Finite Group Databasebeta

← #6804736

Subgroups — #6804736

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36608C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#32560C214yesmaximal, complemented
5#31823C214yesmaximal, complemented
6#30229C211yesmaximal, characteristic, complemented
7#30206C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29191C211yesmaximal, characteristic, complemented
10#29177C211yesmaximal, characteristic, complemented
11#28718C211yesmaximal, characteristic
12#28613C211yesmaximal, characteristic
13#28613C211yesmaximal, characteristic
14#27579C214yesmaximal, complemented
15#27451C214yesmaximal, complemented
16#22268C211yesmaximal, characteristic
17#22219C211yesmaximal, characteristic
18#21646C211yesmaximal, characteristic
19#21340C211yesmaximal, characteristic, complemented
20C2×#492C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#486C2211yescharacteristic
23#2862C2211yescharacteristic
24#2862C2211yescharacteristic
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2581—21no—
30#2581—21no—
31#2581C2212yes—
32#2580—21no—
33#2580—21no—
34#2580C2212yes—
35#2514—21no—
36#2514—21no—
37#2514—21no—
38#2514—21no—
39#2514C2212yes—
40#2514C2212yes—
41#2454—21no—
42#2454—21no—
43#2454—21no—
44#2454—21no—
45#2454C2212yes—
46#2454C2212yes—
47#2356—21no—
48#2356—21no—
49#2356C2212yes—
50#2349—21no—
51#2349—21no—
52#2349C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2101C2214yes—
62#2101C2214yes—
63#2088C2214yes—
64#2074C2214yes—
65#2073C2214yes—
66#2066C2214yes—
67#2055C2214yes—
68#2055C2214yes—
69#2026C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1989C2214yes—
77#1985C2212yes—
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1940C2212yes—
83#1936C2214yes—
84#1935C2214yes—
85#1935C2214yes—
86#1926C2214yes—
87#1919C2214yes—
88#1917C2212yes—
89#1917C2212yes—
90#1880C2214yes—
91#1877C2214yes—
92#1877C2214yes—
93#1877C2214yes—
94#1865C2214yes—
95#1864C2214yes—
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1804C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103#1762C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130#492—24no—
131#486—24no—
132#486—24no—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#401—22no—
150#401—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#399—22no—
155#399—22no—
156#399—22no—
157#396—22no—
158#396—22no—
159#395—22no—
160#393—22no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#390C2312yes—
184#390C2312yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#385—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#383—22no—
220#383—22no—
221#383—22no—
222#382—21no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—21no—
227#382—21no—
228#382C2312yes—
229#382C2312yes—
230#382C2312yes—
231#380—22no—
232#380—22no—
233#380—22no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249#376—22no—
250C2×#95—22no—
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#336C2312yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269C22×Q8—21no—
270C22×Q8—21no—
271C22×Q8C2×D411yescharacteristic
272C22×Q8C2412yes—
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2412yes—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—22no—
279C2×SD16—28no—
280C2×SD16—44no—
281C2×SD16—44no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292#127—22no—
293#127#4114yes—
294#125—44no—
295#125—82no—
296#125—82no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—42no—
300C4×Q8—44no—
301C4×Q8—44no—
302C4×Q8—44no—
303C4×D4—44no—
304C4×D4—44no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—82no—
309C4×D4—82no—
310C4×D4—82no—
311C4×D4—82no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
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×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#106—44no—
380C4⋊C8—42no—
381C4⋊C8—42no—
382C4⋊C8—42no—
383#103—44no—
384#103—44no—
385#103—44no—
386#102—28no—
387#102—28no—
388#102—44no—
389#97—42no—
390#97—42no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—22no—
394C4×C8—22no—
395#95—22no—
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#95C2×D411yescharacteristic
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425C24—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—41no—
431C2×Q8—41no—
432C2×Q8—41no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4C2×#4114yes—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2511yescharacteristic
480SD16—88no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—44no—
559C42—44no—
560C42—44no—
561C42C2×#4112yes—
562C422−1+412yes—
563C23—24no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23C23×D411yescharacteristic
571Q8—44no—
572Q8—44no—
573Q8—44no—
574Q8—84no—
575Q8—84no—
576D4—28no—
577D4—48no—
578D4—48no—
579D4—44no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
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—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
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—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—164no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4#54812yes—
641C2×C4#54812yes—
642C2×C4#57512yes—
643C2×C4#57512yes—
644C2×C4#57712yes—
645C2×C4#57712yes—
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×2−1+411yescharacteristic
648C2×C4#58512yes—
649C2×C4#58512yes—
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22#344911yescharacteristic
662C22C2×#54811yescharacteristic
663C22#350011yescharacteristic
664C22C2×#57511yescharacteristic
665C22C2×#57511yescharacteristic
666C22C2×#57711yescharacteristic
667C22C2×#57711yescharacteristic
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#337614yes—
688C4#337614yes—
689C4#352112yes—
690C4#355712yes—
691C2—84no—
692C2#3916311yescharacteristic
693C2#6097211yescharacteristic
694C2#6101211yescharacteristic
695C2C2×#337611yescharacteristic
696C2C2×#337611yescharacteristic
697C2#6254411yescharacteristic
698C2#6254411yescharacteristic
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. #36608C2semidirect
3. #33287C2semidirect
4. #32560C2semidirect
5. #31823C2semidirect
6. #30229C2semidirect
7. #30206C2semidirect
8. #29220C2semidirect
9. #29191C2semidirect
10. #29177C2semidirect
14. #27579C2semidirect
15. #27451C2semidirect
19. #21340C2semidirect

Maximal subgroups

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

subgroupindex
#366082
#365732
#332872
#325602
#318232
#302292
#302062
#292202
#291912
#291772
#287182
#286132
#275792
#274512
#222682
#222192
#216462
#213402