Finite Group Databasebeta

← #6480735

Subgroups — #6480735

Order 512.

1416 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#36635C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33258C211yesmaximal, characteristic, complemented
6#33256C211yesmaximal, characteristic, complemented
7#30380C214yesmaximal, complemented
8#29577C214yesmaximal, complemented
9#29193C211yesmaximal, characteristic, complemented
10#29191C211yesmaximal, characteristic, complemented
11#29178C211yesmaximal, characteristic, complemented
12#29176C211yesmaximal, characteristic, complemented
13#27975C211yesmaximal, characteristic
14#27962C211yesmaximal, characteristic
15#27949C211yesmaximal, characteristic
16#27908C211yesmaximal, characteristic
17#24604C214yesmaximal
18#22077C214yesmaximal
19C4×#431C211yesmaximal, characteristic
20C2×#485C2214yes—
21C2×#431C2212yes—
22#2864—21no—
23#2864C2212yes—
24#2864C2212yes—
25#2863—21no—
26#2863C2212yes—
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2859C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2855—21no—
36#2855C2212yes—
37#2851—21no—
38#2851—21no—
39#2851—21no—
40#2851—21no—
41#2851—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×#125—21no—
45C4×#125—21no—
46C4×#125—21no—
47C4×#125—21no—
48C4×#125—21no—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8C2212yes—
51#2296C2212yes—
52#2296C2212yes—
53#2295—21no—
54#2295C2212yes—
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57#2283—21no—
58C2×C43C2211yescharacteristic
59#2101C2214yes—
60#2097C2214yes—
61#2088C2214yes—
62#2086C2214yes—
63#2085C2214yes—
64#2074C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#1990C2211yescharacteristic
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1956C2211yescharacteristic
72#1956C2211yescharacteristic
73#1956C2211yescharacteristic
74#1956C2211yescharacteristic
75#1942C2214yes—
76#1942C2214yes—
77#1941C2214yes—
78#1941C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1933C2214yes—
82#1933C2214yes—
83#1920C2214yes—
84#1918C2214yes—
85#1873C2214yes—
86#1865C2214yes—
87#1863C2214yes—
88#1854C2214yes—
89#1845C2211yescharacteristic
90#1845C2211yescharacteristic
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1825C2214yes—
98C4×C4⋊C8C2211yescharacteristic
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2212yes—
101C4×C22⋊C8C2211yescharacteristic
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2212yes—
104#1744C2212yes—
105C42×C8C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#125—21no—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C4×Q8—21no—
117C2×C4×Q8C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#485—216no—
128#431—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2314yes—
132C2×#106C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2312yes—
148C2×C4×C8C2312yes—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391C2312yes—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#383—22no—
214#382—21no—
215#382—21no—
216#382—21no—
217#382—21no—
218#382—21no—
219#382—21no—
220#382—21no—
221#382C2312yes—
222#382C2312yes—
223#381—22no—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380C2312yes—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—22no—
243C4×C4⋊C4—22no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4C2312yes—
246C4×C22⋊C4—22no—
247#376—22no—
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43D411yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268C22×Q8—21no—
269C22×Q8C2412yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285#125—44no—
286#125—44no—
287#125—44no—
288#125—44no—
289#125—44no—
290C22⋊Q8—28no—
291C22⋊Q8—44no—
292C4×Q8—44no—
293C4×Q8—44no—
294#117—28no—
295#117—44no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—42no—
347C2×C42—42no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2×D411yescharacteristic
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—28no—
377#107—44no—
378#106—28no—
379#106—44no—
380C4⋊C8—24no—
381C4⋊C8—24no—
382#103—28no—
383#103—28no—
384#103—44no—
385#103—44no—
386C22⋊C8—24no—
387C22⋊C8#4118yes—
388C4×C8—24no—
389C4×C8—24no—
390C4×C8—22no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—22no—
394C4×C8—42no—
395C4×C8—42no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—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—
422C24C22×D411yescharacteristic
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
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—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
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×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4114yes—
473C22×C4C2511yescharacteristic
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×C8C2×#4114yes—
485C2×C8C2×#4114yes—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—21no—
545C42—21no—
546C42—22no—
547C42—22no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—44no—
553C42—44no—
554C42—44no—
555C42—44no—
556C42#13514yes—
557C42#13514yes—
558C42#13714yes—
559C422−1+412yes—
560C422−1+412yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23#53212yes—
568C23C23×D411yescharacteristic
569Q8—44no—
570Q8—44no—
571Q8—84no—
572C2×C4—28no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
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—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
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#53212yes—
631C2×C4#53612yes—
632C2×C4#53612yes—
633C2×C4C2×#13512yes—
634C2×C4C2×#13512yes—
635C2×C4C2×#13512yes—
636C2×C4C2×#13512yes—
637C2×C4C2×#13512yes—
638C2×C4C2×#13512yes—
639C2×C4C2×#13712yes—
640C2×C4C2×#13712yes—
641C2×C4C2×#13712yes—
642C2×C4#57512yes—
643C2×C4C22×#4112yes—
644C2×C4C2×2−1+411yescharacteristic
645C2×C4C2×2−1+411yescharacteristic
646C2×C4#58512yes—
647C8—24no—
648C8—24no—
649C8—42no—
650C8—42no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22C2×#53211yescharacteristic
659C22C2×#53611yescharacteristic
660C22#344911yescharacteristic
661C22C22×#13511yescharacteristic
662C22C22×#13511yescharacteristic
663C22C22×#13711yescharacteristic
664C22C2×#57511yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#323114yes—
685C4#323314yes—
686C4#351512yes—
687C4#351512yes—
688C2—42no—
689C2#3513911yescharacteristic
690C2#6067311yescharacteristic
691C2#6070111yescharacteristic
692C2C2×#323111yescharacteristic
693C2C2×#323311yescharacteristic
694C2#6094411yescharacteristic
695C2#6094711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33278C2semidirect
4. #33278C2semidirect
5. #33258C2semidirect
6. #33256C2semidirect
7. #30380C2semidirect
8. #29577C2semidirect
9. #29193C2semidirect
10. #29191C2semidirect
11. #29178C2semidirect
12. #29176C2semidirect

Maximal subgroups

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

subgroupindex
#366402
#366352
#332782
#332582
#332562
#303802
#295772
#291932
#291912
#291782
#291762
#279752
#279622
#279492
#279082
#246042
#220772
C4×#4312