Finite Group Databasebeta

← #6565766

Subgroups — #6565766

Order 512.

1795 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3369C212yesmaximal, complemented
2#36594C211yesmaximal, characteristic
3#34335C211yesmaximal, characteristic, complemented
4#29801C212yesmaximal
5#29799C211yesmaximal, characteristic, complemented
6#28753C212yesmaximal
7#28748C211yesmaximal, characteristic, complemented
8#28205C212yesmaximal, complemented
9#28202C211yesmaximal, characteristic
10#25287C214yesmaximal, complemented
11#25273C212yesmaximal, complemented
12#24641C214yesmaximal, complemented
13#24629C212yesmaximal, complemented
14#23685C212yesmaximal, complemented
15#23678C211yesmaximal, characteristic
16#22736C211yesmaximal, characteristic, complemented
17#22544C212yesmaximal, complemented
18C2×Q82C2211yescharacteristic
19C2×#556C2212yes—
20#3369—216no—
21C2×#489C2214yes—
22C2×#487C2212yes—
23C2×#475C2214yes—
24C2×#476C2212yes—
25C2×#464C2214yes—
26C2×#462C2212yes—
27C2×#446C2212yes—
28C2×#441C2214yes—
29C2×C4×SD16C2212yes—
30#2868C2211yescharacteristic
31#2858C2212yes—
32#2547C2212yes—
33#2545C2214yes—
34#2538—21no—
35#2538—21no—
36#2512—22no—
37#2512—22no—
38#2369C2211yescharacteristic
39#2369C2212yes—
40#2365—21no—
41#2365—22no—
42#2355C2211yescharacteristic
43#2355C2212yes—
44#2354—21no—
45#2354—21no—
46#2351—21no—
47Q8×C4⋊C4C2211yescharacteristic
48#2331—21no—
49#2331—22no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2212yes—
52C4×C4⋊Q8C2212yes—
53C4×#127C2211yescharacteristic
54#2295C2211yescharacteristic
55#2295C2212yes—
56#2276—21no—
57C42×Q8C2211yescharacteristic
58#2084C2212yes—
59#2084C2211yescharacteristic
60#2072C2211yescharacteristic
61#2066C2214yes—
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#1959C2214yes—
82#1956C2212yes—
83#1940C2214yes—
84#1939C2212yes—
85#1927C2214yes—
86#1927C2214yes—
87#1926C2214yes—
88#1831C2211yescharacteristic
89#1831C2212yes—
90#1803C2214yes—
91#1802C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94#1763C2214yes—
95C4×#102C2212yes—
96C22×Q16C2314yes—
97C22×SD16C2312yes—
98Q82—24no—
99Q82D418yescomplemented
100#556—28no—
101#556—28no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#126—24no—
111C2×#124C2314yes—
112C2×#124C2314yes—
113C2×C22⋊Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128#489—216no—
129#487—28no—
130#476—28no—
131#475—216no—
132#464—216no—
133#462—28no—
134#446—28no—
135#441—216no—
136C4×SD16—28no—
137C2×#106C2312yes—
138C2×#106C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2312yes—
156C2×#97C2314yes—
157C2×C4×C8C2312yes—
158#397—24no—
159#396—24no—
160#396—24no—
161#391—22no—
162#391—21no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#391C2311yescharacteristic
168#391C2312yes—
169#391C2311yescharacteristic
170#390—21no—
171#390—21no—
172#390—22no—
173#390—22no—
174#390—22no—
175#390—22no—
176#390C2312yes—
177#389—21no—
178#389—22no—
179#389—21no—
180#389—22no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389C2312yes—
186#389C2311yescharacteristic
187#389C2311yescharacteristic
188#389C2312yes—
189#388—24no—
190#388—24no—
191#387—22no—
192#387—21no—
193#387—22no—
194#387—21no—
195#387—22no—
196#387—22no—
197#387—22no—
198#384—21no—
199#384—21no—
200#384—21no—
201#384—21no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#382—21no—
206#382—22no—
207#380—22no—
208#380—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4D412yes—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D412yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230#376—22no—
231#376—21no—
232C43C2311yescharacteristic
233C43C2312yes—
234#340C2312yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2311yescharacteristic
239#336C2312yes—
240#336C2312yes—
241#336C2314yes—
242C22×Q8—22no—
243C22×Q8—21no—
244C22×Q8C2×D411yescharacteristic
245C22×Q8C2414yes—
246C22×Q8C2411yescharacteristic
247C22×D4—21no—
248C22×D4—21no—
249C22×D4C2412yes—
250C23×C4—22no—
251C2×Q16—28no—
252C2×Q16—44no—
253C2×Q16—44no—
254C2×SD16—24no—
255C2×SD16—42no—
256C2×SD16—42no—
257C22×C8C2412yes—
258C22×C8C2414yes—
259C22×C8C2412yes—
260C4⋊Q8—28no—
261C4⋊Q8—216no—
262C4⋊Q8—24no—
263C4⋊Q8—22no—
264C4⋊Q8—24no—
265C4⋊Q8—24no—
266C4⋊Q8—44no—
267C4⋊Q8—44no—
268C4⋊Q8C2×D418yes—
269C4⋊Q8C2×D414yes—
270#127—42no—
271#126—416no—
272#124—28no—
273#124—28no—
274#124—44no—
275#124—44no—
276C22⋊Q8—48no—
277C4×Q8—28no—
278C4×Q8—24no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—82no—
286C4×Q8—84no—
287C4×Q8C2×D414yes—
288C4×Q8C2×D418yes—
289C4×Q8C2×D414yes—
290C4×Q8#4114yes—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—44no—
294C4×D4—44no—
295#117—48no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C2×D411yescharacteristic
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2411yescharacteristic
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C42—22no—
337C2×C42—21no—
338C2×C42—24no—
339C2×C42—22no—
340C2×C42—21no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D412yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356C2×C42C2412yes—
357#106—24no—
358#106—28no—
359C4⋊C8—24no—
360C4⋊C8—28no—
361C4⋊C8—44no—
362C4⋊C8—42no—
363#103—24no—
364#103—28no—
365#103—28no—
366#103—28no—
367#103—44no—
368#103—44no—
369#103—44no—
370#103—44no—
371#102—24no—
372#102—28no—
373#102—42no—
374#102—44no—
375#102—42no—
376#97—28no—
377C4×C8—24no—
378#95—21no—
379#95—22no—
380#95—21no—
381#95—22no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—24no—
387#95—24no—
388#95—22no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—21no—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C2×Q8—28no—
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—82no—
413C2×Q8—81no—
414C2×Q8—82no—
415C2×Q8—81no—
416C2×Q8#12114yes—
417C2×Q8C22×D412yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—42no—
425C2×D4—42no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—42no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D412yes—
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C2511yescharacteristic
452Q16—416no—
453SD16—48no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C4⋊C4—28no—
464C4⋊C4—24no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—24no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—81no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4#12114yes—
496C4⋊C4#12118yes—
497C4⋊C4C22×D412yes—
498C4⋊C4C22×D414yes—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—28no—
507C42—24no—
508C42—24no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—22no—
514C42—22no—
515C42—42no—
516C42—41no—
517C42—42no—
518C42—44no—
519C42—48no—
520C42—42no—
521C42—82no—
522C42—82no—
523C42—82no—
524C42#13714yes—
525C42C22×D412yes—
526C42C22×D414yes—
527C42C2×#4112yes—
528C42C2×#4114yes—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23C23×D411yescharacteristic
537Q8—28no—
538Q8—24no—
539Q8—42no—
540Q8—44no—
541Q8—44no—
542Q8—44no—
543Q8—44no—
544Q8—88no—
545Q8—84no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—84no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—88no—
598C2×C4C2×#12114yes—
599C2×C4C2×#12114yes—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#13712yes—
603C2×C4C2×#13712yes—
604C2×C4C2×#13712yes—
605C2×C4#57512yes—
606C2×C4#57712yes—
607C2×C4C23×D411yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4#58512yes—
610C2×C4#58512yes—
611C8—42no—
612C8—42no—
613C8—44no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22C22×#12111yescharacteristic
622C22#348711yescharacteristic
623C22#350011yescharacteristic
624C22C22×#13711yescharacteristic
625C22C2×#57511yescharacteristic
626C22C2×#57711yescharacteristic
627C22#358611yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—41no—
639C4—42no—
640C4—81no—
641C4—84no—
642C4—82no—
643C4—81no—
644C4—82no—
645C4#303714yes—
646C4#348712yes—
647C4#352412yes—
648C2—162no—
649C2#3601811yescharacteristic
650C2C2×#303711yescharacteristic
651C2#6042911yescharacteristic
652C2#6188911yescharacteristic
653C2#6189911yescharacteristic
654C2#6254211yescharacteristic
655C2#6258111yescharacteristic
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×#3369C2semidirect
3. #34335C2semidirect
5. #29799C2semidirect
7. #28748C2semidirect
8. #28205C2semidirect
10. #25287C2semidirect
11. #25273C2semidirect
12. #24641C2semidirect
13. #24629C2semidirect
14. #23685C2semidirect
16. #22736C2semidirect
17. #22544C2semidirect
99. Q82D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33692
#365942
#343352
#298012
#297992
#287532
#287482
#282052
#282022
#252872
#252732
#246412
#246292
#236852
#236782
#227362
#225442