Finite Group Databasebeta

← #5227949

Subgroups — #5227949

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17144C211yesmaximal, characteristic, complemented
2#16917C212yesmaximal
3#16750C212yesmaximal, complemented
4#16746C212yesmaximal, complemented
5#16712C212yesmaximal, complemented
6#16447C212yesmaximal
7#16387C212yesmaximal, complemented
8#16244C211yesmaximal, characteristic, complemented
9#16235C212yesmaximal, complemented
10#16224C212yesmaximal
11#16017C212yesmaximal
12#15845C212yesmaximal
13#15799C212yesmaximal, complemented
14#15797C212yesmaximal
15#15671C211yesmaximal, characteristic
16#15670C211yesmaximal, characteristic
17#14660C211yesmaximal, characteristic
18#14614C211yesmaximal, characteristic, complemented
19#14599C211yesmaximal, characteristic, complemented
20C22×C22⋊Q8C2211yescharacteristic
21C2×#400C2211yescharacteristic
22C2×#400C2212yes—
23C2×#397C2211yescharacteristic
24C2×#395C2211yescharacteristic
25C2×#393C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#388C2212yes—
28C2×#382C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1446C2212yes—
32#1446C2212yes—
33#1446C2212yes—
34#1446C2212yes—
35#1446C2211yescharacteristic
36#1446C2211yescharacteristic
37#1445C2212yes—
38#1445C2211yescharacteristic
39#1445C2212yes—
40#1445C2211yescharacteristic
41#1445C2211yescharacteristic
42#1445C2211yescharacteristic
43#1445C2212yes—
44#1445C2212yes—
45#1444C2212yes—
46#1443C2212yes—
47#1443C2212yes—
48#1443C2212yes—
49#1443C2212yes—
50#1443C2212yes—
51#1443C2212yes—
52#1443C2212yes—
53#1443C2212yes—
54#1443C2212yes—
55#1443C2212yes—
56#1443C2212yes—
57#1443C2212yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2212yes—
66#1442C2212yes—
67#1442C2212yes—
68#1442C2212yes—
69#1442C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2212yes—
74#1440C2212yes—
75#1440C2212yes—
76#1440C2212yes—
77#1440C2212yes—
78#1440C2212yes—
79#1440C2212yes—
80#1439C2212yes—
81#1439C2212yes—
82#1439C2212yes—
83#1439C2212yes—
84#1437C2211yescharacteristic
85#1437C2211yescharacteristic
86#1437C2212yes—
87#1436C2212yes—
88#1436C2212yes—
89#1436C2211yescharacteristic
90#1436C2212yes—
91#1436C2212yes—
92#1436C2211yescharacteristic
93#1436C2212yes—
94#1436C2212yes—
95#1436C2212yes—
96#1436C2212yes—
97#1436C2212yes—
98#1436C2212yes—
99C4⋊#95C2212yes—
100#1434C2212yes—
101#1434C2212yes—
102#1434C2212yes—
103#1434C2212yes—
104#1434C2212yes—
105C22⋊#95C2212yes—
106C22⋊#95C2212yes—
107C23×Q8C2311yescharacteristic
108C23×Q8C2311yescharacteristic
109C24×C4C2311yescharacteristic
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—42no—
112C2×C22⋊Q8—42no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#400—24no—
136#400—44no—
137#397—42no—
138#395—24no—
139#393—42no—
140#391—42no—
141#388—44no—
142#382—42no—
143#381—42no—
144#380—42no—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2311yescharacteristic
206C2×#95C2311yescharacteristic
207C25C2411yescharacteristic
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—41no—
211C22×Q8—41no—
212C22×Q8—42no—
213C22×Q8—41no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—42no—
217C22×Q8—41no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C22⋊Q8—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C42—41no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—41no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306#95—22no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—41no—
353#95—41no—
354#95—42no—
355#95—41no—
356#95—41no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—41no—
362#95—42no—
363#95—41no—
364#95—42no—
365#95—42no—
366#95—42no—
367#95—42no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24H(5,𝔽2)12yes—
377C24C2511yescharacteristic
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×Q8—82no—
386C2×Q8—84no—
387C2×Q8—84no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4—84no—
501C22×C4—84no—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+414yes—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—82no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—48no—
520C22⋊C4—84no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C42—44no—
526C42—82no—
527C42—84no—
528C42—84no—
529C23—22no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23C22×#4111yescharacteristic
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×2−1+412yes—
554C23#58512yes—
555C23#58512yes—
556C23#58512yes—
557C23#58512yes—
558C23#58512yes—
559Q8—88no—
560Q8—88no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
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—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—88no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—44no—
680C22—44no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22#349011yescharacteristic
686C22#351011yescharacteristic
687C22#351011yescharacteristic
688C22#351612yes—
689C22#352112yes—
690C22#353312yes—
691C22#354011yescharacteristic
692C22#354011yescharacteristic
693C22#354411yescharacteristic
694C22#354512yes—
695C22#354511yescharacteristic
696C22#355312yes—
697C22#355412yes—
698C22#356012yes—
699C22#356412yes—
700C22#356512yes—
701C22#356512yes—
702C22#356612yes—
703C22#356612yes—
704C22#356612yes—
705C22#356612yes—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—81no—
719C4—81no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C2—82no—
725C2#4703511yescharacteristic
726C2#5050312yes—
727C2#5099012yes—
728C2#5180311yescharacteristic
729C2#5184011yescharacteristic
730C2#5437112yes—
731C2#5440212yes—
732C2#5527512yes—
733C2#5602312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17144C2semidirect
3. #16750C2semidirect
4. #16746C2semidirect
5. #16712C2semidirect
7. #16387C2semidirect
8. #16244C2semidirect
9. #16235C2semidirect
13. #15799C2semidirect
18. #14614C2semidirect
19. #14599C2semidirect

Maximal subgroups

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

subgroupindex
#171442
#169172
#167502
#167462
#167122
#164472
#163872
#162442
#162352
#162242
#160172
#158452
#157992
#157972
#156712
#156702
#146602
#146142
#145992