Finite Group Databasebeta

← #482687

Subgroups — #482687

Order 512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35703C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic, complemented
3#17369C211yesmaximal, characteristic, complemented
4#11898C212yesmaximal, complemented
5#11876C212yesmaximal, complemented
6#11825C212yesmaximal, complemented
7#11823C212yesmaximal, complemented
8#11759C212yesmaximal, complemented
9#11753C212yesmaximal, complemented
10C2×#548C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2975—21no—
13C22×#99—21no—
14#2638C411yescharacteristic, complemented
15#2631C2211yescharacteristic
16#2617C411yescharacteristic, complemented
17#2610C411yescharacteristic, complemented
18#2599C411yescharacteristic, complemented
19#2561C2211yescharacteristic
20#2543C2211yescharacteristic
21#2477—21no—
22#2472—21no—
23#2463—21no—
24#2459—21no—
25#2444—21no—
26#2410—21no—
27#2409C2211yescharacteristic
28#2404C2211yescharacteristic
29#2398C2211yescharacteristic
30#2388C411yescharacteristic, complemented
31C2×#393C411yescharacteristic, complemented
32#2364—21no—
33#2358—21no—
34#2353C411yescharacteristic, complemented
35#2315—21no—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40C2×#342C2212yes—
41#1465—21no—
42#1465C2212yes—
43#1462—21no—
44#1462C2212yes—
45#1460C2212yes—
46#1458—21no—
47#1351C2214yescomplemented
48#1351C2214yescomplemented
49#1344C2214yescomplemented
50#1343C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic, complemented
53C23×D4D411yescharacteristic
54#548—44no—
55C2×#124—21no—
56C2×#124C2×C411yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C411yescharacteristic, complemented
63C2×#120—21no—
64C2×#120—21no—
65#518—44no—
66C2×C4×Q8C2×C411yescharacteristic, complemented
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85#436—44no—
86#431—22no—
87C2×#99—22no—
88C2×#99—22no—
89C2×#99—44no—
90#407—22no—
91#407—22no—
92#400—21no—
93#398C2×C411yescharacteristic, complemented
94#397—21no—
95#397—21no—
96#397C2×C411yescharacteristic
97#397C2×C411yescharacteristic, complemented
98#397C2311yescharacteristic
99#395C2×C411yescharacteristic, complemented
100#395C2×C411yescharacteristic, complemented
101#394—21no—
102#394—21no—
103#394—21no—
104#394—21no—
105#394—21no—
106#394C2×C411yescharacteristic, complemented
107#393—21no—
108#393—21no—
109#393C2×C411yescharacteristic
110#393C2×C411yescharacteristic
111#393C2×C411yescharacteristic, complemented
112#393C2×C411yescharacteristic, complemented
113#393C2311yescharacteristic
114#393C2311yescharacteristic
115#392—21no—
116#392—21no—
117#392C2×C411yescharacteristic
118#392C2×C411yescharacteristic, complemented
119#392C2311yescharacteristic
120#391—21no—
121#390—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—44no—
148#383C2×C411yescharacteristic, complemented
149#381—21no—
150#381C2×C411yescharacteristic
151#380—22no—
152#380—22no—
153#380C2×C411yescharacteristic, complemented
154#379—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4C2×C411yescharacteristic
160#376C2×C411yescharacteristic, complemented
161C2×#95—21no—
162C2×#95C2×C411yescharacteristic, complemented
163#342—22no—
164#342C2314yes—
165#324—22no—
166#323—22no—
167#323—22no—
168#323C2314yes—
169C25C22⋊C411yescharacteristic
170C25C2×D411yescharacteristic
171C2×#41—42no—
172C2×#41—82no—
173C22×Q8—21no—
174C22×Q8—42no—
175C22×Q8—82no—
176C22×Q8C22×C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4C22⋊C411yescharacteristic
188C22×D4C22⋊C411yescharacteristic
189C22×D4C22×C411yescharacteristic
190C22×D4C2×D411yescharacteristic
191C22×D4C2×D411yescharacteristic
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C2×M16—42no—
206C2×M16—42no—
207#124—42no—
208#124—44no—
209C22⋊Q8—44no—
210C22⋊Q8—42no—
211C22⋊Q8—82no—
212#121—44no—
213#121—42no—
214#121—42no—
215#121—42no—
216#120—44no—
217#120—82no—
218C4×Q8—42no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—44no—
228C4×D4—82no—
229#117—42no—
230#117—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C22⋊C4—24no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22×C411yescharacteristic
281C2×C22⋊C4C22×C411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2411yescharacteristic
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C4⋊C8—42no—
294#99—44no—
295#99—44no—
296C22⋊C8—24no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C4×C8—42no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—42no—
318#95—42no—
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323C24—22no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—41no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C22×D411yescharacteristic
340#41—84no—
341#41—84no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—42no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—42no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4AGL(1,ℤ/8)12yes—
384C2×D4#13712yes—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—84no—
422C22×C4—162no—
423C22×C4—162no—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C23×C411yescharacteristic
433C22×C4C22×D411yescharacteristic
434M16—162no—
435M16—162no—
436C2×C8—44no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—162no—
449C4⋊C4—162no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—84no—
486C22⋊C4#9914yes—
487C42—22no—
488C42—22no—
489C42—41no—
490C42—41no—
491C42—42no—
492C42—81no—
493C42—81no—
494C23—24no—
495C23—24no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—41no—
508C23—41no—
509C23—44no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—82no—
526C23—82no—
527C23—84no—
528C23—161no—
529C23—161no—
530C23—162no—
531C23—161no—
532C23—161no—
533C23—162no—
534C23C2×#9912yes—
535C23#41712yes—
536C23#42112yes—
537C23C22×C22⋊C411yescharacteristic
538C23C2×AGL(1,ℤ/8)11yescharacteristic
539C23C2×#13711yescharacteristic
540Q8—42no—
541Q8—42no—
542Q8—84no—
543Q8—82no—
544D4—42no—
545D4—42no—
546D4—44no—
547D4—44no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—82no—
552D4—84no—
553D4—84no—
554D4—161no—
555D4—161no—
556D4—162no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
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—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
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—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—161no—
610C2×C4—161no—
611C2×C4—162no—
612C2×C4—168no—
613C2×C4C2×#9912yes—
614C2×C4C2×#9912yes—
615C2×C4#41712yes—
616C8—84no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—21no—
623C22—22no—
624C22—21no—
625C22—22no—
626C22—42no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—84no—
651C22—84no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—164no—
656C22C22×#9911yescharacteristic
657C22C2×#41711yescharacteristic
658C22C2×#42111yescharacteristic
659C4—22no—
660C4—22no—
661C4—22no—
662C4—42no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—164no—
677C2—21no—
678C2—21no—
679C2—41no—
680C2—42no—
681C2—42no—
682C2—41no—
683C2—42no—
684C2—81no—
685C2—161no—
686C2#1759211yescharacteristic
687C2#3334811yescharacteristic
688C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35703C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20633C2semidirect
3. #17369C2semidirect
4. #11898C2semidirect
5. #11876C2semidirect
6. #11825C2semidirect
7. #11823C2semidirect
8. #11759C2semidirect
9. #11753C2semidirect
10. C2×#548C4semidirect
14. #2638C4semidirect
16. #2617C4semidirect
17. #2610C4semidirect
18. #2599C4semidirect
30. #2388C4semidirect
31. C2×#393C4semidirect
34. #2353C4semidirect
47. #1351C22semidirect
48. #1351C22semidirect
49. #1344C22semidirect
50. #1343C22semidirect
52. C23×Q8C2×C4semidirect
56. C2×#124C2×C4semidirect
62. C2×#121C2×C4semidirect
66. C2×C4×Q8C2×C4semidirect
93. #398C2×C4semidirect
97. #397C2×C4semidirect
99. #395C2×C4semidirect
100. #395C2×C4semidirect
106. #394C2×C4semidirect
111. #393C2×C4semidirect
112. #393C2×C4semidirect
118. #392C2×C4semidirect
148. #383C2×C4semidirect
153. #380C2×C4semidirect
160. #376C2×C4semidirect
162. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357032
#206332
#173692
#118982
#118762
#118252
#118232
#117592
#117532