Finite Group Databasebeta

← #6606603

Subgroups — #6606603

Order 512.

1857 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3356C212yesmaximal
2#36188C211yesmaximal, characteristic
3C4×#554C211yesmaximal, characteristic
4#33196C211yesmaximal, characteristic, complemented
5#33108C212yesmaximal, complemented
6#28687C211yesmaximal, characteristic, complemented
7#28284C211yesmaximal, characteristic, complemented
8#27649C212yesmaximal, complemented
9#25454C212yesmaximal, complemented
10#25220C214yesmaximal, complemented
11#24760C212yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#24590C214yesmaximal, complemented
14#22650C211yesmaximal, characteristic
15#22504C212yesmaximal
16C4×#462C211yesmaximal, characteristic, complemented
17#21170C212yesmaximal, complemented
18C2×#556C2212yes—
19C2×#554C2211yescharacteristic
20#3356—216no—
21C2×#498C2212yes—
22C2×#485C2214yes—
23C2×#483C2214yes—
24C2×#477C2212yes—
25C2×#475C2212yes—
26C2×#470C2214yes—
27C2×#462C2211yescharacteristic
28C2×C4×Q16C2212yes—
29C2×#436C2212yes—
30C2×#428C2214yes—
31C2×#423C2212yes—
32#2868C2211yescharacteristic
33#2857C2212yes—
34#2568C2212yes—
35#2547—21no—
36#2547—21no—
37#2499C2214yes—
38#2496—22no—
39#2496—22no—
40#2370—21no—
41#2369C2211yescharacteristic
42#2368—22no—
43#2365—21no—
44#2365—21no—
45#2360C2212yes—
46#2355C2211yescharacteristic
47#2349—21no—
48D4×C4⋊C4C2211yescharacteristic
49#2345C2212yes—
50C4×C4⋊Q8C2211yescharacteristic
51C4×C4⋊Q8C2211yescharacteristic
52C4×#125—21no—
53C4×C22⋊Q8—22no—
54C4×C22⋊Q8C2212yes—
55#2285C2212yes—
56C42×Q8C2211yescharacteristic
57C42×D4—21no—
58C42×D4C2211yescharacteristic
59C2×C4×C4⋊C4C2212yes—
60#2085C2214yes—
61#2084C2214yes—
62#2074C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#1990C2212yes—
67#1988C2212yes—
68#1981C2211yescharacteristic
69#1981C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1948C2212yes—
73#1948C2212yes—
74#1948C2212yes—
75#1948C2212yes—
76#1947C2212yes—
77#1947C2211yescharacteristic
78#1939C2212yes—
79#1928C2212yes—
80#1927C2214yes—
81#1927C2214yes—
82#1894C2214yes—
83#1878C2212yes—
84#1872C2214yes—
85#1851C2214yes—
86#1842C2211yescharacteristic
87#1825C2214yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91#1803C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#1736C2214yes—
96C22×Q16C2312yes—
97C22×M16C2314yes—
98#556—28no—
99#556—28no—
100#554—24no—
101#554D418yes—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#126—24no—
108C2×#125—21no—
109C2×#124—22no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C2314yes—
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—22no—
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42—22no—
128C22×C42C2312yes—
129#498—28no—
130#485—216no—
131#483—216no—
132#477—28no—
133#475—28no—
134#470—216no—
135#462—24no—
136C4×Q16—28no—
137#436—28no—
138#428—216no—
139C2×#107C2314yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144#423—28no—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C4×C8C2312yes—
160#400—24no—
161#397—24no—
162#397—24no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391—22no—
167#391C2312yes—
168#391C2312yes—
169#391C2314yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2312yes—
179#389C2312yes—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—24no—
186#387—24no—
187#385—22no—
188#385—22no—
189#385—22no—
190#384—21no—
191#384—21no—
192#384C2311yescharacteristic
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#382C2314yes—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380C2312yes—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D411yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231C4×C22⋊C4C2312yes—
232C2×#95—22no—
233C43—21no—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2312yes—
242#336C2314yes—
243#336C2312yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4C2×D411yescharacteristic
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2×D412yes—
253C23×C4C2412yes—
254C2×Q16—24no—
255C2×Q16—24no—
256C2×Q16—42no—
257C2×M16—28no—
258C2×M16—28no—
259C2×M16—44no—
260C22×C8C2414yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C4⋊Q8—24no—
264C4⋊Q8—22no—
265C4⋊Q8—24no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270C4⋊Q8C2×D414yes—
271#126—416no—
272#125—28no—
273#124—48no—
274C22⋊Q8—24no—
275C22⋊Q8—216no—
276C22⋊Q8—28no—
277C22⋊Q8—28no—
278C22⋊Q8C2×D418yes—
279C4×Q8—22no—
280C4×Q8—22no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—41no—
286C4×Q8—48no—
287C4×Q8—82no—
288C4×Q8C2×D414yes—
289C4×D4—28no—
290C4×D4—24no—
291C4×D4—42no—
292C4×D4—44no—
293C4×D4C2×D414yes—
294C4×D4#4114yes—
295#117—28no—
296#117—28no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D418yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2414yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—21no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—21no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—84no—
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D412yes—
357C2×C42C2×D412yes—
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364#107—24no—
365#107—28no—
366#106—28no—
367#106—28no—
368#106—28no—
369C4⋊C8—28no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—42no—
374C4⋊C8#4114yes—
375#103—24no—
376#103—28no—
377#103—24no—
378#103—24no—
379#103—28no—
380#103—24no—
381#103—44no—
382C22⋊C8—28no—
383C4×C8—24no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—21no—
392#95—21no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24C22×D412yes—
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—21no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—22no—
417C2×Q8—44no—
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8C22×D412yes—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4#12114yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D414yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C2×#4114yes—
462C22×C4C2511yescharacteristic
463Q16—48no—
464M16—416no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—24no—
473C2×C8C2×#4114yes—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—82no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4#12114yes—
510C4⋊C4C22×D412yes—
511C4⋊C4C22×D414yes—
512C4⋊C42−1+412yes—
513C22⋊C4—28no—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4#12118yes—
520C42—24no—
521C42—24no—
522C42—24no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42—42no—
531C42—48no—
532C42—44no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—44no—
537C42—81no—
538C42—82no—
539C42—82no—
540C42#13714yes—
541C42C22×D412yes—
542C42C2×#4112yes—
543C42C2×#4112yes—
544C422−1+412yes—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23C2×#12114yes—
552C23C23×D411yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—42no—
558Q8—42no—
559Q8—42no—
560Q8—42no—
561Q8—84no—
562Q8—88no—
563D4—28no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—84no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4C2×#12114yes—
617C2×C4C2×#12112yes—
618C2×C4C2×#12112yes—
619C2×C4#54712yes—
620C2×C4#54712yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57512yes—
625C2×C4#57712yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C22×#4111yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4#58511yescharacteristic
632C8—24no—
633C8—44no—
634C8—42no—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22C22×#12111yescharacteristic
643C22C2×#54711yescharacteristic
644C22#348911yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57511yescharacteristic
647C22C2×#57711yescharacteristic
648C22#358611yescharacteristic
649C4—22no—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—82no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4#303714yes—
666C4#335614yescomplemented
667C4D4×#4111yescharacteristic
668C4#348211yescharacteristic
669C4#349911yescharacteristic
670C4#351511yescharacteristic
671C2—44no—
672C2#3625911yescharacteristic
673C2C2×#303711yescharacteristic
674C2#6042911yescharacteristic
675C2C2×#335611yescharacteristic
676C2#6125711yescharacteristic
677C2#6211311yescharacteristic
678C2#6214411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33196C2semidirect
5. #33108C2semidirect
6. #28687C2semidirect
7. #28284C2semidirect
8. #27649C2semidirect
9. #25454C2semidirect
10. #25220C2semidirect
11. #24760C2semidirect
12. #24683C2semidirect
13. #24590C2semidirect
16. C4×#462C2semidirect
17. #21170C2semidirect
666. C4#3356semidirect

Maximal subgroups

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

subgroupindex
C2×#33562
#361882
C4×#5542
#331962
#331082
#286872
#282842
#276492
#254542
#252202
#247602
#246832
#245902
#226502
#225042
C4×#4622
#211702