Finite Group Databasebeta

← #164808

Subgroups — #164808

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36368C211yesmaximal, characteristic
2#21998C211yesmaximal, characteristic, complemented
3#21998C211yesmaximal, characteristic, complemented
4#7733C211yesmaximal, characteristic, complemented
5#7729C211yesmaximal, characteristic, complemented
6#7721C211yesmaximal, characteristic, complemented
7#7715C211yesmaximal, characteristic, complemented
8#7688C211yesmaximal, characteristic, complemented
9#7685C211yesmaximal, characteristic, complemented
10#7647C211yesmaximal, characteristic, complemented
11#7642C211yesmaximal, characteristic, complemented
12#7631C211yesmaximal, characteristic, complemented
13#7599C211yesmaximal, characteristic, complemented
14#7594C211yesmaximal, characteristic, complemented
15#7563C211yesmaximal, characteristic, complemented
16#2863C2211yescharacteristic
17#2857C2211yescharacteristic
18#2855C2211yescharacteristic
19#2578—21no—
20#2573—21no—
21#2572—21no—
22#2511C411yescharacteristic
23#2492C411yescharacteristic
24#2485C411yescharacteristic
25#2477—21no—
26#2463—21no—
27#2457—21no—
28#2452C411yescharacteristic
29#2422C411yescharacteristic
30#2407C411yescharacteristic
31#2368—21no—
32#2359—21no—
33#2347C411yescharacteristic
34#2339C411yescharacteristic
35C4×#121C2211yescharacteristic
36#2297C2211yescharacteristic
37#2293C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1923—21no—
40#1923—21no—
41#1916—21no—
42#1916—21no—
43#1870—21no—
44#1870—21no—
45#1851—21no—
46#1851—21no—
47#1839C2211yescharacteristic
48#1839C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1817C2211yescharacteristic
52#1817C2211yescharacteristic
53#1797C2211yescharacteristic
54#1797C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
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
76C22×M16—21no—
77C22×M16—21no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#123—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×#117—21no—
91C2×#117C2×C411yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106#400—21no—
107#400—21no—
108#400—21no—
109#397—21no—
110#397—21no—
111#396—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#393—21no—
116#393—21no—
117#392—21no—
118#391—21no—
119#391—21no—
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2311yescharacteristic
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#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#386—21no—
151#386—21no—
152#386—21no—
153#386—21no—
154#386—21no—
155#386—21no—
156#385—21no—
157#385—21no—
158#385—21no—
159#385—21no—
160#384C2×C411yescharacteristic
161#383—21no—
162#383—21no—
163#381—21no—
164#381—21no—
165#380—21no—
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169#380C2×C411yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C2×#95—21no—
187C43C2311yescharacteristic
188#336—21no—
189#336—21no—
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×Q8—21no—
211C22×Q8C22×C411yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4C22⋊C411yescharacteristic
215C22×D4C22×C411yescharacteristic
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C2×M16—42no—
226C2×M16—42no—
227C2×M16—81no—
228C2×M16—81no—
229C2×M16—81no—
230C2×M16—81no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C4⋊Q8—81no—
240#123—44no—
241C22⋊Q8—44no—
242C22⋊Q8—81no—
243C22⋊Q8—81no—
244C22⋊Q8—81no—
245#121—22no—
246#121—41no—
247C4×D4—22no—
248C4×D4—41no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4—81no—
255#117—42no—
256#117—81no—
257#117—81no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—81no—
302C2×C42—81no—
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2411yescharacteristic
316C4⋊C8—81no—
317C4⋊C8—81no—
318C4⋊C8—81no—
319C4⋊C8—81no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—81no—
323C22⋊C8—81no—
324C22⋊C8—81no—
325C22⋊C8—81no—
326C4×C8—81no—
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#95—21no—
349#95—21no—
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356C24—21no—
357C24C2×C22⋊C411yescharacteristic
358C24C2×C22⋊C411yescharacteristic
359#41—84no—
360#41—84no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—42no—
366C2×Q8—81no—
367C2×Q8—81no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4#13112yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434M16—84no—
435M16—84no—
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—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—42no—
523C42—81no—
524C42—81no—
525C23—22no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23C22×C22⋊C411yescharacteristic
547Q8—81no—
548Q8—81no—
549Q8—162no—
550D4—42no—
551D4—42no—
552D4—44no—
553D4—84no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
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—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
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—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—162no—
624C2×C4#41312yes—
625C2×C4#42112yes—
626C2×C4#42112yes—
627C2×C4C2×#13111yescharacteristic
628C2×C4#56811yescharacteristic
629C2×C4#57611yescharacteristic
630C2×C4#57711yescharacteristic
631C2×C4#57711yescharacteristic
632C2×C4#57811yescharacteristic
633C8—81no—
634C8—81no—
635C8—81no—
636C8—81no—
637C8—81no—
638C8—81no—
639C8—81no—
640C8—81no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—42no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#288011yescharacteristic
663C22#288311yescharacteristic
664C22C2×#41311yescharacteristic
665C22#289711yescharacteristic
666C22#289811yescharacteristic
667C22C2×#42111yescharacteristic
668C22C2×#42111yescharacteristic
669C4—22no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—22no—
675C4—21no—
676C4—21no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—81no—
692C2—81no—
693C2—81no—
694C2—82no—
695C2#1742211yescharacteristic
696C2#1752211yescharacteristic
697C2#1752311yescharacteristic
698C2#1759711yescharacteristic
699C2#1759811yescharacteristic
700C2#1766511yescharacteristic
701C2#1777211yescharacteristic
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. #21998C2semidirect
3. #21998C2semidirect
4. #7733C2semidirect
5. #7729C2semidirect
6. #7721C2semidirect
7. #7715C2semidirect
8. #7688C2semidirect
9. #7685C2semidirect
10. #7647C2semidirect
11. #7642C2semidirect
12. #7631C2semidirect
13. #7599C2semidirect
14. #7594C2semidirect
15. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#363682
#219982
#77332
#77292
#77212
#77152
#76882
#76852
#76472
#76422
#76312
#75992
#75942
#75632