Finite Group Databasebeta

← #6482028

Subgroups — #6482028

Order 512.

1913 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35236C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3#32985C212yesmaximal, complemented
4#32981C212yesmaximal, complemented
5#29895C212yesmaximal, complemented
6#29890C212yesmaximal, complemented
7#26311C214yesmaximal, complemented
8#26243C214yesmaximal, complemented
9#25017C212yesmaximal, complemented
10#25011C212yesmaximal, complemented
11#24970C212yesmaximal, complemented
12#24968C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21867C211yesmaximal, characteristic, complemented
15#21788C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic, complemented
17#21365C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#485C2214yescomplemented
22C2×#482C2214yescomplemented
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2848C2212yes—
26#2847C2212yes—
27#2636—22no—
28#2562C2212yes—
29#2540C2212yes—
30#2538C2211yescharacteristic
31#2538C2211yescharacteristic
32#2537C2211yescharacteristic
33#2537C2211yescharacteristic
34#2487—21no—
35#2411—22no—
36#2408—21no—
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2214yes—
41#2354C2212yes—
42#2354C2212yes—
43#2324—21no—
44#2324—21no—
45C2×#390—21no—
46C2×#388—22no—
47#2285C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2076C2214yescomplemented
53#2076C2214yescomplemented
54#2075C2214yescomplemented
55#2075C2214yescomplemented
56#2036C2214yescomplemented
57#2036C2214yescomplemented
58#2033C2214yescomplemented
59#2033C2214yescomplemented
60#1992C2212yescomplemented
61#1989C2212yescomplemented
62#1983C2212yescomplemented
63#1959C2212yescomplemented
64#1955C2214yes—
65#1954C2212yes—
66#1954C2212yes—
67#1948C2212yescomplemented
68#1947C2212yescomplemented
69#1947C2214yescomplemented
70#1875C2214yescomplemented
71#1874C2212yes—
72#1874C2211yescharacteristic
73#1874C2211yescharacteristic
74#1873C2214yescomplemented
75#1872C2212yes—
76#1872C2211yescharacteristic
77#1872C2211yescharacteristic
78#1854C2214yescomplemented
79#1854C2214yescomplemented
80#1853C2212yes—
81#1852C2212yes—
82#1852C2212yes—
83#1852C2212yes—
84#1845C2212yescomplemented
85#1834C2212yescomplemented
86#1825C2214yescomplemented
87#1825C2214yescomplemented
88#1803C2212yescomplemented
89#1802C2212yescomplemented
90#1779C2212yes—
91#1779C2212yes—
92#1776C2214yescomplemented
93C4×#103C2212yescomplemented
94C4×#102C2212yescomplemented
95C23×D4D411yescharacteristic
96C24×C4C2312yes—
97#529—24no—
98#529—24no—
99C2×#126—21no—
100C2×#125C2311yescharacteristic
101C2×#124C2311yescharacteristic
102C2×#123—21no—
103C2×C22⋊Q8C2311yescharacteristic
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2311yescharacteristic
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4—24no—
115C2×C4×D4—24no—
116C2×C4×D4D411yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—48no—
128C2×#117C2314yes—
129C2×#117C2314yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4D411yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—22no—
137C22×C22⋊C4—22no—
138C22×C22⋊C4D412yes—
139C22×C42—22no—
140C22×C42—21no—
141C22×C42C2311yescharacteristic
142#485—48no—
143#482—48no—
144#432—24no—
145#431—24no—
146C2×#107C2314yes—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2312yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#102C2312yes—
156C2×#102C2314yes—
157C2×#102C2312yes—
158C2×C22⋊C8C2311yescharacteristic
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2314yes—
162C2×C22⋊C8C2314yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2314yes—
165C2×#97C2312yes—
166C2×C4×C8C2312yes—
167#400—22no—
168#399—22no—
169#396—22no—
170#396—22no—
171#394—22no—
172#394—22no—
173#394—22no—
174#392—22no—
175#390—21no—
176#390—21no—
177#390—22no—
178#390—22no—
179#390—24no—
180#390—22no—
181#390—22no—
182#389—21no—
183#389—21no—
184#388—22no—
185#388—22no—
186#388—24no—
187#388—24no—
188#388—24no—
189#388—24no—
190#386—22no—
191#386—22no—
192#386—22no—
193#385—22no—
194#385—22no—
195#385—22no—
196#384—21no—
197#384—21no—
198#384C2312yes—
199#384C2311yescharacteristic
200#384C2312yes—
201#384C2311yescharacteristic
202#382—21no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—21no—
207#380—21no—
208#380C2312yes—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212#380C2314yes—
213#380C2314yes—
214#380C2312yes—
215#379—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—24no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C2×#95—22no—
226C2×#95—21no—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237C25C2×D412yes—
238C22×Q8C2411yescharacteristic
239C22×D4—21no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—24no—
243C22×D4—42no—
244C22×D4—42no—
245C22×D4C2×D411yescharacteristic
246C22×D4C2×D411yescharacteristic
247C22×D4C2×D412yes—
248C22×D4C2411yescharacteristic
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C23×C4C2412yes—
264C23×C4C2411yescharacteristic
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2414yes—
268#126—44no—
269#125—22no—
270#125—22no—
271#124—22no—
272#124—22no—
273#123—44no—
274C22⋊Q8—22no—
275C22⋊Q8—22no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278#121—22no—
279#121—22no—
280#121—44no—
281#121—44no—
282#121—44no—
283#121—44no—
284#120—84no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4—28no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—24no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—44no—
299#117—28no—
300#117—28no—
301#117—44no—
302#117—44no—
303#117—48no—
304#117—48no—
305#117—44no—
306#117—82no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
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—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2411yescharacteristic
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4—84no—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C22⋊C4C2×D412yes—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—48no—
372C2×C42—44no—
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42#4114yes—
376C2×C42#4114yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—44no—
380#107—44no—
381#107—44no—
382#106—28no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385#103—24no—
386#103—42no—
387#103—44no—
388#102—24no—
389#102—42no—
390#102—44no—
391C22⋊C8—22no—
392C22⋊C8—22no—
393C22⋊C8—24no—
394C22⋊C8—44no—
395C22⋊C8—44no—
396C22⋊C8—44no—
397C22⋊C8—44no—
398#97—24no—
399C4×C8—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—24no—
411C24—24no—
412C24—24no—
413C24—21no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24C22×D412yes—
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—416no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—82no—
445C2×D4—82no—
446C2×D4—82no—
447C2×D4—82no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—21no—
472C22×C4—24no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4#12314yes—
494C22×C4#12314yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2511yescharacteristic
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—24no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4H(5,𝔽2)14yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—24no—
549C22⋊C4—24no—
550C22⋊C4—48no—
551C22⋊C4—48no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—84no—
563C22⋊C4—84no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C42—48no—
569C42—48no—
570C42—44no—
571C42—42no—
572C42—44no—
573C42—82no—
574C42#12314yes—
575C42#12314yes—
576C42C2×#4112yes—
577C42C2×#4112yes—
578C42C2×#4112yes—
579C42C2×#4112yes—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—48no—
599C23—48no—
600C23—48no—
601C23—41no—
602C23—41no—
603C23—41no—
604C23—41no—
605C23—41no—
606C23—41no—
607C23—41no—
608C23—84no—
609C23—84no—
610C23—84no—
611C23—84no—
612C23C2×#12312yes—
613C23#57612yes—
614C23#57812yes—
615C23C23×D411yescharacteristic
616Q8—42no—
617Q8—42no—
618D4—48no—
619D4—44no—
620D4—44no—
621D4—42no—
622D4—42no—
623D4—84no—
624D4—164no—
625D4—164no—
626C2×C4—28no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—48no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—44no—
675C2×C4—41no—
676C2×C4—84no—
677C2×C4—88no—
678C2×C4—84no—
679C2×C4C2×#12312yes—
680C2×C4C2×#12312yes—
681C2×C4C2×#12312yes—
682C2×C4C2×#12312yes—
683C2×C4C2×#12312yes—
684C2×C4#57512yes—
685C2×C4#57512yes—
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C2×H(5,𝔽2)12yes—
689C2×C4#58512yes—
690C8—44no—
691C8—42no—
692C8—42no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—42no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22—42no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—816no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22—81no—
720C22—81no—
721C22#309814yes—
722C22C22×#12311yescharacteristic
723C22#348411yescharacteristic
724C22#348511yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57511yescharacteristic
727C22C2×#57611yescharacteristic
728C22C2×#57811yescharacteristic
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—42no—
734C4—42no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4—81no—
739C4—81no—
740C4—81no—
741C4—84no—
742C4—82no—
743C4#309314yes—
744C4#346012yes—
745C4#350612yes—
746C2—24no—
747C2—82no—
748C2—82no—
749C2—161no—
750C2#3515211yescharacteristic
751C2C2×#309311yescharacteristic
752C2C2×#309811yescharacteristic
753C2#6159511yescharacteristic
754C2#6160511yescharacteristic
755C2#6173011yescharacteristic
756C2#6174011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35236C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2354C2semidirect
3. #32985C2semidirect
4. #32981C2semidirect
5. #29895C2semidirect
6. #29890C2semidirect
7. #26311C2semidirect
8. #26243C2semidirect
9. #25017C2semidirect
10. #25011C2semidirect
11. #24970C2semidirect
12. #24968C2semidirect
13. #21982C2semidirect
14. #21867C2semidirect
15. #21788C2semidirect
16. #21388C2semidirect
17. #21365C2semidirect
21. C2×#485C22semidirect
22. C2×#482C22semidirect
52. #2076C22semidirect
53. #2076C22semidirect
54. #2075C22semidirect
55. #2075C22semidirect
56. #2036C22semidirect
57. #2036C22semidirect
58. #2033C22semidirect
59. #2033C22semidirect
60. #1992C22semidirect
61. #1989C22semidirect
62. #1983C22semidirect
63. #1959C22semidirect
67. #1948C22semidirect
68. #1947C22semidirect
69. #1947C22semidirect
70. #1875C22semidirect
74. #1873C22semidirect
78. #1854C22semidirect
79. #1854C22semidirect
84. #1845C22semidirect
85. #1834C22semidirect
86. #1825C22semidirect
87. #1825C22semidirect
88. #1803C22semidirect
89. #1802C22semidirect
92. #1776C22semidirect
93. C4×#103C22semidirect
94. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#352362
C2×#23542
#329852
#329812
#298952
#298902
#263112
#262432
#250172
#250112
#249702
#249682
#219822
#218672
#217882
#213882
#213652