Finite Group Databasebeta

← #642604

Subgroups — #642604

Order 512.

3986 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
1C22×#529C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2660C211yesmaximal, characteristic, complemented
4C2×#2640C211yesmaximal, characteristic, complemented
5#16656C214yesmaximal, complemented
6#16359C214yesmaximal, complemented
7#15219C212yesmaximal
8#15182C212yesmaximal, complemented
9#14907C214yesmaximal, complemented
10#14723C214yesmaximal, complemented
11#14283C211yesmaximal, characteristic, complemented
12#14273C211yesmaximal, characteristic, complemented
13#13978C212yesmaximal, complemented
14#13977C212yesmaximal, complemented
15#13753C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#529—28no—
18C2×#529—28no—
19C2×#529—28no—
20C22×#126C2212yes—
21C22×#125C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#123C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×#121C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2212yes—
29C22×C4×D4C2211yescharacteristic
30C22×#117C2212yes—
31C23×C42C2211yescharacteristic
32#2857—28no—
33#2660—28no—
34#2640—28no—
35C2×#398C2214yes—
36C2×#397C2214yes—
37C2×#395C2214yes—
38C2×#393C2214yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#390C2212yes—
43C2×#390C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#385C2218yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2211yescharacteristic
54C2×#376C2211yescharacteristic
55#1442C2218yes—
56#1441C2218yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2218yes—
60#1439C2214yes—
61#1439C2218yes—
62#1439C2214yes—
63#1439C2218yes—
64#1438C2218yes—
65#1436C2218yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84#529—2128no—
85C2×#126—28no—
86C2×#126—28no—
87C2×#126—28no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125D418yes—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124D418yes—
94C2×#123—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—48no—
102C2×#121—24no—
103C2×#121—24no—
104C2×#121—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×D4—24no—
109C2×C4×D4—28no—
110C2×C4×D4—28no—
111C2×C4×D4—24no—
112C2×C4×D4—28no—
113C2×C4×D4—24no—
114C2×#117—28no—
115C2×#117—28no—
116C2×#117—28no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2318yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2318yes—
137C22×C22⋊C4C2314yes—
138C22×C22⋊C4C2318yes—
139C22×C42—24no—
140C22×C42—24no—
141C22×C42D418yes—
142C22×C42C2312yes—
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149#398—216no—
150#397—216no—
151#395—216no—
152#393—216no—
153#391—28no—
154#391—28no—
155#391—28no—
156#390—28no—
157#390—28no—
158#389—28no—
159#389—28no—
160#385—416no—
161#384—28no—
162#384—24no—
163#382—28no—
164#380—28no—
165#380—28no—
166C4×C22⋊C4—28no—
167#376—24no—
168#376—24no—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2312yes—
172C2×#95C2318yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2318yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8C2×D414yes—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—44no—
204C22×D4C2×D414yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C4⋊Q8—416no—
239#126—264no—
240#125—232no—
241#124—232no—
242#123—264no—
243C22⋊Q8—232no—
244C22⋊Q8—432no—
245#121—232no—
246C4×Q8—232no—
247C4×D4—264no—
248C4×D4—232no—
249#117—264no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D414yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—216no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×D418yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—232no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42#4118yes—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—216no—
341#95—216no—
342#95—28no—
343#95—216no—
344#95—28no—
345#95—44no—
346#95—48no—
347#95—48no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24C22×D414yes—
364C24C22×D412yes—
365C242−1+414yes—
366C24C2511yescharacteristic
367C2×Q8—28no—
368C2×Q8—216no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×D4—216no—
378C2×D4—28no—
379C2×D4—216no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—48no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4118yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C42−1+414yes—
476C4⋊C4—216no—
477C4⋊C4—216no—
478C4⋊C4—216no—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—816no—
489C22⋊C4—216no—
490C22⋊C4—216no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—216no—
494C22⋊C4—416no—
495C22⋊C4—432no—
496C22⋊C4—416no—
497C22⋊C4—816no—
498C42—216no—
499C42—28no—
500C42—28no—
501C42—28no—
502C42—216no—
503C42—48no—
504C42#120116yes—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—48no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23#53614yes—
539C23C23×D411yescharacteristic
540C23C23×D411yescharacteristic
541C23C23×D411yescharacteristic
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×2−1+412yes—
545C23C2×2−1+412yes—
546C23#58512yes—
547C23#58512yes—
548Q8—416no—
549Q8—816no—
550D4—432no—
551D4—416no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—432no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—416no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4—48no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4C2×#12014yes—
618C2×C4C2×#12014yes—
619C2×C4C2×#12014yes—
620C2×C4#53214yes—
621C2×C4#53514yes—
622C2×C4#53614yes—
623C2×C4#54718yes—
624C2×C4#54818yes—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—42no—
629C22—48no—
630C22—48no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—416no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—82no—
639C22—88no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C22×#12011yescharacteristic
644C22C2×#53211yescharacteristic
645C22C2×#53511yescharacteristic
646C22C2×#53611yescharacteristic
647C22C2×#53611yescharacteristic
648C22#344911yescharacteristic
649C22#344911yescharacteristic
650C22#346014yes—
651C22C2×#54712yes—
652C22C2×#54812yes—
653C22#348812yes—
654C22#348812yes—
655C22#348912yes—
656C22#348912yes—
657C22#349514yes—
658C22#355214yes—
659C22#356514yes—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—48no—
664C4—42no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—84no—
670C4—84no—
671C4—84no—
672C4#243218yes—
673C2—44no—
674C2—84no—
675C2#3462411yescharacteristic
676C2C2×#243211yescharacteristic
677C2#3525011yescharacteristic
678C2#3530511yescharacteristic
679C2#3532611yescharacteristic
680C2#3754511yescharacteristic
681C2#3761511yescharacteristic
682C2#3775914yes—
683C2#3776714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2857C2semidirect
3. C2×#2660C2semidirect
4. C2×#2640C2semidirect
5. #16656C2semidirect
6. #16359C2semidirect
8. #15182C2semidirect
9. #14907C2semidirect
10. #14723C2semidirect
11. #14283C2semidirect
12. #14273C2semidirect
13. #13978C2semidirect
14. #13977C2semidirect
15. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#28572
C2×#26602
C2×#26402
#166562
#163592
#152192
#151822
#149072
#147232
#142832
#142732
#139782
#139772
#137532