Finite Group Databasebeta

← #6499237

Subgroups — #6499237

Order 512.

1406 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#33077C214yesmaximal, complemented
4#30005C214yesmaximal, complemented
5#29253C214yesmaximal
6#29252C214yesmaximal
7#29193C211yesmaximal, characteristic, complemented
8#29193C211yesmaximal, characteristic, complemented
9#29193C211yesmaximal, characteristic, complemented
10#29191C211yesmaximal, characteristic, complemented
11#29179C211yesmaximal, characteristic, complemented
12#29178C211yesmaximal, characteristic, complemented
13#29178C211yesmaximal, characteristic, complemented
14#29177C211yesmaximal, characteristic, complemented
15#21984C211yesmaximal, characteristic
16#21982C211yesmaximal, characteristic
17#21864C211yesmaximal, characteristic
18#21860C211yesmaximal, characteristic
19#20809C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2647C2212yes—
28#2647C2212yes—
29#2571—21no—
30#2571—21no—
31#2552—21no—
32#2552—21no—
33#2442—21no—
34#2442—21no—
35#2442—21no—
36#2442—21no—
37#2370—21no—
38#2370—21no—
39#2362—21no—
40#2362—21no—
41#2356—21no—
42#2351—21no—
43#2347—21no—
44#2346—21no—
45#2323C2212yes—
46#2323C2212yes—
47#2320C2212yes—
48#2317C2212yes—
49#2316C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54#2285C2211yescharacteristic
55#2285C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2276C2212yes—
58C4×#117C2211yescharacteristic
59#2088C2214yes—
60#2087C2214yes—
61#2086C2214yes—
62#2086C2214yes—
63#2068C2214yes—
64#2068C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1956C2211yescharacteristic
72#1956C2211yescharacteristic
73#1956C2211yescharacteristic
74#1956C2211yescharacteristic
75#1949C2214yes—
76#1947C2214yes—
77#1921C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1873C2214yes—
82#1872C2214yes—
83#1865C2214yes—
84#1863C2214yes—
85#1863C2214yes—
86#1863C2214yes—
87#1854C2214yes—
88#1852C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1843C2212yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1832C2212yes—
99#1805C2214yes—
100#1803C2214yes—
101#1798C2212yes—
102#1798C2212yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C4⋊C8C2211yescharacteristic
105C4×C4⋊C8C2211yescharacteristic
106C4×C4⋊C8C2211yescharacteristic
107#533—24no—
108#533Q818yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—21no—
115C2×#125—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4C2312yes—
132C22×C42C2311yescharacteristic
133#431—24no—
134#431—24no—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#391—21no—
164#391—21no—
165#391C2312yes—
166#391C2312yes—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#385—22no—
204#385—22no—
205#384—22no—
206#384—21no—
207#384—21no—
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2312yes—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#383—22no—
216#382—22no—
217#382—21no—
218#382—21no—
219#382—21no—
220#382—21no—
221#382—21no—
222#382—22no—
223#382C2312yes—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—22no—
228#380—22no—
229#380C2312yes—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—22no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253#376—22no—
254#376C2311yescharacteristic
255#376C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8C2412yes—
272C23×C4—22no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8C2×Q814yes—
283C4⋊Q8C2×Q814yes—
284#125—28no—
285#125—82no—
286#125—82no—
287#125—82no—
288#125—82no—
289C22⋊Q8—28no—
290C22⋊Q8—24no—
291C22⋊Q8C2×Q818yes—
292C4×Q8—28no—
293C4×Q8—24no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×Q8—82no—
297C4×Q8—82no—
298C4×Q8C2×Q818yes—
299#117—28no—
300#117—42no—
301#117—42no—
302#117—44no—
303#117—82no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×Q814yes—
375C2×C42#4114yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—44no—
383#107—44no—
384#107—44no—
385#106—44no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388C4⋊C8—42no—
389C4⋊C8—42no—
390C4⋊C8—42no—
391C4⋊C8—42no—
392#103—44no—
393#103—44no—
394#103—44no—
395#103—44no—
396C22⋊C8—24no—
397C22⋊C8—24no—
398C4×C8—22no—
399C4×C8—22no—
400C4×C8—22no—
401C4×C8—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430C24C22×D411yescharacteristic
431C2×Q8—24no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—24no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8C22×Q814yes—
438C22×C4—22no—
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—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4C22⋊Q814yes—
456C22×C4C22⋊Q814yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×Q812yes—
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4C22×Q814yes—
529C22⋊C4—28no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—44no—
535C22⋊C4—82no—
536C42—28no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—44no—
550C42—44no—
551C42—44no—
552C42—82no—
553C42—82no—
554C42—82no—
555C42C22⋊Q814yes—
556C42C22⋊Q814yes—
557C42C22×Q812yes—
558C42C22×Q812yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23C2×C22⋊Q812yes—
568C23C23×D411yescharacteristic
569Q8—28no—
570Q8—48no—
571Q8—44no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
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—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—164no—
631C2×C4C2×C22⋊Q812yes—
632C2×C4C2×C22⋊Q812yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×C22⋊Q812yes—
636C2×C4#57612yes—
637C2×C4#57712yes—
638C2×C4#57812yes—
639C2×C4#57812yes—
640C2×C4C23×Q811yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4#58512yes—
643C2×C4#58512yes—
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22C22×C22⋊Q811yescharacteristic
656C22#344911yescharacteristic
657C22#344911yescharacteristic
658C22C2×#57611yescharacteristic
659C22C2×#57711yescharacteristic
660C22C2×#57811yescharacteristic
661C22C2×#57811yescharacteristic
662C4—22no—
663C4—22no—
664C4—22no—
665C4—44no—
666C4—44no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4#308214yes—
682C4#308314yes—
683C4#350412yes—
684C4#351212yes—
685C2—42no—
686C2#3533311yescharacteristic
687C2C2×#308211yescharacteristic
688C2C2×#308311yescharacteristic
689C2#6100611yescharacteristic
690C2#6100811yescharacteristic
691C2#6101311yescharacteristic
692C2#6101411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #33077C2semidirect
4. #30005C2semidirect
7. #29193C2semidirect
8. #29193C2semidirect
9. #29193C2semidirect
10. #29191C2semidirect
11. #29179C2semidirect
12. #29178C2semidirect
13. #29178C2semidirect
14. #29177C2semidirect

Maximal subgroups

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

subgroupindex
#352862
#342252
#330772
#300052
#292532
#292522
#291932
#291912
#291792
#291782
#291772
#219842
#219822
#218642
#218602
#208092