Finite Group Databasebeta

← #6565754

Subgroups — #6565754

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3364C212yesmaximal, complemented
2#36594C211yesmaximal, characteristic
3#34328C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal
5#29799C211yesmaximal, characteristic, complemented
6#28751C212yesmaximal
7#28748C211yesmaximal, characteristic, complemented
8#28209C212yesmaximal, complemented
9#28201C211yesmaximal, characteristic
10#25283C214yesmaximal, complemented
11#25275C212yesmaximal, complemented
12#24637C214yesmaximal, complemented
13#24629C212yesmaximal, complemented
14#23679C212yesmaximal, complemented
15#23678C211yesmaximal, characteristic
16#22734C211yesmaximal, characteristic, complemented
17#22541C212yesmaximal, complemented
18C2×#557C2211yescharacteristic
19C2×#556C2212yes—
20#3364—216no—
21C2×#488C2214yes—
22C2×#487C2212yes—
23C2×#479C2214yes—
24C2×#474C2212yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×C8×Q8C2212yes—
28C2×C4×Q16C2214yes—
29C2×C4×SD16C2212yes—
30#2868C2211yescharacteristic
31#2858C2212yes—
32#2547C2212yes—
33#2545C2214yes—
34#2538—21no—
35#2538—21no—
36#2512—22no—
37#2512—22no—
38#2371—22no—
39#2370C2212yes—
40#2365—21no—
41#2365C2211yescharacteristic
42#2356—21no—
43#2356C2212yes—
44#2355C2211yescharacteristic
45#2355C2211yescharacteristic
46#2354—21no—
47#2354—21no—
48#2336—22no—
49#2327—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2212yes—
52C4×#127C2211yescharacteristic
53C4×#125C2212yes—
54#2295C2211yescharacteristic
55#2295C2212yes—
56#2276—21no—
57C42×Q8C2211yescharacteristic
58#2084C2211yescharacteristic
59#2083C2212yes—
60#2072C2211yescharacteristic
61#2068C2214yes—
62#2065C2212yes—
63#2056C2214yes—
64#2055C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2042C2214yes—
70#2041C2212yes—
71#2041C2211yescharacteristic
72#2016C2212yes—
73#2016C2212yes—
74#2015C2214yes—
75#2014C2211yescharacteristic
76#1983C2214yes—
77#1983C2214yes—
78#1983C2214yes—
79#1981C2212yes—
80#1981C2214yes—
81#1958C2214yes—
82#1956C2212yes—
83#1939C2212yes—
84#1939C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1925C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1805C2214yes—
91#1802C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2214yes—
95C4×#102C2212yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98#557—24no—
99#557D418yescomplemented
100#556—28no—
101#556—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#126—24no—
109C2×#125—22no—
110C2×#125C2312yes—
111C2×#124C2314yes—
112C2×#124C2314yes—
113C2×C22⋊Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127#488—216no—
128#487—28no—
129#479—216no—
130#474—28no—
131#464—216no—
132#462—28no—
133C8×Q8—28no—
134C4×Q16—216no—
135C4×SD16—28no—
136C2×#107C2314yes—
137C2×#106C2312yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#102C2312yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2312yes—
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#397—24no—
158#396—24no—
159#396—24no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391C2314yes—
164#391C2311yescharacteristic
165#391C2312yes—
166#391C2311yescharacteristic
167#390—21no—
168#390—21no—
169#390—22no—
170#390—22no—
171#390—22no—
172#390—22no—
173#390C2312yes—
174#389—22no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2311yescharacteristic
181#389C2312yes—
182#388—24no—
183#388—24no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—21no—
192#387—22no—
193#387—21no—
194#387—22no—
195#387—22no—
196#387—22no—
197#384—21no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384C2311yescharacteristic
202#384C2311yescharacteristic
203#383—22no—
204#382—21no—
205#382C2312yes—
206#380—22no—
207#380—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D411yescharacteristic
221C4×C4⋊C4D412yes—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2311yescharacteristic
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2312yes—
229#376—21no—
230#376—22no—
231#376—22no—
232C43C2312yes—
233C43C2311yescharacteristic
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2312yes—
241#336C2314yes—
242C22×Q8—21no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2414yes—
245C22×Q8C2411yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C23×C4—22no—
250C2×Q16—28no—
251C2×Q16—44no—
252C2×Q16—44no—
253C2×SD16—24no—
254C2×SD16—42no—
255C2×SD16—42no—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C22×C8C2412yes—
259C4⋊Q8—28no—
260C4⋊Q8—24no—
261C4⋊Q8—22no—
262C4⋊Q8—24no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8C2×D414yes—
266#127—42no—
267#126—416no—
268#125—216no—
269#125—24no—
270#125C2×D418yes—
271#124—28no—
272#124—28no—
273#124—44no—
274#124—44no—
275C22⋊Q8—48no—
276C4×Q8—28no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—42no—
283C4×Q8—44no—
284C4×Q8—44no—
285C4×Q8C2×D414yes—
286C4×Q8C2×D418yes—
287C4×Q8C2×D414yes—
288C4×Q8#4114yes—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—44no—
293#117—48no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D411yescharacteristic
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2411yescharacteristic
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C42—22no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—21no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354#107—28no—
355#106—24no—
356C4⋊C8—24no—
357C4⋊C8—28no—
358C4⋊C8—44no—
359C4⋊C8—42no—
360#103—24no—
361#103—28no—
362#103—28no—
363#103—28no—
364#103—44no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—24no—
369#102—28no—
370#102—42no—
371#102—44no—
372#102—42no—
373C4×C8—24no—
374C4×C8—28no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—21no—
379#95—21no—
380#95—22no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—22no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—22no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—21no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8#12114yes—
413C2×Q8C22×D412yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—42no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448Q16—416no—
449SD16—48no—
450C2×C8—22no—
451C2×C8—24no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—24no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—24no—
458C2×C8—22no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—28no—
462C4⋊C4—24no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—22no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—42no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4#12114yes—
491C4⋊C4#12118yes—
492C4⋊C4C22×D412yes—
493C4⋊C4C22×D414yes—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C42—28no—
502C42—24no—
503C42—24no—
504C42—22no—
505C42—22no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—42no—
511C42—41no—
512C42—42no—
513C42—44no—
514C42—48no—
515C42—44no—
516C42—42no—
517C42—82no—
518C42—82no—
519C42#13514yes—
520C42C22×D412yes—
521C42C22×D414yes—
522C42C2×#4112yes—
523C42C2×#4114yes—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C23×D411yescharacteristic
532Q8—28no—
533Q8—24no—
534Q8—42no—
535Q8—48no—
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—84no—
545D4—84no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—88no—
592C2×C4C2×#12114yes—
593C2×C4C2×#12112yes—
594C2×C4C2×#12114yes—
595C2×C4C2×#12112yes—
596C2×C4C2×#13512yes—
597C2×C4C2×#13512yes—
598C2×C4C2×#13512yes—
599C2×C4#57512yes—
600C2×C4#57812yes—
601C2×C4C23×D411yescharacteristic
602C2×C4C22×#4111yescharacteristic
603C2×C4#58512yes—
604C2×C4#58512yes—
605C8—42no—
606C8—42no—
607C8—44no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22C22×#12111yescharacteristic
616C22#348711yescharacteristic
617C22#350011yescharacteristic
618C22C22×#13511yescharacteristic
619C22C2×#57511yescharacteristic
620C22C2×#57811yescharacteristic
621C22#358611yescharacteristic
622C4—24no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—24no—
628C4—42no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—44no—
634C4—81no—
635C4—84no—
636C4—82no—
637C4—81no—
638C4—82no—
639C4#303514yes—
640C4#348712yes—
641C4#352412yes—
642C2—162no—
643C2#3601811yescharacteristic
644C2C2×#303511yescharacteristic
645C2#6043211yescharacteristic
646C2#6188211yescharacteristic
647C2#6190511yescharacteristic
648C2#6252811yescharacteristic
649C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3364C2semidirect
3. #34328C2semidirect
5. #29799C2semidirect
7. #28748C2semidirect
8. #28209C2semidirect
10. #25283C2semidirect
11. #25275C2semidirect
12. #24637C2semidirect
13. #24629C2semidirect
14. #23679C2semidirect
16. #22734C2semidirect
17. #22541C2semidirect
99. #557D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33642
#365942
#343282
#298012
#297992
#287512
#287482
#282092
#282012
#252832
#252752
#246372
#246292
#236792
#236782
#227342
#225412