Finite Group Databasebeta

← #6792522

Subgroups — #6792522

Order 512.

1905 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3#32902C211yesmaximal, characteristic
4#31643C214yesmaximal, complemented
5#31527C214yesmaximal, complemented
6#29912C212yesmaximal, complemented
7#28399C212yesmaximal, complemented
8#27814C212yesmaximal, complemented
9#25903C211yesmaximal, characteristic, complemented
10#25321C212yesmaximal, complemented
11#24814C212yesmaximal, complemented
12#24500C212yesmaximal, complemented
13#24302C211yesmaximal, characteristic, complemented
14#23936C212yesmaximal, complemented
15#22599C211yesmaximal, characteristic, complemented
16#21861C212yesmaximal, complemented
17C4×#450C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#529C2212yes—
20C22×C4×D4C2211yescharacteristic
21C2×#486C2212yescomplemented
22C2×#477C2212yescomplemented
23C2×#459C2212yes—
24C2×#450C2211yescharacteristic
25C2×#447C2212yes—
26C2×C4×SD16C2212yes—
27C2×#431C2212yes—
28#2856C2211yescharacteristic
29#2854C2212yes—
30#2563—21no—
31#2563—22no—
32#2563—21no—
33#2563C2214yes—
34#2563C2212yes—
35#2563C2212yes—
36#2552C2211yescharacteristic, complemented
37#2548C2211yescharacteristic, complemented
38#2512—21no—
39#2501—21no—
40#2479—22no—
41#2454C2212yescomplemented
42#2448C2212yescomplemented
43#2425—22no—
44C2×#393—22no—
45#2366C2211yescharacteristic, complemented
46#2361—21no—
47#2354—21no—
48Q8×C4⋊C4C2211yescharacteristic, complemented
49C4×C22⋊Q8C2211yescharacteristic
50C2×#390C2212yes—
51C2×#382C2212yes—
52C2×#380—21no—
53C2×#380—21no—
54C2×C4×C4⋊C4C2211yescharacteristic
55#2101C2214yescomplemented
56#2092C2214yescomplemented
57#2085C2214yescomplemented
58#2073C2214yescomplemented
59#2067C2214yes—
60#2057C2212yescomplemented
61#2054C2214yes—
62#2025C2214yes—
63#2024C2212yescomplemented
64#2023C2214yescomplemented
65#2012C2214yes—
66#1991C2214yescomplemented
67#1990C2212yescomplemented
68#1987C2212yescomplemented
69#1970C2214yes—
70#1958C2214yescomplemented
71#1939C2212yescomplemented
72#1934C2214yes—
73#1916C2214yescomplemented
74#1880C2214yes—
75#1879C2214yes—
76#1879C2214yes—
77#1877C2212yes—
78#1874C2212yes—
79#1874C2212yes—
80#1874C2212yes—
81#1872C2211yescharacteristic
82#1872C2212yes—
83#1864C2212yes—
84#1853C2211yescharacteristic
85#1852C2212yes—
86#1817C2212yes—
87#1805C2214yescomplemented
88#1804C2214yescomplemented
89#1802C2212yes—
90#1797C2211yescharacteristic
91#1786C2211yescharacteristic
92C8×C4⋊C4C2212yescomplemented
93C4×#102C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×SD16C2312yes—
99C22×D8C2314yes—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106#529—28no—
107#529—44no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#123—22no—
112C2×#123—22no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121—21no—
115C2×#121—21no—
116C2×#121C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—44no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C22⋊C4—21no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#486—44no—
139#477—44no—
140#459—44no—
141#450—24no—
142#447—44no—
143C4×SD16—44no—
144#431—44no—
145C2×#107C2314yes—
146C2×#106C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2314yes—
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2311yescharacteristic
161C2×C4×C8C2312yes—
162C2×C4×C8C2314yes—
163#399—22no—
164#399—22no—
165#398—22no—
166#397—22no—
167#396—22no—
168#394—22no—
169#393—24no—
170#393—24no—
171#393—24no—
172#393—24no—
173#391C2311yescharacteristic
174#390—21no—
175#390—21no—
176#390—21no—
177#390—22no—
178#390—22no—
179#390—21no—
180#390—21no—
181#390C2314yes—
182#390C2314yes—
183#390C2312yes—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—22no—
188#389C2311yescharacteristic
189#389C2312yes—
190#389C2312yes—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—21no—
196#386—22no—
197#386—22no—
198#385—21no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384C2311yescharacteristic
203#382—22no—
204#382—22no—
205#382C2314yes—
206#382C2314yes—
207#380—22no—
208#380—22no—
209#380—22no—
210#380—22no—
211#380—24no—
212#380—24no—
213#380—22no—
214#380C2311yescharacteristic
215C4×C4⋊C4—21no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4D411yescharacteristic
218C4×C4⋊C4D412yes—
219C4×C4⋊C4D411yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2311yescharacteristic
224C4×C22⋊C4—21no—
225C2×#95—22no—
226C2×#95—22no—
227C2×#95D412yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2312yes—
239C25—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—44no—
243C22×Q8—82no—
244C22×Q8—82no—
245C22×Q8C2411yescharacteristic
246C22×D4—24no—
247C22×D4—22no—
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—42no—
254C22×D4C2412yes—
255C22×D4C2412yes—
256C23×C4—22no—
257C23×C4—21no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—44no—
262C23×C4—44no—
263C23×C4—44no—
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D412yes—
268C23×C4C2411yescharacteristic
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—42no—
272C2×D8—28no—
273C2×D8—44no—
274C2×D8—44no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—24no—
279C4⋊Q8—24no—
280C4⋊Q8—44no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283#127—24no—
284#126—48no—
285#125—24no—
286#125—42no—
287#124—24no—
288#124—42no—
289#123—48no—
290#123—84no—
291C22⋊Q8—22no—
292C22⋊Q8#4114yes—
293#121—24no—
294#121—24no—
295#121—44no—
296#121—82no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—42no—
300C4×D4—24no—
301C4×D4—24no—
302C4×D4—24no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—48no—
309C4×D4—44no—
310#117—48no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4—84no—
362C2×C22⋊C4—84no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2411yescharacteristic
377#107—44no—
378#106—42no—
379C4⋊C8—24no—
380C4⋊C8—44no—
381#103—42no—
382#103—44no—
383#102—24no—
384#102—24no—
385#102—42no—
386#102—44no—
387#102—44no—
388#102—44no—
389#102—44no—
390C22⋊C8—22no—
391C22⋊C8—22no—
392C22⋊C8—24no—
393C4×C8—24no—
394C4×C8—44no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—24no—
399#95—21no—
400#95—21no—
401#95—22no—
402#95—21no—
403#95—21no—
404#95—21no—
405#95—22no—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409#95C2×D412yes—
410C24—24no—
411C24—22no—
412C24—22no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24—42no—
418C24—42no—
419C24C22×D411yescharacteristic
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—82no—
426C2×Q8—82no—
427C2×Q8—82no—
428C2×Q8—82no—
429C2×Q8—816no—
430C2×Q8C2×#4112yes—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—24no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—24no—
441C2×D4—48no—
442C2×D4—48no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—81no—
453C2×D4—81no—
454C2×D42−1+414yes—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—82no—
490C22×C4—816no—
491C22×C4#13514yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4C2511yescharacteristic
504SD16—48no—
505D8—88no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—22no—
513C2×C8—24no—
514C2×C8—22no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—81no—
541C4⋊C4—81no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4C2×#4112yes—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—42no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—82no—
556C22⋊C4—84no—
557C22⋊C4—84no—
558C22⋊C4—84no—
559C22⋊C4—84no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—22no—
564C42—22no—
565C42—24no—
566C42—24no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—44no—
573C42—42no—
574C42—42no—
575C42H(5,𝔽2)14yes—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—24no—
581C23—48no—
582C23—48no—
583C23—48no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—84no—
601C23#54612yes—
602C23C2×#13512yes—
603C23#57612yes—
604C23C23×D411yescharacteristic
605Q8—24no—
606Q8—42no—
607Q8—88no—
608Q8—88no—
609D4—24no—
610D4—24no—
611D4—48no—
612D4—42no—
613D4—44no—
614D4—42no—
615D4—44no—
616D4—88no—
617D4—84no—
618D4—162no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—44no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—48no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—88no—
667C2×C4—84no—
668C2×C4—88no—
669C2×C4—84no—
670C2×C4—88no—
671C2×C4—88no—
672C2×C4#53512yes—
673C2×C4#53512yes—
674C2×C4#54612yes—
675C2×C4C2×#13512yes—
676C2×C4C2×#13512yes—
677C2×C4#57512yes—
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)12yes—
680C2×C4C2×H(5,𝔽2)11yescharacteristic
681C2×C4C2×2−1+412yes—
682C8—42no—
683C8—42no—
684C8—44no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—816no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22#330514yes—
707C22C2×#53511yescharacteristic
708C22C2×#54611yescharacteristic
709C22#348611yescharacteristic
710C22C22×#13511yescharacteristic
711C22C2×#57511yescharacteristic
712C22C2×#57611yescharacteristic
713C22#358511yescharacteristic
714C4—22no—
715C4—22no—
716C4—22no—
717C4—41no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4—42no—
723C4—42no—
724C4—82no—
725C4—81no—
726C4—84no—
727C4—81no—
728C4—82no—
729C4#321114yes—
730C4#352012yes—
731C4#353612yes—
732C2—24no—
733C2—84no—
734C2—162no—
735C2#3890511yescharacteristic
736C2C2×#321111yescharacteristic
737C2#6081111yescharacteristic
738C2C2×#330511yescharacteristic
739C2#6116611yescharacteristic
740C2#6181911yescharacteristic
741C2#6183511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36117C2semidirect
2. C2×#2563C2semidirect
4. #31643C2semidirect
5. #31527C2semidirect
6. #29912C2semidirect
7. #28399C2semidirect
8. #27814C2semidirect
9. #25903C2semidirect
10. #25321C2semidirect
11. #24814C2semidirect
12. #24500C2semidirect
13. #24302C2semidirect
14. #23936C2semidirect
15. #22599C2semidirect
16. #21861C2semidirect
17. C4×#450C2semidirect
21. C2×#486C22semidirect
22. C2×#477C22semidirect
36. #2552C22semidirect
37. #2548C22semidirect
41. #2454C22semidirect
42. #2448C22semidirect
45. #2366C22semidirect
48. Q8×C4⋊C4C22semidirect
55. #2101C22semidirect
56. #2092C22semidirect
57. #2085C22semidirect
58. #2073C22semidirect
60. #2057C22semidirect
63. #2024C22semidirect
64. #2023C22semidirect
66. #1991C22semidirect
67. #1990C22semidirect
68. #1987C22semidirect
70. #1958C22semidirect
71. #1939C22semidirect
73. #1916C22semidirect
87. #1805C22semidirect
88. #1804C22semidirect
92. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#361172
C2×#25632
#329022
#316432
#315272
#299122
#283992
#278142
#259032
#253212
#248142
#245002
#243022
#239362
#225992
#218612
C4×#4502