Finite Group Databasebeta

← #6853753

Subgroups — #6853753

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C211yesmaximal, characteristic, complemented
2#36573C211yesmaximal, characteristic
3#33258C211yesmaximal, characteristic, complemented
4#32569C214yesmaximal, complemented
5#31871C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#30307C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29190C211yesmaximal, characteristic
10#29190C211yesmaximal, characteristic
11#29177C211yesmaximal, characteristic, complemented
12#29176C211yesmaximal, characteristic, complemented
13#29143C211yesmaximal, characteristic
14#27302C214yesmaximal, complemented
15#27006C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic
17#22220C211yesmaximal, characteristic
18C4×#486C211yesmaximal, characteristic
19#21339C211yesmaximal, characteristic, complemented
20C2×#486C2211yescharacteristic
21#2859C2211yescharacteristic
22#2859C2211yescharacteristic
23#2859C2211yescharacteristic
24#2859C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2584—21no—
28#2584C2212yes—
29#2581—21no—
30#2580—21no—
31#2580—21no—
32#2580C2212yes—
33#2509—21no—
34#2509C2212yes—
35#2508—21no—
36#2508C2212yes—
37#2454—21no—
38#2454—21no—
39#2454—21no—
40#2454—21no—
41#2454C2212yes—
42#2454C2212yes—
43#2453—21no—
44#2453—21no—
45#2370—21no—
46#2370C2212yes—
47#2351—21no—
48#2349—21no—
49#2349—21no—
50#2349C2212yes—
51C4×C4⋊Q8C2211yescharacteristic
52C4×C4⋊Q8C2211yescharacteristic
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C42×Q8C2211yescharacteristic
58#2101C2214yes—
59#2088C2214yes—
60#2088C2214yes—
61#2085C2214yes—
62#2085C2214yes—
63#2065C2214yes—
64#2065C2214yes—
65#2059C2214yes—
66#2059C2214yes—
67#2025C2214yes—
68#1990C2211yescharacteristic
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1989C2214yes—
72#1984C2212yes—
73#1957C2211yescharacteristic
74#1957C2211yescharacteristic
75#1957C2211yescharacteristic
76#1942C2214yes—
77#1942C2214yes—
78#1939C2212yes—
79#1934C2214yes—
80#1925C2214yes—
81#1925C2214yes—
82#1920C2214yes—
83#1919C2214yes—
84#1916C2212yes—
85#1916C2212yes—
86#1864C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1862C2214yes—
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1804C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#102C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#125—21no—
110C2×#125—21no—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#125C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D411yescharacteristic
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#486Q818yescomplemented
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2312yes—
146C2×C4×C8C2312yes—
147C2×C4×C8C2311yescharacteristic
148#400—22no—
149#400—22no—
150#400—22no—
151#399—22no—
152#399—22no—
153#399—22no—
154#398—22no—
155#398—22no—
156#396—22no—
157#396—22no—
158#395—22no—
159#393—22no—
160#391—21no—
161#391—21no—
162#391—21no—
163#390—21no—
164#390—21no—
165#390—21no—
166#390—21no—
167#390—21no—
168#390—21no—
169#390C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
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#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#385—22no—
211#385—22no—
212#385—22no—
213#385—22no—
214#385—22no—
215#385—22no—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#382—21no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382—22no—
227#382C2312yes—
228#382C2312yes—
229#380—22no—
230#380—22no—
231#380—22no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4D411yescharacteristic
237C4×C4⋊C4D411yescharacteristic
238C4×C4⋊C4D411yescharacteristic
239C4×C4⋊C4D411yescharacteristic
240C4×C4⋊C4D411yescharacteristic
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2311yescharacteristic
246C4×C4⋊C4C2311yescharacteristic
247C4×C4⋊C4C2311yescharacteristic
248C2×#95—22no—
249C43C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#336C2312yes—
261#336C2312yes—
262#336C2312yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267C22×Q8—21no—
268C22×Q8C2×D411yescharacteristic
269C22×D4—21no—
270C22×D4—21no—
271C22×D4C2412yes—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C4⋊Q8—22no—
280C4⋊Q8C2×Q814yes—
281#127—22no—
282#127C2×Q814yes—
283#125—44no—
284#125—44no—
285#125—44no—
286#125—82no—
287#125—82no—
288#125—82no—
289C4×Q8—24no—
290C4×Q8—42no—
291C4×Q8—42no—
292C4×Q8—44no—
293C4×Q8—82no—
294C4×Q8—82no—
295C4×D4—44no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—44no—
299C4×D4—44no—
300C4×D4—82no—
301C4×D4—82no—
302C4×D4—82no—
303C4×D4—82no—
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—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
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#107—44no—
380#107—44no—
381#106—44no—
382#106—44no—
383C4⋊C8—42no—
384C4⋊C8—42no—
385C4⋊C8—42no—
386#102—44no—
387#102—44no—
388#102—44no—
389#102C2×Q8116yes—
390C4×C8—22no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—42no—
394C4×C8—42no—
395C4×C8C2×Q814yes—
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#95C2×D411yescharacteristic
415#95C2×D411yescharacteristic
416#95C2×D411yescharacteristic
417#95C2×D411yescharacteristic
418#95C2×D411yescharacteristic
419#95C2×D411yescharacteristic
420C24—22no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—41no—
424C2×Q8—41no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C22×Q814yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8C22×Q814yes—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4C22×Q814yes—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—44no—
552C42—82no—
553C42—82no—
554C42#13514yes—
555C42C22×Q812yes—
556C422−1+412yes—
557C23—24no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23C23×D411yescharacteristic
565Q8—44no—
566Q8—48no—
567D4—28no—
568D4—48no—
569D4—48no—
570D4—44no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—24no—
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—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—164no—
627C2×C4—164no—
628C2×C4—164no—
629C2×C4D4×Q812yes—
630C2×C4D4×Q812yes—
631C2×C4C2×#13512yes—
632C2×C4C2×#13512yes—
633C2×C4C2×#13512yes—
634C2×C4#57512yes—
635C2×C4#57612yes—
636C2×C4#57712yes—
637C2×C4C23×Q811yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58512yes—
640C2×C4#58512yes—
641C8—24no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22#344911yescharacteristic
653C22C2×D4×Q811yescharacteristic
654C22#350011yescharacteristic
655C22C22×#13511yescharacteristic
656C22C2×#57511yescharacteristic
657C22C2×#57611yescharacteristic
658C22C2×#57711yescharacteristic
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—44no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4#338014yes—
679C4#338814yes—
680C4#350412yes—
681C4#356512yes—
682C2—84no—
683C2#3961311yescharacteristic
684C2#6094411yescharacteristic
685C2#6100611yescharacteristic
686C2C2×#338011yescharacteristic
687C2C2×#338811yescharacteristic
688C2#6253111yescharacteristic
689C2#6254611yescharacteristic
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. #36574C2semidirect
3. #33258C2semidirect
4. #32569C2semidirect
5. #31871C2semidirect
6. #30307C2semidirect
7. #30307C2semidirect
8. #29220C2semidirect
11. #29177C2semidirect
12. #29176C2semidirect
14. #27302C2semidirect
15. #27006C2semidirect
19. #21339C2semidirect
130. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
#365732
#332582
#325692
#318712
#303072
#292202
#291902
#291772
#291762
#291432
#273022
#270062
#222202
C4×#4862
#213392