Finite Group Databasebeta

← #6493631

Subgroups — #6493631

Order 512.

1944 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic
3#33109C212yesmaximal, complemented
4#33108C212yesmaximal, complemented
5#29926C214yesmaximal, complemented
6#27983C212yesmaximal, complemented
7#27919C212yesmaximal, complemented
8#27896C214yesmaximal, complemented
9#27589C214yesmaximal, complemented
10#24006C212yesmaximal
11#24003C212yesmaximal
12#21987C212yesmaximal, complemented
13#21811C212yesmaximal, complemented
14#20715C211yesmaximal, characteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C23×C42C2211yescharacteristic
18C2×#484C2214yes—
19C2×#483C2214yes—
20C2×#432C2212yes—
21#2857—21no—
22#2857—21no—
23#2857—21no—
24#2857—21no—
25#2857C2212yes—
26#2857C2212yes—
27#2857C2212yes—
28#2857C2212yes—
29C2×#391C2212yes—
30C2×#389—21no—
31C2×#389—21no—
32C2×#384C2211yescharacteristic
33#2283—21no—
34#2283—21no—
35#2283—21no—
36#2283—21no—
37#2283C2212yes—
38#2283C2212yes—
39#2283C2212yes—
40#2283C2212yes—
41C2×#383—21no—
42C2×#382C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#381—21no—
45C2×#380—21no—
46C2×#380—21no—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52C2×#376—21no—
53#2097C2218yescomplemented
54#2074C2218yescomplemented
55#2036C2218yes—
56#2035C2218yes—
57#1981C2214yescomplemented
58#1981C2214yescomplemented
59#1960C2214yescomplemented
60#1960C2214yescomplemented
61#1948C2214yescomplemented
62#1947C2214yescomplemented
63#1928C2218yescomplemented
64#1927C2218yescomplemented
65#1872C2214yes—
66#1872C2214yes—
67#1853C2214yes—
68#1852C2214yes—
69#1847C2214yescomplemented
70#1836C2214yescomplemented
71#1817C2212yes—
72#1797C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1787C2214yes—
76#1779C2212yes—
77#1749C2212yes—
78#1749C2212yes—
79C23×Q8C2312yes—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8C2314yes—
92C2×C22⋊Q8C2314yes—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—22no—
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C42—28no—
107C22×C42—21no—
108C22×C42—21no—
109C22×C42D411yescharacteristic
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D411yescharacteristic
113C22×C42D411yescharacteristic
114C22×C42D411yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121#484—48no—
122#483—48no—
123#432—28no—
124C2×#107C2318yes—
125C2×#106C2318yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2318yes—
128C2×#103C2318yes—
129C2×#103C2318yes—
130C2×#103C2318yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135#391—22no—
136#391—22no—
137#391C2314yes—
138#391C2314yes—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#384—21no—
148#384—21no—
149#384C2312yes—
150#384C2312yes—
151#383—22no—
152#383—22no—
153#383—22no—
154#383—22no—
155#382—21no—
156#382—21no—
157#382—21no—
158#382—21no—
159#382C2312yes—
160#382C2312yes—
161#382C2312yes—
162#382C2312yes—
163#381—22no—
164#381—22no—
165#381—22no—
166#381—22no—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380—22no—
175#380—21no—
176#380—21no—
177#380—21no—
178#380—21no—
179#380C2312yes—
180#380C2312yes—
181#380C2312yes—
182#380C2312yes—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C4⋊C4C2312yes—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195#376—22no—
196#376—22no—
197#376—22no—
198#376—22no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—22no—
208C2×#95—22no—
209#340C2318yes—
210#340C2318yes—
211#336C2314yes—
212#336C2314yes—
213C25C2×D411yescharacteristic
214C22×Q8—22no—
215C22×Q8—28no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8C2414yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22×C8C2418yes—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255C4⋊Q8—44no—
256C4⋊Q8—44no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C22⋊Q8—48no—
262C22⋊Q8—48no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—44no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42#4118yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2411yescharacteristic
336#107—48no—
337#106—48no—
338C4⋊C8—28no—
339#103—48no—
340#103—48no—
341#103—48no—
342#103—48no—
343C22⋊C8—28no—
344C22⋊C8—28no—
345C22⋊C8—44no—
346#97—28no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—41no—
372C24—41no—
373C24#13714yes—
374C24C22×D411yescharacteristic
375C24C22×D411yescharacteristic
376C24C22×D411yescharacteristic
377C2×Q8—28no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—28no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—816no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4#12314yes—
462C22×C4#12314yes—
463C22×C4#13714yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4C2511yescharacteristic
478C2×C8—28no—
479C2×C8—28no—
480C2×C8—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—44no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42#12314yes—
525C42#12314yes—
526C42C2×#4114yes—
527C42C2×#4114yes—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—28no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23C2×#12312yes—
550C23#53512yes—
551C23C2×#13712yes—
552C23C2×#13712yes—
553C23C2×#13712yes—
554C23C2×#13712yes—
555C23#57512yes—
556C23C23×D411yescharacteristic
557Q8—48no—
558Q8—48no—
559Q8—88no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
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—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
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—44no—
619C2×C4—44no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4C2×#12312yes—
627C2×C4C2×#12312yes—
628C2×C4C2×#12312yes—
629C2×C4C2×#12312yes—
630C2×C4C2×#12312yes—
631C2×C4#53512yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13712yes—
634C2×C4#57512yes—
635C2×C4C22×#4112yes—
636C2×C4C2×H(5,𝔽2)12yes—
637C2×C4#58512yes—
638C8—48no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—416no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22#309514yes—
660C22#321814yes—
661C22C22×#12311yescharacteristic
662C22C2×#53511yescharacteristic
663C22#344911yescharacteristic
664C22C22×#13711yescharacteristic
665C22C22×#13711yescharacteristic
666C22C2×#57511yescharacteristic
667C22C2×#57511yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—22no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—84no—
684C4#309514yes—
685C4#349414yes—
686C2—24no—
687C2—24no—
688C2—42no—
689C2#3526211yescharacteristic
690C2C2×#309511yescharacteristic
691C2C2×#309511yescharacteristic
692C2C2×#321811yescharacteristic
693C2#6080511yescharacteristic
694C2#6095011yescharacteristic
695C2#6096011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33109C2semidirect
4. #33108C2semidirect
5. #29926C2semidirect
6. #27983C2semidirect
7. #27919C2semidirect
8. #27896C2semidirect
9. #27589C2semidirect
12. #21987C2semidirect
13. #21811C2semidirect
53. #2097C22semidirect
54. #2074C22semidirect
57. #1981C22semidirect
58. #1981C22semidirect
59. #1960C22semidirect
60. #1960C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1928C22semidirect
64. #1927C22semidirect
69. #1847C22semidirect
70. #1836C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#22832
#331092
#331082
#299262
#279832
#279192
#278962
#275892
#240062
#240032
#219872
#218112
#207152