Finite Group Databasebeta

← #164856

Subgroups — #164856

Order 512.

781 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36499C211yesmaximal, characteristic
2#22000C211yesmaximal, characteristic, complemented
3#21998C211yesmaximal, characteristic, complemented
4#7718C211yesmaximal, characteristic, complemented
5#7716C211yesmaximal, characteristic, complemented
6#7716C211yesmaximal, characteristic, complemented
7#7714C211yesmaximal, characteristic, complemented
8#7681C211yesmaximal, characteristic, complemented
9#7675C211yesmaximal, characteristic, complemented
10#7652C211yesmaximal, characteristic, complemented
11#7648C211yesmaximal, characteristic, complemented
12#7633C211yesmaximal, characteristic, complemented
13#7604C211yesmaximal, characteristic, complemented
14#7600C211yesmaximal, characteristic, complemented
15#7565C211yesmaximal, characteristic, complemented
16#2861C2211yescharacteristic
17#2855C2211yescharacteristic
18#2854C2211yescharacteristic
19#2596—21no—
20#2589—21no—
21#2562C411yescharacteristic
22#2540C411yescharacteristic
23#2516—21no—
24#2492C411yescharacteristic
25#2491C411yescharacteristic
26#2459—21no—
27#2458—21no—
28#2455—21no—
29#2423C411yescharacteristic
30#2422C411yescharacteristic
31#2374—21no—
32#2370C411yescharacteristic
33#2367—21no—
34#2367C411yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C22⋊Q8C2211yescharacteristic
37C4×#121C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1870—21no—
44#1869—21no—
45#1851—21no—
46#1850—21no—
47#1839C2211yescharacteristic
48#1839C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1817C2211yescharacteristic
52#1817C2211yescharacteristic
53#1797C2211yescharacteristic
54#1797C2211yescharacteristic
55#1749C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1728C2211yescharacteristic
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×M16—21no—
76C23×C8—21no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×#125C2×C411yescharacteristic
79C2×#124—21no—
80C2×#124C2×C411yescharacteristic
81C2×#123—21no—
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×C4×Q8—21no—
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108#400—21no—
109#400—21no—
110#399—21no—
111#399—21no—
112#398—21no—
113#397—21no—
114#394—21no—
115#394—21no—
116#394—21no—
117#393—21no—
118#393—21no—
119#392—21no—
120#391—21no—
121#391—21no—
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2311yescharacteristic
125#390—21no—
126#390—21no—
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2311yescharacteristic
130#389—21no—
131#389—21no—
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2311yescharacteristic
135#389C2311yescharacteristic
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#388—21no—
148#388—21no—
149#388—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#386—21no—
157#386—21no—
158#385—21no—
159#385—21no—
160#385—21no—
161#385—21no—
162#384C2×C411yescharacteristic
163#381—21no—
164#380C2×C411yescharacteristic
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
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
206C22×Q8—21no—
207C22×Q8C22×C411yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4C22⋊C411yescharacteristic
211C22×D4C22×C411yescharacteristic
212C23×C4—21no—
213C23×C4C22×C411yescharacteristic
214C23×C4C22×C411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C2×M16—42no—
220C2×M16—81no—
221C2×M16—81no—
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—81no—
232C22×C8—81no—
233C4⋊Q8—22no—
234C4⋊Q8—41no—
235#125—42no—
236#124—42no—
237#124—44no—
238#123—44no—
239#123—81no—
240#123—81no—
241C22⋊Q8—22no—
242C22⋊Q8—41no—
243C22⋊Q8—81no—
244#121—22no—
245#121—41no—
246#121—81no—
247C4×Q8—81no—
248C4×Q8—81no—
249C4×Q8—81no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—81no—
254C4×D4—81no—
255C4×D4—81no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
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⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—81no—
306C2×C42—81no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2411yescharacteristic
320C4⋊C8—81no—
321C4⋊C8—81no—
322C4⋊C8—81no—
323C4⋊C8—81no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—81no—
327C22⋊C8—81no—
328C22⋊C8—81no—
329C22⋊C8—81no—
330#97—81no—
331C4×C8—81no—
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
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8—81no—
366C2×D4—22no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—41no—
373C2×D4—41no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—81no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
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—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—81no—
412C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426M16—84no—
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—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—84no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4#13112yes—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—41no—
482C22⋊C4—41no—
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—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—81no—
520C42—81no—
521C42—81no—
522C42—81no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23#42112yes—
544C23C22×C22⋊C411yescharacteristic
545Q8—81no—
546Q8—81no—
547Q8—84no—
548D4—44no—
549D4—81no—
550D4—81no—
551D4—84no—
552D4—162no—
553C2×C4—21no—
554C2×C4—21no—
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—22no—
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—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—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—162no—
621C2×C4#41312yes—
622C2×C4#42112yes—
623C2×C4C2×#13111yescharacteristic
624C2×C4#56811yescharacteristic
625C2×C4#57611yescharacteristic
626C2×C4#57711yescharacteristic
627C2×C4#57711yescharacteristic
628C2×C4#57811yescharacteristic
629C8—81no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8—81no—
635C8—81no—
636C8—81no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22#288011yescharacteristic
659C22#288311yescharacteristic
660C22C2×#41311yescharacteristic
661C22#289711yescharacteristic
662C22#289811yescharacteristic
663C22C2×#42111yescharacteristic
664C22C2×#42111yescharacteristic
665C4—21no—
666C4—21no—
667C4—22no—
668C4—21no—
669C4—21no—
670C4—22no—
671C4—21no—
672C4—21no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—42no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C2—42no—
689C2—81no—
690C2—161no—
691C2#1742911yescharacteristic
692C2#1752111yescharacteristic
693C2#1752511yescharacteristic
694C2#1759711yescharacteristic
695C2#1759811yescharacteristic
696C2#1766211yescharacteristic
697C2#1777511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2540C4
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. #22000C2semidirect
3. #21998C2semidirect
4. #7718C2semidirect
5. #7716C2semidirect
6. #7716C2semidirect
7. #7714C2semidirect
8. #7681C2semidirect
9. #7675C2semidirect
10. #7652C2semidirect
11. #7648C2semidirect
12. #7633C2semidirect
13. #7604C2semidirect
14. #7600C2semidirect
15. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#364992
#220002
#219982
#77182
#77162
#77142
#76812
#76752
#76522
#76482
#76332
#76042
#76002
#75652