Finite Group Databasebeta

← #6793016

Subgroups — #6793016

Order 512.

1917 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic, complemented
2#34932C211yesmaximal, characteristic, complemented
3#33195C211yesmaximal, characteristic, complemented
4#31602C214yesmaximal, complemented
5#31550C214yesmaximal, complemented
6#29936C212yesmaximal, complemented
7#29219C211yesmaximal, characteristic, complemented
8#28279C211yesmaximal, characteristic, complemented
9#28140C212yesmaximal, complemented
10#26106C212yesmaximal, complemented
11#25338C212yesmaximal, complemented
12#24835C212yesmaximal, complemented
13#24811C212yesmaximal, complemented
14#24150C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21884C212yesmaximal, complemented
17C4×#459C211yesmaximal, characteristic, complemented
18C2×#547—21no—
19C2×#532C2212yes—
20C2×#529—21no—
21C2×#474C2212yes—
22C2×#474C2212yes—
23C2×#459C2211yescharacteristic
24C2×C4×D8C2212yes—
25C2×C8×D4C2212yes—
26C22×C4⋊C8C2212yes—
27#2868C2211yescharacteristic
28#2857C2212yes—
29#2553—22no—
30#2552C2211yescharacteristic
31#2547—21no—
32#2547C2211yescharacteristic
33#2542C2211yescharacteristic
34#2537—21no—
35#2537C2212yes—
36#2537C2211yescharacteristic
37#2496C2212yes—
38#2489—21no—
39#2432—22no—
40#2427—21no—
41#2422—22no—
42#2422C2214yes—
43#2422C2212yes—
44#2395—22no—
45D4×C4⋊C4—21no—
46D4×C4⋊C4C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C4×C22⋊Q8—21no—
50C4×#121C2212yes—
51#2285C2212yes—
52C42×D4C2211yescharacteristic
53C42×D4C2211yescharacteristic
54#2099C2214yes—
55#2099C2214yes—
56#2086C2214yes—
57#2083C2214yes—
58#2075C2214yes—
59#2075C2214yes—
60#2067C2214yes—
61#2067C2214yes—
62#2033C2214yes—
63#1999C2214yes—
64#1990C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1988C2212yes—
68#1982C2212yes—
69#1982C2211yescharacteristic
70#1947C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1925C2212yes—
76#1893C2214yes—
77#1893C2214yes—
78#1893C2214yes—
79#1879C2212yes—
80#1879C2214yes—
81#1862C2214yes—
82#1850C2214yes—
83#1842C2211yescharacteristic
84#1834C2212yes—
85#1808C2214yes—
86#1802C2211yescharacteristic
87#1802C2212yes—
88#1802C2214yes—
89#1802C2214yes—
90#1802C2212yes—
91#1802C2212yes—
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95C22×#41—22no—
96C22×D8C2312yes—
97C23×C8C2314yes—
98#547—48no—
99#532—28no—
100#532—44no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#127C2311yescharacteristic
106#529—48no—
107C2×#126—22no—
108C2×#125—21no—
109C2×#124—21no—
110C2×#124—22no—
111C2×#123—22no—
112C2×#123—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×#121—21no—
117C2×#121—22no—
118C2×#121—22no—
119C2×#121C2314yes—
120C2×#121C2312yes—
121C2×C4×Q8—21no—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—22no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4D411yescharacteristic
128C2×C4×D4C2312yes—
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×C4×D4C2314yes—
132C2×C4×D4C2311yescharacteristic
133C2×C4×D4C2311yescharacteristic
134C2×#117—22no—
135C22×C4⋊C4—22no—
136C22×C42—21no—
137C22×C42C2312yes—
138#474—44no—
139#474—44no—
140#459—24no—
141C4×D8—44no—
142C8×D4—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×C4⋊C8—44no—
147C2×C4⋊C8C2311yescharacteristic
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2311yescharacteristic
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2312yes—
157C2×#102C2312yes—
158C2×#102C2312yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C4×C8C2312yes—
162#400—22no—
163#399—22no—
164#398—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#397—24no—
169#394—22no—
170#394—22no—
171#394—24no—
172#394—24no—
173#391—21no—
174#391—21no—
175#391—22no—
176#390—21no—
177#390—21no—
178#390C2312yes—
179#390C2312yes—
180#389—21no—
181#389—21no—
182#389—22no—
183#389C2312yes—
184#389C2314yes—
185#389C2312yes—
186#388—22no—
187#387—22no—
188#386—22no—
189#385—24no—
190#385—24no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#384C2312yes—
197#384C2312yes—
198#384C2312yes—
199#384C2312yes—
200#382—22no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380—21no—
206#380C2312yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2312yes—
238C2×#41—48no—
239C2×#41—48no—
240C2×#41—84no—
241C22×Q8—21no—
242C22×Q8—22no—
243C22×D4—22no—
244C22×D4—24no—
245C22×D4—21no—
246C22×D4—21no—
247C22×D4C2×D411yescharacteristic
248C22×D4C2412yes—
249C22×D4C2411yescharacteristic
250C23×C4—22no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2412yes—
258C2×D8—24no—
259C2×D8—24no—
260C2×D8—42no—
261C22×C8—44no—
262C22×C8—44no—
263C22×C8—44no—
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—44no—
272C4⋊Q8—82no—
273#127—22no—
274#127#4114yes—
275#126—48no—
276#125—44no—
277#124—48no—
278#124—44no—
279#123—48no—
280#123—416no—
281C22⋊Q8—44no—
282C22⋊Q8—48no—
283C22⋊Q8—44no—
284#121—28no—
285#121—24no—
286#121—42no—
287#121—44no—
288#121—48no—
289#121—44no—
290#121—48no—
291C4×Q8—44no—
292C4×D4—28no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—24no—
301C4×D4—22no—
302C4×D4—42no—
303C4×D4—41no—
304C4×D4—44no—
305C4×D4—48no—
306C4×D4—44no—
307C4×D4—48no—
308C4×D4—44no—
309C4×D4D818yescomplemented
310#117—48no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
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—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—48no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2414yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—21no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—44no—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—42no—
375#107—44no—
376#107—44no—
377C4⋊C8—22no—
378C4⋊C8—22no—
379C4⋊C8—24no—
380C4⋊C8—24no—
381C4⋊C8—44no—
382C4⋊C8—42no—
383#102—24no—
384#102—42no—
385#102—44no—
386#102—42no—
387#102—42no—
388#102—44no—
389#102—44no—
390C22⋊C8—44no—
391C4×C8—24no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—24no—
402#95—24no—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—21no—
407C24—22no—
408C24C22×D412yes—
409#41—832no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×D4—22no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—22no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—21no—
430C2×D4—42no—
431C2×D4—42no—
432C2×D4—48no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—48no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—82no—
439C2×D4—84no—
440C2×D4C2×D814yes—
441C2×D4C2×#4114yes—
442C2×D42−1+412yes—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—44no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—84no—
476C22×C4C2×D818yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C2511yescharacteristic
487D8—48no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—24no—
496C2×C8—22no—
497C2×C8—88no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—22no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—81no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4C2×D814yes—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—48no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4C2×D818yes—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—21no—
557C42—21no—
558C42—41no—
559C42—42no—
560C42—42no—
561C42—42no—
562C42—44no—
563C42C2×D814yes—
564C42C2×#4112yes—
565C42H(5,𝔽2)12yes—
566C42H(5,𝔽2)12yes—
567C422−1+412yes—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—22no—
576C23—41no—
577C23—42no—
578C23—42no—
579C23—41no—
580C23—41no—
581C23—42no—
582C23—41no—
583C23—42no—
584C23—41no—
585C23—42no—
586C23—41no—
587C23—42no—
588C23C22×D814yes—
589C23C23×D411yescharacteristic
590Q8—84no—
591Q8—88no—
592D4—28no—
593D4—24no—
594D4—22no—
595D4—24no—
596D4—22no—
597D4—48no—
598D4—41no—
599D4—41no—
600D4—42no—
601D4—48no—
602D4—42no—
603D4—88no—
604D4—816no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—216no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—21no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—816no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4—84no—
654C2×C4#53712yes—
655C2×C4#53712yes—
656C2×C4#54612yes—
657C2×C4#54612yes—
658C2×C4C22×D812yes—
659C2×C4C22×D814yes—
660C2×C4C22×D812yes—
661C2×C4C22×D812yes—
662C2×C4#57512yes—
663C2×C4#57712yes—
664C2×C4C22×#4111yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C2×C4#58511yescharacteristic
670C8—42no—
671C8—44no—
672C8—42no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—42no—
686C22—81no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—81no—
691C22—81no—
692C22—82no—
693C22—81no—
694C22C2×#53711yescharacteristic
695C22C2×#54611yescharacteristic
696C22#348911yescharacteristic
697C22C23×D811yescharacteristic
698C22C2×#57511yescharacteristic
699C22C2×#57711yescharacteristic
700C22#358611yescharacteristic
701C4—22no—
702C4—22no—
703C4—24no—
704C4—22no—
705C4—41no—
706C4—42no—
707C4—42no—
708C4—81no—
709C4—82no—
710C4—81no—
711C4—84no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4#324514yes—
716C4#329414yes—
717C4#351511yescharacteristic
718C4#352011yescharacteristic
719C4#354611yescharacteristic
720C4#355211yescharacteristic
721C2—44no—
722C2—82no—
723C2—84no—
724C2#3890811yescharacteristic
725C2C2×#324511yescharacteristic
726C2#6093111yescharacteristic
727C2C2×#329411yescharacteristic
728C2#6115911yescharacteristic
729C2#6209511yescharacteristic
730C2#6213411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #36127C2semidirect
2. #34932C2semidirect
3. #33195C2semidirect
4. #31602C2semidirect
5. #31550C2semidirect
6. #29936C2semidirect
7. #29219C2semidirect
8. #28279C2semidirect
9. #28140C2semidirect
10. #26106C2semidirect
11. #25338C2semidirect
12. #24835C2semidirect
13. #24811C2semidirect
14. #24150C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21884C2semidirect
17. C4×#459C2semidirect
309. C4×D4D8semidirect

Maximal subgroups

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

subgroupindex
#361272
#349322
#331952
#316022
#315502
#299362
#292192
#282792
#281402
#261062
#253382
#248352
#248112
#241502
D4×C4⋊C82
#218842
C4×#4592