Finite Group Databasebeta

← #482458

Subgroups — #482458

Order 512.

960 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35836C211yesmaximal, characteristic
2#20724C211yesmaximal, characteristic
3#17369C211yesmaximal, characteristic
4#11902C212yesmaximal, complemented
5#11899C212yesmaximal, complemented
6#11811C212yesmaximal, complemented
7#11809C212yesmaximal, complemented
8#11788C212yesmaximal, complemented
9#11784C212yesmaximal, complemented
10C2×#554C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2975—21no—
13#2628C411yescharacteristic, complemented
14#2626C411yescharacteristic, complemented
15#2626C2211yescharacteristic
16#2580C2211yescharacteristic
17#2564C411yescharacteristic, complemented
18#2564C2211yescharacteristic
19#2542C2211yescharacteristic
20#2487C411yescharacteristic, complemented
21#2487C2211yescharacteristic
22#2454—21no—
23#2449—21no—
24#2449—21no—
25#2441—21no—
26#2437—21no—
27#2437—21no—
28#2396C411yescharacteristic, complemented
29C2×#399C411yescharacteristic, complemented
30C2×#399C2211yescharacteristic
31#2349C411yescharacteristic, complemented
32D4×C4⋊C4—21no—
33D4×C22⋊C4—21no—
34#2312—21no—
35#1819C2212yes—
36#1799—21no—
37#1799C2212yes—
38#1781—21no—
39C4×#99—21no—
40C4×#99C2212yes—
41#1465—21no—
42#1465C2212yes—
43#1462—21no—
44#1462C2212yes—
45#1460C2212yes—
46#1458—21no—
47#1352C2214yescomplemented
48#1352C2214yescomplemented
49#1352C2214yescomplemented
50#1351C2214yescomplemented
51C23×D4D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic
61#518—44no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
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—
89#407—22no—
90#407—22no—
91#400C2×C411yescharacteristic
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#399—21no—
95#399—21no—
96#399—21no—
97#399—21no—
98#399C2×C411yescharacteristic
99#399C2×C411yescharacteristic
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2×C411yescharacteristic
103#399C2×C411yescharacteristic
104#399C2311yescharacteristic
105#399C2311yescharacteristic
106#399C2311yescharacteristic
107#398—21no—
108#396—21no—
109#396—21no—
110#396—21no—
111#396—21no—
112#396—21no—
113#395—21no—
114#395C2×C411yescharacteristic
115#393—21no—
116#393C2×C411yescharacteristic
117#393C2×C411yescharacteristic
118#393C2311yescharacteristic
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#388—44no—
127#387—21no—
128#387—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#384—21no—
144#382C2×C411yescharacteristic
145#380—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—22no—
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic
153#379—21no—
154#379—21no—
155#379—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C2×#95C2×C411yescharacteristic
161C2×#95C2×C411yescharacteristic
162#342—22no—
163#342C2314yes—
164#324—22no—
165#323—22no—
166#323—22no—
167#323C2314yes—
168C25C22⋊C411yescharacteristic
169C25C2×D411yescharacteristic
170C22×Q8—21no—
171C22×Q8C22×C411yescharacteristic
172C22×D4—24no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C2×D411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—42no—
193C2×M16—42no—
194C4⋊Q8—42no—
195#125—44no—
196C22⋊Q8—44no—
197C22⋊Q8—42no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C22⋊Q8—82no—
201C22⋊Q8—82no—
202C4×Q8—42no—
203C4×D4—24no—
204C4×D4—24no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—44no—
212C4×D4—44no—
213#117—42no—
214#117—42no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C22⋊C4—24no—
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—24no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
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—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
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⋊C4C2×D411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2411yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—42no—
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C4⋊C8—42no—
292#99—44no—
293#99—44no—
294C22⋊C8—24no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297C4×C8—42no—
298#95—21no—
299#95—21no—
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—42no—
312#95—42no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—22no—
318C24—22no—
319C24—22no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—81no—
324C24—81no—
325C24—81no—
326C24—81no—
327C24C2×C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C22×D411yescharacteristic
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—81no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—416no—
349C2×D4#13714yes—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400M16—162no—
401M16—162no—
402C2×C8—44no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—162no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C22⋊C4—162no—
461C22⋊C4—162no—
462C22⋊C4#9914yes—
463C42—22no—
464C42—22no—
465C42—41no—
466C42—41no—
467C42—42no—
468C42—42no—
469C42—162no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—41no—
483C23—41no—
484C23—44no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—44no—
494C23—82no—
495C23—82no—
496C23—84no—
497C23—84no—
498C23—161no—
499C23—161no—
500C23—162no—
501C23—162no—
502C23C2×#9912yes—
503C23#41912yes—
504C23C22×C22⋊C411yescharacteristic
505C23C2×#13712yes—
506Q8—82no—
507Q8—162no—
508D4—28no—
509D4—44no—
510D4—44no—
511D4—44no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—161no—
567C2×C4—161no—
568C2×C4—162no—
569C2×C4—162no—
570C2×C4C2×#9912yes—
571C2×C4C2×#9912yes—
572C2×C4#41912yes—
573C2×C4#42112yes—
574C8—84no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—164no—
603C22—164no—
604C22C22×#9911yescharacteristic
605C22C2×#41911yescharacteristic
606C22C2×#42111yescharacteristic
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—164no—
626C2—21no—
627C2—21no—
628C2—42no—
629C2—42no—
630C2—42no—
631C2—81no—
632C2—81no—
633C2#1757811yescharacteristic
634C2#3335011yescharacteristic
635C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
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
4. #11902C2semidirect
5. #11899C2semidirect
6. #11811C2semidirect
7. #11809C2semidirect
8. #11788C2semidirect
9. #11784C2semidirect
10. C2×#554C4semidirect
13. #2628C4semidirect
14. #2626C4semidirect
17. #2564C4semidirect
20. #2487C4semidirect
28. #2396C4semidirect
29. C2×#399C4semidirect
31. #2349C4semidirect
47. #1352C22semidirect
48. #1352C22semidirect
49. #1352C22semidirect
50. #1351C22semidirect

Maximal subgroups

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

subgroupindex
#358362
#207242
#173692
#119022
#118992
#118112
#118092
#117882
#117842