Finite Group Databasebeta

← #162480

Subgroups — #162480

Order 512.

839 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35046C211yesmaximal, characteristic
2#22785C211yesmaximal, characteristic, complemented
3#20556C211yesmaximal, characteristic, complemented
4#7712C211yesmaximal, characteristic, complemented
5#7710C211yesmaximal, characteristic, complemented
6#7702C211yesmaximal, characteristic, complemented
7#7701C211yesmaximal, characteristic, complemented
8#7692C211yesmaximal, characteristic, complemented
9#7690C211yesmaximal, characteristic, complemented
10#7682C211yesmaximal, characteristic, complemented
11#7681C211yesmaximal, characteristic, complemented
12#7583C211yesmaximal, characteristic, complemented
13#7578C211yesmaximal, characteristic, complemented
14#7566C211yesmaximal, characteristic, complemented
15#7563C211yesmaximal, characteristic, complemented
16C2×#435—21no—
17#2862C2211yescharacteristic
18#2860C2211yescharacteristic
19#2855C2211yescharacteristic
20#2850C2211yescharacteristic
21#2653—21no—
22#2636—21no—
23#2589—21no—
24#2576—21no—
25#2518—21no—
26#2514C411yescharacteristic
27#2491C411yescharacteristic
28#2459—21no—
29#2448C411yescharacteristic
30#2439C411yescharacteristic
31#2432C411yescharacteristic
32#2429C411yescharacteristic
33#2415C411yescharacteristic
34#2407C411yescharacteristic
35#2337—21no—
36#2334—21no—
37C4×C22⋊Q8C2211yescharacteristic
38#2297C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1923—21no—
41#1916—21no—
42#1869—21no—
43#1839C2211yescharacteristic
44#1817C2211yescharacteristic
45#1797C2211yescharacteristic
46#1771—21no—
47#1767C2211yescharacteristic
48#1753—21no—
49#1752—21no—
50C8×C22⋊C4—21no—
51#1749C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1745C2211yescharacteristic
55#1744C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C22×M16—21no—
77C23×C8—21no—
78C23×C8—21no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#126—21no—
82C2×#124—21no—
83C2×#124C2×C411yescharacteristic
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×#117C2×C411yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#435—28no—
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×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109#401—21no—
110#400—21no—
111#399—21no—
112#398—21no—
113#397—21no—
114#397—21no—
115#396—21no—
116#396—21no—
117#394—21no—
118#394—21no—
119#394—21no—
120#394—21no—
121#394—21no—
122#393—21no—
123#393—21no—
124#392—21no—
125#391—21no—
126#391—21no—
127#391—21no—
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#391C2×C411yescharacteristic
132#391C2311yescharacteristic
133#391C2311yescharacteristic
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#390C2311yescharacteristic
138#389—21no—
139#389—21no—
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2311yescharacteristic
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
152#388—21no—
153#388—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#386—21no—
160#386—21no—
161#386—21no—
162#386—21no—
163#386—21no—
164#385—21no—
165#385—21no—
166#385—21no—
167#385—21no—
168#385—21no—
169#383—21no—
170#382—21no—
171#382—21no—
172#382C2×C411yescharacteristic
173#381—21no—
174#380C2×C411yescharacteristic
175#380C2×C411yescharacteristic
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C43C2311yescharacteristic
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C2×#41—22no—
208C2×#41—22no—
209C2×#41—81no—
210C2×#41—81no—
211C22×Q8—21no—
212C22×Q8C22×C411yescharacteristic
213C22×Q8C22×C411yescharacteristic
214C22×D4—21no—
215C22×D4—21no—
216C22×D4C22⋊C411yescharacteristic
217C22×D4C22×C411yescharacteristic
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C2×M16—24no—
226C2×M16—81no—
227C2×M16—81no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—24no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—42no—
238C22×C8—42no—
239C22×C8—81no—
240C22×C8—81no—
241C22×C8—81no—
242C4⋊Q8—81no—
243C4⋊Q8—81no—
244#126—42no—
245#126—42no—
246#124—44no—
247#124—81no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—81no—
251C22⋊Q8—81no—
252#121—81no—
253C4×Q8—81no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—24no—
257C4×D4—41no—
258C4×D4—41no—
259#117—81no—
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⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
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×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
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—24no—
321C4⋊C8—81no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8—42no—
325C22⋊C8—42no—
326C22⋊C8—81no—
327#97—24no—
328#97—81no—
329#97—81no—
330C4×C8—42no—
331C4×C8—81no—
332C4×C8—81no—
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#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359#95C22⋊C411yescharacteristic
360C24—21no—
361C24C2×C22⋊C411yescharacteristic
362C24C2×C22⋊C411yescharacteristic
363#41—84no—
364#41—84no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—21no—
369C2×Q8—41no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×D4—22no—
377C2×D4—22no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4—81no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
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×C4#13112yes—
432C22×C4#13112yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436M16—84no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
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—
460C2×C8—41no—
461C2×C8—84no—
462C2×C8—84no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—81no—
478C4⋊C4—81no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4#10414yes—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—22no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—42no—
526C42—81no—
527C42—81no—
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—
546C23—41no—
547C23#40812yes—
548C23C2×#10412yes—
549C23C22×C22⋊C411yescharacteristic
550Q8—81no—
551Q8—81no—
552Q8—81no—
553Q8—81no—
554Q8—162no—
555D4—44no—
556D4—81no—
557D4—81no—
558D4—84no—
559D4—162no—
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—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
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—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—84no—
627C2×C4#40812yes—
628C2×C4C2×#10412yes—
629C2×C4C2×#10412yes—
630C2×C4C2×#13111yescharacteristic
631C2×C4C2×#13111yescharacteristic
632C2×C4#57511yescharacteristic
633C2×C4#57511yescharacteristic
634C2×C4#57711yescharacteristic
635C2×C4#57811yescharacteristic
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C8—81no—
642C8—81no—
643C8—81no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22C2×#40811yescharacteristic
666C22#288311yescharacteristic
667C22#288811yescharacteristic
668C22#289611yescharacteristic
669C22#289811yescharacteristic
670C22C22×#10411yescharacteristic
671C22#290111yescharacteristic
672C4—21no—
673C4—21no—
674C4—22no—
675C4—21no—
676C4—21no—
677C4—22no—
678C4—21no—
679C4—21no—
680C4—22no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—81no—
695C2—42no—
696C2—42no—
697C2—161no—
698C2#1738611yescharacteristic
699C2#1746211yescharacteristic
700C2#1748411yescharacteristic
701C2#1754811yescharacteristic
702C2#1762911yescharacteristic
703C2#1765211yescharacteristic
704C2#1776811yescharacteristic
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. #22785C2semidirect
3. #20556C2semidirect
4. #7712C2semidirect
5. #7710C2semidirect
6. #7702C2semidirect
7. #7701C2semidirect
8. #7692C2semidirect
9. #7690C2semidirect
10. #7682C2semidirect
11. #7681C2semidirect
12. #7583C2semidirect
13. #7578C2semidirect
14. #7566C2semidirect
15. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#350462
#227852
#205562
#77122
#77102
#77022
#77012
#76922
#76902
#76822
#76812
#75832
#75782
#75662
#75632