Finite Group Databasebeta

← #6795535

Subgroups — #6795535

Order 512.

1575 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#32306C214yesmaximal, complemented
5#31954C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28539C212yesmaximal, complemented
9#28284C211yesmaximal, characteristic, complemented
10#27738C212yesmaximal, complemented
11#27677C212yesmaximal, complemented
12#25706C212yesmaximal, complemented
13#24659C212yesmaximal, complemented
14#24118C212yesmaximal
15#22799C211yesmaximal, characteristic, complemented
16#21987C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#500C2212yes—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2572—22no—
28#2571C2211yescharacteristic
29#2567C2212yes—
30#2563C2211yescharacteristic
31#2563C2211yescharacteristic
32#2552—21no—
33#2552—21no—
34#2537C2211yescharacteristic
35#2502C2214yes—
36#2495—21no—
37#2495C2212yes—
38#2440—22no—
39#2440—22no—
40#2420—21no—
41#2420C2212yes—
42#2398—22no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×#121—21no—
48C4×#121C2212yes—
49C2×#384C2212yes—
50#2283—21no—
51#2283C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2097C2214yes—
56#2093C2214yes—
57#2083C2214yes—
58#2074C2214yes—
59#2068C2214yes—
60#2036C2214yes—
61#2036C2214yes—
62#2035C2214yes—
63#2000C2214yes—
64#1990C2212yes—
65#1987C2212yes—
66#1981C2212yes—
67#1981C2211yescharacteristic
68#1973C2214yes—
69#1954C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1947C2212yes—
76#1935C2214yes—
77#1927C2212yes—
78#1924C2214yes—
79#1877C2212yes—
80#1874C2214yes—
81#1853C2214yes—
82#1852C2214yes—
83#1852C2214yes—
84#1847C2214yes—
85#1842C2211yescharacteristic
86#1836C2214yes—
87#1803C2212yes—
88#1802C2211yescharacteristic
89#1802C2212yes—
90#1802C2212yes—
91#1797C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#554—44no—
97#554—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#125—22no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8C2314yes—
106C2×#121—21no—
107C2×#121C2312yes—
108C2×C4×Q8—21no—
109C2×C4×Q8C2312yes—
110C2×C4×D4—22no—
111C2×C4×D4—22no—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C22×C4⋊C4—22no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4C2314yes—
121C22×C42—21no—
122C22×C42C2312yes—
123#500—44no—
124#477—44no—
125#476—44no—
126#461—24no—
127C4×SD16—44no—
128#432—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#102C2314yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×#97C2314yes—
149C2×C4×C8C2312yes—
150#400—22no—
151#400—22no—
152#398—24no—
153#398—24no—
154#397—22no—
155#397—22no—
156#395—24no—
157#394—22no—
158#394—22no—
159#393—22no—
160#393—22no—
161#391—21no—
162#391C2312yes—
163#391C2314yes—
164#390—21no—
165#390C2312yes—
166#389—22no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389C2312yes—
175#389C2312yes—
176#388—22no—
177#388—22no—
178#386—22no—
179#386—22no—
180#385—24no—
181#385—24no—
182#384—21no—
183#384—21no—
184#384—22no—
185#384—22no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2314yes—
189#384C2312yes—
190#384C2312yes—
191#383—22no—
192#383—22no—
193#382—21no—
194#382—21no—
195#382C2312yes—
196#382C2312yes—
197#381—22no—
198#381—22no—
199#380—22no—
200#380—22no—
201#380—21no—
202#380—21no—
203#380—21no—
204#380C2312yes—
205#380C2312yes—
206#380C2312yes—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2312yes—
225#376—22no—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—24no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4C2×D412yes—
247C23×C4C2×D412yes—
248C23×C4C2412yes—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8C2412yes—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C4⋊Q8—22no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—22no—
261C4⋊Q8—82no—
262C4⋊Q8—82no—
263#125—48no—
264C22⋊Q8—416no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267#121—24no—
268#121—42no—
269#121—44no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—82no—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—24no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—41no—
283C4×D4—48no—
284C4×D4—84no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—84no—
307C2×C4⋊C4—84no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C22⋊C4C2×D412yes—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—21no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—44no—
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2412yes—
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—44no—
350#107—44no—
351#107—44no—
352#106—42no—
353#106—44no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—22no—
357C4⋊C8—42no—
358C4⋊C8#4114yes—
359#103—24no—
360#103—42no—
361#103—44no—
362#103—42no—
363#103—44no—
364#102—42no—
365#102—44no—
366C22⋊C8—28no—
367C22⋊C8—44no—
368#97—28no—
369C4×C8—24no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C24C22×D412yes—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—42no—
399C2×Q8—44no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—82no—
403C2×Q8—82no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—82no—
414C2×D4#13714yes—
415C2×D42−1+412yes—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—81no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C2511yescharacteristic
458SD16—48no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8C2×#4114yes—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—48no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—82no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C22⋊C4—82no—
516C22⋊C4—82no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—44no—
527C42—42no—
528C42—42no—
529C42—42no—
530C42—44no—
531C42#13714yes—
532C42C2×#4112yes—
533C42H(5,𝔽2)12yes—
534C42H(5,𝔽2)12yes—
535C422−1+412yes—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×#13714yes—
550C23C23×D411yescharacteristic
551Q8—42no—
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—84no—
556Q8—164no—
557D4—28no—
558D4—22no—
559D4—22no—
560D4—41no—
561D4—41no—
562D4—88no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4—168no—
615C2×C4#53512yes—
616C2×C4#53512yes—
617C2×C4#54712yes—
618C2×C4#54712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4#57612yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C2×H(5,𝔽2)11yescharacteristic
626C2×C4C2×H(5,𝔽2)11yescharacteristic
627C2×C4C2×2−1+411yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58511yescharacteristic
630C8—24no—
631C8—44no—
632C8—42no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#53511yescharacteristic
648C22C2×#54711yescharacteristic
649C22#348911yescharacteristic
650C22C22×#13711yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57611yescharacteristic
653C22#358511yescharacteristic
654C4—22no—
655C4—22no—
656C4—22no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—41no—
661C4—82no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—82no—
666C4—84no—
667C4—82no—
668C4—82no—
669C4#321814yes—
670C4#335714yes—
671C4#351711yescharacteristic
672C4#352011yescharacteristic
673C4#354711yescharacteristic
674C4#355211yescharacteristic
675C2—44no—
676C2—82no—
677C2#3892011yescharacteristic
678C2C2×#321811yescharacteristic
679C2#6083211yescharacteristic
680C2C2×#335711yescharacteristic
681C2#6125411yescharacteristic
682C2#6210811yescharacteristic
683C2#6212711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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
1. #36180C2semidirect
3. #33197C2semidirect
4. #32306C2semidirect
5. #31954C2semidirect
6. #29926C2semidirect
7. #28689C2semidirect
8. #28539C2semidirect
9. #28284C2semidirect
10. #27738C2semidirect
11. #27677C2semidirect
12. #25706C2semidirect
13. #24659C2semidirect
15. #22799C2semidirect
16. #21987C2semidirect
17. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#349432
#331972
#323062
#319542
#299262
#286892
#285392
#282842
#277382
#276772
#257062
#246592
#241182
#227992
#219872
C4×#4612