes—
108C22×SD16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—42no—
114#544—42no—
115#535—42no—
116#535—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×#127C2312yes—
120C2×#125—21no—
121C2×#125C2312yes—
122C2×#124—24no—
123C2×#124C2312yes—
124C2×C22⋊Q8—22no—
125C2×#121—21no—
126C2×#121—21no—
127C2×#120—22no—
128#519—24no—
129#519—24no—
130C2×C4×Q8—22no—
131C2×C4×Q8C2314yes—
132C2×#117—22no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C22⋊C4—22no—
141C22×C42C2311yescharacteristic
142#428—42no—
143C2×#107C2312yes—
144C2×#106—42no—
145C2×#106C2312yes—
146C2×#106C2314yes—
147#419—42no—
148#418—24no—
149#418—24no—
150#417—42no—
151#416—24no—
152#416—24no—
153C2×#103—42no—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#102—42no—
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×#102C2312yes—
167C4×M16—24no—
168C2×#97C2312yes—
169C2×C4×C8—24no—
170C2×C4×C8C2312yes—
171C2×C4×C8C2312yes—
172C2×C4×C8C2312yes—
173#397—24no—
174#394—24no—
175#394—24no—
176#393—24no—
177#391—21no—
178#391—21no—
179#391—22no—
180#391C2314yes—
181#391C2312yes—
182#391C2312yes—
183#391C2311yescharacteristic
184#391C2311yescharacteristic
185#391C2312yes—
186#390—22no—
187#390—24no—
188#390—22no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2311yescharacteristic
193#390C2312yes—
194#390C2312yes—
195#390C2311yescharacteristic
196#389C2314yes—
197#389C2312yes—
198#389C2312yes—
199#388—24no—
200#388—24no—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#386—22no—
206#386—22no—
207#385—22no—
208#385—22no—
209#384—21no—
210#383—24no—
211#382—21no—
212#382—24no—
213#381—22no—
214#380—22no—
215#379—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—24no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D412yes—
220C4×C4⋊C4D412yes—
221C4×C4⋊C4C2314yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D414yes—
225#376—24no—
226#376C2312yes—
227C2×#95—22no—
228C43C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#336C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C22×Q8—21no—
247C22×Q8—22no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8#4114yes—
254C22×Q8C2411yescharacteristic
255C22×Q8C2412yes—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4C2412yes—
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×D412yes—
273C23×C4C2411yescharacteristic
274#137—416no—
275C2×Q16—24no—
276C2×Q16—24no—
277C2×Q16—42no—
278C2×SD16—28no—
279C2×SD16—24no—
280C2×SD16—28no—
281C2×SD16—24no—
282C2×SD16—44no—
283C2×SD16—42no—
284C2×SD16—416no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16#4114yes—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8#4114yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C4⋊Q8—42no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304#127—42no—
305#127—42no—
306#125—42no—
307#125—42no—
308#125—44no—
309#124—42no—
310#124—42no—
311#124—88no—
312C22⋊Q8—48no—
313#121—44no—
314#121—44no—
315#120—48no—
316C4×Q8—216no—
317C4×Q8—28no—
318C4×Q8—28no—
319#117—216no—
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—22no—
329#117#4114yes—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—84no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D412yes—
367C2×C42—24no—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42C2×D414yes—
375C2×C42C2×D412yes—
376C2×C42C2×D414yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381#107—42no—
382#106—44no—
383#106—42no—
384#103—24no—
385#103—24no—
386#103—44no—
387#103—42no—
388#103—42no—
389#103—42no—
390#102—24no—
391#102—28no—
392#102—24no—
393#102—28no—
394#102—44no—
395#102—42no—
396#97—24no—
397C4×C8—24no—
398C4×C8—24no—
399C4×C8—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—22no—
410#95—24no—
411#95C2×D414yes—
412#95C2×D414yes—
413C24—22no—
414C24—21no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—82no—
419C24—82no—
420C24C22×D411yescharacteristic
421#41—44no—
422#41—44no—
423#41—44no—
424#41—44no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—44no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—84no—
445C2×Q8C2×#4114yes—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—21no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—84no—
471C2×D4—84no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C4H(5,𝔽2)12yes—
505C22×C4C2511yescharacteristic
506Q16—48no—
507SD16—416no—
508SD16—48no—
509M16—28no—
510M16—44no—
511C2×C8—28no—
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—44no—
523C2×C8C2×#4114yes—
524C2×C8C2×#4114yes—
525C4⋊C4—216no—
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—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—44no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—84no—
547C4⋊C4—82no—
548C4⋊C4C2×#4114yes—
549C22⋊C4—24no—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—42no—
556C22⋊C4—48no—
557C22⋊C4—44no—
558C22⋊C4—84no—
559C22⋊C4—82no—
560C22⋊C4—84no—
561C22⋊C4—84no—
562C22⋊C4—84no—
563C42—216no—
564C42—24no—
565C42—24no—
566C42—28no—
567C42—21no—
568C42—21no—
569C42—21no—
570C42—21no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—44no—
576C42—42no—
577C42—42no—
578C42—84no—
579C23—22no—
580C23—22no—
581C23—42no—
582C23—44no—
583C23—41no—
584C23—42no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—41no—
589C23—42no—
590C23—44no—
591C23—88no—
592C23—84no—
593C23—164no—
594C23—162no—
595C23—162no—
596C23#52512yes—
597C23#53212yes—
598C23#53512yes—
599C23C23×D411yescharacteristic
600Q8—24no—
601Q8—24no—
602Q8—48no—
603Q8—42no—
604Q8—42no—
605Q8—41no—
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—164no—
610D4—42no—
611D4—41no—
612D4—42no—
613D4—41no—
614D4—42no—
615D4—42no—
616D4—41no—
617D4—41no—
618D4—84no—
619D4—84no—
620D4—164no—
621D4—164no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—48no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—84no—
657C2×C4—162no—
658C2×C4—162no—
659C2×C4#52512yes—
660C2×C4#53212yes—
661C2×C4#53512yes—
662C2×C4#55114yes—
663C2×C4#55114yes—
664C2×C4#55514yes—
665C2×C4#55514yes—
666C2×C4#57714yes—
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C22×#4111yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×H(5,𝔽2)11yescharacteristic
673C8—24no—
674C8—24no—
675C8—42no—
676C8—42no—
677C22—24no—
678C22—22no—
679C22—22no—
680C22—24no—
681C22—82no—
682C22—84no—
683C22—81no—
684C22—82no—
685C22—81no—
686C22—82no—
687C22—81no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—168no—
692C22#261014yes—
693C22C2×#52511yescharacteristic
694C22C2×#53211yescharacteristic
695C22C2×#53511yescharacteristic
696C22C2×#57712yes—
697C22#358412yes—
698C4—24no—
699C4—42no—
700C4—44no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—44no—
705C4—82no—
706C4—81no—
707C4—82no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4#261014yes—
712C4C2×#55111yescharacteristic
713C4C2×#55111yescharacteristic
714C4C2×#55511yescharacteristic
715C4C2×#55511yescharacteristic
716C2—24no—
717C2—161no—
718C2—162no—
719C2C2×#261011yescharacteristic
720C2#6057212yes—
721C2#6070612yes—
722C2#6081811yescharacteristic
723C2#6081911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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
2. #34254C2semidirect
3. #29571C2semidirect
4. #29569C2semidirect
5. #29557C2semidirect
6. #29555C2semidirect
7. #23556C2semidirect
9. #23457C2semidirect
11. #21237C2semidirect
12. C2×#1968C2semidirect
13. #21095C2semidirect
14. #21083C2semidirect
15. #21068C2semidirect
16. #21063C2semidirect
17. #20936C2semidirect
18. #20930C2semidirect
19. #20366C2semidirect
47. #2320C22semidirect
54. #2295C22semidirect
59. #2059C22semidirect
60. #2058C22semidirect
63. #2026C22semidirect
64. #2024C22semidirect
67. #1973C22semidirect
68. #1971C22semidirect
69. #1969C22semidirect
72. #1968C22semidirect
75. #1934C22semidirect
76. #1934C22semidirect
77. #1933C22semidirect
78. #1933C22semidirect
79. #1927C22semidirect
81. #1926C22semidirect
85. #1880C22semidirect
86. #1878C22semidirect
87. #1746C22semidirect
88. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#352642
#342542
#295712
#295692
#295572
#295552
#235562
#235092
#234572
#234232
#212372
C2×#19682
#210952
#210832
#210682
#210632
#209362
#209302
#203662