Finite Group Databasebeta

← #480570

Subgroups — #480570

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35349C211yesmaximal, characteristic
2#17367C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#11579C212yesmaximal, complemented
5#11578C212yesmaximal, complemented
6#11577C212yesmaximal, complemented
7#11575C212yesmaximal, complemented
8#11453C212yesmaximal, complemented
9C2×#1326C212yesmaximal, complemented
10C22×C22⋊Q8C411yescharacteristic
11C2×#518C2211yescharacteristic
12#2973—21no—
13#2972—21no—
14#2579C411yescharacteristic
15#2579C411yescharacteristic
16#2579C2211yescharacteristic
17#2579C2211yescharacteristic
18#2487C411yescharacteristic
19#2487C2211yescharacteristic
20#2450—21no—
21#2450—21no—
22#2449—21no—
23#2449—21no—
24#2438—21no—
25#2438—21no—
26#2437—21no—
27#2437—21no—
28#2409C411yescharacteristic
29#2409C2211yescharacteristic
30#2315C411yescharacteristic
31#2315C2211yescharacteristic
32#2308C411yescharacteristic
33C2×#386C411yescharacteristic
34C2×#385C2211yescharacteristic
35#1466C2212yes—
36#1465C2212yes—
37#1464—21no—
38#1462—21no—
39#1460C2212yes—
40#1459—21no—
41#1459—21no—
42#1459C2212yes—
43#1458—21no—
44C2×#323—21no—
45C2×#323C2212yes—
46C2×#323C2212yes—
47#1326C2214yescomplemented
48#1326C2214yescomplemented
49#1326C2214yescomplemented
50#1326C2214yescomplemented
51C23×Q8C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C24×C4C2×C411yescharacteristic
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—44no—
61#518—44no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4C2311yescharacteristic
84#435—44no—
85C8×D4—44no—
86#432—22no—
87#431—22no—
88#407—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#400—21no—
93#400—21no—
94#399—21no—
95#399—21no—
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#396—21no—
101#396—21no—
102#396—21no—
103#396—21no—
104#395—21no—
105#395—21no—
106#393—21no—
107#393—21no—
108#391—21no—
109#391—21no—
110#389—21no—
111#389—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#386—21no—
117#386—21no—
118#386—21no—
119#386—21no—
120#386C2×C411yescharacteristic
121#386C2×C411yescharacteristic
122#386C2×C411yescharacteristic
123#386C2×C411yescharacteristic
124#386C2×C411yescharacteristic
125#386C2×C411yescharacteristic
126#386C2×C411yescharacteristic
127#386C2×C411yescharacteristic
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385C2×C411yescharacteristic
141#385C2×C411yescharacteristic
142#385C2×C411yescharacteristic
143#385C2×C411yescharacteristic
144#385C2311yescharacteristic
145#385C2311yescharacteristic
146#385C2311yescharacteristic
147#385C2311yescharacteristic
148#381—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380C2×C411yescharacteristic
154#380C2×C411yescharacteristic
155#380C2×C411yescharacteristic
156#380C2×C411yescharacteristic
157#379—21no—
158C2×#95—21no—
159C2×#95—21no—
160#324—22no—
161#324—22no—
162#323—22no—
163#323—22no—
164#323—22no—
165#323—22no—
166#323C2314yes—
167#323C2314yes—
168C25C22⋊C411yescharacteristic
169C25C22×C411yescharacteristic
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8—42no—
173C22×Q8—82no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—42no—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—42no—
186C23×C4—82no—
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C22×C411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C2×M16—42no—
201C22×C8—42no—
202C22×C8—42no—
203C22×C8—42no—
204C22⋊Q8—44no—
205C22⋊Q8—44no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208C22⋊Q8—82no—
209C22⋊Q8—82no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218#117—42no—
219#117—42no—
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—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
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—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—82no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22×C411yescharacteristic
273C2×C22⋊C4C22×C411yescharacteristic
274C2×C22⋊C4C22×C411yescharacteristic
275C2×C22⋊C4C22×C411yescharacteristic
276C2×C22⋊C4C22×C411yescharacteristic
277C2×C22⋊C4C2×D411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C22⋊C4C2×D411yescharacteristic
281C2×C22⋊C4C2411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C4⋊C8—42no—
287C4⋊C8—42no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293C22⋊C8—42no—
294#97—42no—
295C4×C8—42no—
296#95—21no—
297#95—21no—
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—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—41no—
321C24—81no—
322C24—81no—
323C24—81no—
324C24—81no—
325C24C2×C22⋊C411yescharacteristic
326C24C2×C22⋊C411yescharacteristic
327C24C2×C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24#13114yes—
330C24C23×C411yescharacteristic
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—81no—
336C2×Q8—81no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—416no—
351C2×D4#10114yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
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×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411M16—162no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—162no—
415C2×C8—162no—
416C2×C8—162no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4#9914yes—
467C42—22no—
468C42—22no—
469C42—41no—
470C42—41no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—28no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—41no—
482C23—41no—
483C23—44no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—82no—
493C23—82no—
494C23—84no—
495C23—84no—
496C23—84no—
497C23—161no—
498C23—161no—
499C23—162no—
500C23#40812yes—
501C23#40812yes—
502C23C2×#9912yes—
503C23C2×#10112yes—
504C23C2×#10112yes—
505C23C22×C22⋊C411yescharacteristic
506C23C2×#13112yes—
507Q8—84no—
508Q8—84no—
509D4—28no—
510D4—44no—
511D4—44no—
512D4—44no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
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—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—161no—
572C2×C4—161no—
573C2×C4—162no—
574C2×C4C2×#9912yes—
575C2×C4C2×#9912yes—
576C2×C4C2×#10112yes—
577C8—84no—
578C8—84no—
579C22—21no—
580C22—22no—
581C22—21no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—81no—
604C22—88no—
605C22—81no—
606C22—164no—
607C22#147214yes—
608C22C2×#40811yescharacteristic
609C22C22×#9911yescharacteristic
610C22C22×#10111yescharacteristic
611C4—22no—
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C2—21no—
630C2—21no—
631C2—24no—
632C2—42no—
633C2—42no—
634C2—42no—
635C2—81no—
636C2C2×#147211yescharacteristic
637C2#3334811yescharacteristic
638C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ 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. #11579C2semidirect
5. #11578C2semidirect
6. #11577C2semidirect
7. #11575C2semidirect
8. #11453C2semidirect
9. C2×#1326C2semidirect
47. #1326C22semidirect
48. #1326C22semidirect
49. #1326C22semidirect
50. #1326C22semidirect

Maximal subgroups

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

subgroupindex
#353492
#173672
#173502
#115792
#115782
#115772
#115752
#114532
C2×#13262