Finite Group Databasebeta

← #164999

Subgroups — #164999

Order 512.

783 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36401C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22785C211yesmaximal, characteristic, complemented
4#7731C211yesmaximal, characteristic, complemented
5#7718C211yesmaximal, characteristic, complemented
6#7691C211yesmaximal, characteristic, complemented
7#7674C211yesmaximal, characteristic, complemented
8#7672C211yesmaximal, characteristic, complemented
9#7666C211yesmaximal, characteristic, complemented
10#7644C211yesmaximal, characteristic, complemented
11#7623C211yesmaximal, characteristic, complemented
12#7612C211yesmaximal, characteristic, complemented
13#7583C211yesmaximal, characteristic, complemented
14#7578C211yesmaximal, characteristic, complemented
15#7562C211yesmaximal, characteristic, complemented
16#2860C2211yescharacteristic
17#2854C2211yescharacteristic
18#2851C2211yescharacteristic
19#2603—21no—
20#2602—21no—
21#2597—21no—
22#2595—21no—
23#2578—21no—
24#2577—21no—
25#2513C411yescharacteristic
26#2490C411yescharacteristic
27#2485C411yescharacteristic
28#2453C411yescharacteristic
29#2421C411yescharacteristic
30#2407C411yescharacteristic
31#2374—21no—
32#2373—21no—
33#2347C411yescharacteristic
34#2341C411yescharacteristic
35C4×C22⋊Q8C2211yescharacteristic
36#2297C2211yescharacteristic
37#2295C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1924—21no—
41#1924—21no—
42#1923—21no—
43#1916—21no—
44#1869—21no—
45#1839C2211yescharacteristic
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770—21no—
52C4×C4⋊C8C2211yescharacteristic
53C8×C22⋊C4—21no—
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1744C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C23×C8—21no—
77C2×#126—21no—
78C2×#125C2×C411yescharacteristic
79C2×#123—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×#117C2×C411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105#400—21no—
106#400—21no—
107#398—21no—
108#397—21no—
109#397—21no—
110#397—21no—
111#397—21no—
112#397—21no—
113#394—21no—
114#394—21no—
115#394—21no—
116#392—21no—
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2311yescharacteristic
120#390—21no—
121#390—21no—
122#390—21no—
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2311yescharacteristic
128#389—21no—
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2311yescharacteristic
134#389C2311yescharacteristic
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#386—21no—
156#386—21no—
157#386—21no—
158#386—21no—
159#385—21no—
160#385—21no—
161#385—21no—
162#385—21no—
163#383—21no—
164#382C2×C411yescharacteristic
165#380—21no—
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169#380C2×C411yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188#376—21no—
189C43C2311yescharacteristic
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C2×#41—22no—
207C2×#41—41no—
208C2×#41—42no—
209C2×#41—81no—
210C22×Q8C22×C411yescharacteristic
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C22⋊C411yescharacteristic
214C22×D4C22×C411yescharacteristic
215C23×C4—21no—
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—42no—
231C22×C8—42no—
232C22×C8—81no—
233#126—42no—
234#126—42no—
235#125—81no—
236#123—42no—
237#123—42no—
238C22⋊Q8—22no—
239C22⋊Q8—41no—
240#121—81no—
241C4×Q8—22no—
242C4×Q8—41no—
243C4×Q8—81no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—42no—
249#117—42no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—81no—
299C2×C42—81no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2411yescharacteristic
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—81no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—81no—
321C22⋊C8—81no—
322C22⋊C8—81no—
323#97—81no—
324#97—81no—
325C4×C8—42no—
326C4×C8—42no—
327C4×C8—81no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354C24—21no—
355C24C2×C22⋊C411yescharacteristic
356C24C2×C22⋊C411yescharacteristic
357#41—84no—
358#41—84no—
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8#13112yes—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4—81no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—81no—
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—84no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—42no—
512C42—42no—
513C42—81no—
514C42—81no—
515C42—81no—
516C42#13512yes—
517C42#13512yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#41312yes—
539C23C22×C22⋊C411yescharacteristic
540Q8—42no—
541Q8—42no—
542D4—44no—
543D4—81no—
544D4—81no—
545D4—84no—
546D4—162no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—84no—
615C2×C4#41612yes—
616C2×C4#41612yes—
617C2×C4C2×#13111yescharacteristic
618C2×C4#56811yescharacteristic
619C2×C4C2×#13511yescharacteristic
620C2×C4C2×#13511yescharacteristic
621C2×C4#57511yescharacteristic
622C2×C4#57511yescharacteristic
623C8—42no—
624C8—42no—
625C8—81no—
626C8—81no—
627C8—81no—
628C8—81no—
629C8—81no—
630C8—81no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22#288011yescharacteristic
653C22#288311yescharacteristic
654C22C2×#41311yescharacteristic
655C22C2×#41611yescharacteristic
656C22#289611yescharacteristic
657C22#290111yescharacteristic
658C22#290111yescharacteristic
659C4—21no—
660C4—21no—
661C4—22no—
662C4—22no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—81no—
682C2—42no—
683C2—81no—
684C2—161no—
685C2#1742211yescharacteristic
686C2#1751011yescharacteristic
687C2#1751611yescharacteristic
688C2#1761711yescharacteristic
689C2#1762311yescharacteristic
690C2#1769011yescharacteristic
691C2#1771311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2407C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22839C2semidirect
3. #22785C2semidirect
4. #7731C2semidirect
5. #7718C2semidirect
6. #7691C2semidirect
7. #7674C2semidirect
8. #7672C2semidirect
9. #7666C2semidirect
10. #7644C2semidirect
11. #7623C2semidirect
12. #7612C2semidirect
13. #7583C2semidirect
14. #7578C2semidirect
15. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#364012
#228392
#227852
#77312
#77182
#76912
#76742
#76722
#76662
#76442
#76232
#76122
#75832
#75782
#75622