Finite Group Databasebeta

← #1423749

Subgroups — #1423749

Order 512.

4900 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#559C211yesmaximal, characteristic, complemented
2C2×#2406C212yesmaximal, complemented
3C2×#2431C211yesmaximal, characteristic
4#16032C214yesmaximal, complemented
5#15950C212yesmaximal, complemented
6#15314C214yesmaximal, complemented
7#15262C212yesmaximal, complemented
8#14881C212yesmaximal, complemented
9#14721C214yesmaximal, complemented
10#14505C212yesmaximal, complemented
11#14419C211yesmaximal, characteristic, complemented
12#14172C212yesmaximal, complemented
13#14015C214yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C2×#559—28no—
16C2×#559—216no—
17C22×#127C2211yescharacteristic
18C22×#124C2214yes—
19C22×#124C2211yescharacteristic
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2212yes—
25C22×C4×D4C2214yes—
26C23×C4⋊C4C2212yes—
27#2431—28no—
28#2406—216no—
29C2×#395C2214yes—
30C2×#394C2214yes—
31C2×#393C2212yes—
32C2×#392C2214yes—
33C2×#391C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#386C2214yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#381C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2214yes—
47#1446C2212yes—
48#1444C2214yes—
49#1444C2212yes—
50#1441C2214yes—
51#1440C2212yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1434C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22×#41—28no—
72C22×#41—24no—
73C22×#41—42no—
74C22×#41—42no—
75C22×#41—44no—
76C23×Q8C2311yescharacteristic
77C23×Q8C2312yes—
78C23×D4C2312yes—
79C23×D4C2314yes—
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2312yes—
88#559—2128no—
89C2×#127—28no—
90C2×#127D418yes—
91C2×#124—216no—
92C2×#124—24no—
93C2×#124—216no—
94C2×#124—216no—
95C2×#124—44no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—216no—
101C2×#123—44no—
102C2×#123—44no—
103C2×#123D4116yescomplemented
104C2×#121—24no—
105C2×#121—42no—
106C2×#121—42no—
107C2×C4×Q8—216no—
108C2×C4×Q8—28no—
109C2×C4×D4—216no—
110C2×C4×D4—216no—
111C2×C4×D4—216no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4—44no—
114C22×C4⋊C4—44no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#395—216no—
140#394—216no—
141#393—28no—
142#392—216no—
143#391—24no—
144#390—24no—
145#386—28no—
146#386—216no—
147#386—28no—
148#386—216no—
149#386—44no—
150#385—216no—
151#385—216no—
152#385—44no—
153#385—48no—
154#381—28no—
155C4×C22⋊C4—24no—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25C2412yes—
174C25C2411yescharacteristic
175C25C2412yes—
176C2×#41—432no—
177C2×#41—416no—
178C2×#41—432no—
179C2×#41—416no—
180C2×#41—416no—
181C22×Q8—28no—
182C22×Q8—24no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—44no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—28no—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—41no—
211C22×D4—41no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4C2×D4116yes—
217C22×D4C2×D414yes—
218C22×D4C2×D414yes—
219C22×D4C2×D418yes—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—28no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—28no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—44no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4C2×D418yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2414yes—
253C23×C4C2414yes—
254C23×C4C2414yes—
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259#127—232no—
260#124—2128no—
261#124—416no—
262#123—264no—
263#123—264no—
264#123—432no—
265#121—416no—
266C4×Q8—264no—
267C4×D4—2128no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—432no—
285C2×C4⋊C4C2×D4116yes—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4C2×D4116yes—
327C2×C22⋊C4C2×D418yes—
328C2×C42—28no—
329C2×C42—28no—
330C2×C42—28no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—42no—
334C2×C42C2×D414yes—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95#4118yescomplemented
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—21no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24C22×D414yes—
376C24C22×D414yes—
377C24C22×D414yes—
378C24C22×D414yes—
379C24H(5,𝔽2)12yes—
380C24C2511yescharacteristic
381#41—4128no—
382C2×Q8—216no—
383C2×Q8—28no—
384C2×Q8—416no—
385C2×Q8—416no—
386C2×Q8—48no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×D4—216no—
393C2×D4—232no—
394C2×D4—28no—
395C2×D4—232no—
396C2×D4—416no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—416no—
404C2×D4—416no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—88no—
418C2×D4—84no—
419C2×D4—88no—
420C2×D4#120116yes—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—216no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—216no—
457C22×C4—216no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—416no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—44no—
492C22×C4—88no—
493C22×C4—88no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—84no—
500C22×C4C22×D418yes—
501C22×C4C22×D418yes—
502C22×C4C22×D414yes—
503C22×C4C22×D412yes—
504C22×C4C22×D414yes—
505C22×C4C2×#4118yes—
506C22×C4C2×#4114yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C4H(5,𝔽2)14yes—
509C22×C42−1+414yes—
510C4⋊C4—216no—
511C4⋊C4—232no—
512C4⋊C4—416no—
513C4⋊C4—416no—
514C4⋊C4—416no—
515C4⋊C4—44no—
516C22⋊C4—232no—
517C22⋊C4—216no—
518C22⋊C4—232no—
519C22⋊C4—216no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—416no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—88no—
530C22⋊C4—84no—
531C42—232no—
532C42—28no—
533C42—44no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—22no—
552C23—28no—
553C23—24no—
554C23—24no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—41no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—41no—
572C23—42no—
573C23—41no—
574C23—44no—
575C23—44no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—42no—
589C23—41no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23C2×#12018yes—
594C23C23×D411yescharacteristic
595C23C23×D412yes—
596C23C22×#4112yes—
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)11yescharacteristic
600C23C2×2−1+412yes—
601C23#58514yes—
602Q8—432no—
603Q8—416no—
604D4—416no—
605D4—464no—
606D4—432no—
607D4—88no—
608D4—816no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—416no—
632C2×C4—432no—
633C2×C4—432no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—816no—
671C2×C4—816no—
672C2×C4C2×#12014yes—
673C2×C4#54718yes—
674C22—28no—
675C22—216no—
676C22—44no—
677C22—44no—
678C22—48no—
679C22—44no—
680C22—48no—
681C22—48no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—42no—
686C22—44no—
687C22—48no—
688C22—42no—
689C22—44no—
690C22—48no—
691C22—48no—
692C22—44no—
693C22—42no—
694C22—82no—
695C22—84no—
696C22—82no—
697C22—84no—
698C22—84no—
699C22—82no—
700C22—82no—
701C22—84no—
702C22C22×#12011yescharacteristic
703C22C2×#54712yes—
704C22D4×#4114yes—
705C22#348111yescharacteristic
706C22#348412yes—
707C22#348411yescharacteristic
708C22#348411yescharacteristic
709C22#348412yes—
710C22#348512yes—
711C22#348914yes—
712C22#350614yes—
713C22#352212yes—
714C22#353312yes—
715C22#354012yes—
716C22#355011yescharacteristic
717C22#355414yes—
718C4—24no—
719C4—28no—
720C4—42no—
721C4—44no—
722C4—48no—
723C4—48no—
724C4—48no—
725C4—82no—
726C4—82no—
727C4—84no—
728C2—48no—
729C2—82no—
730C2—84no—
731C2#3534611yescharacteristic
732C2#3621312yes—
733C2#4311014yes—
734C2#4535412yes—
735C2#5131712yes—
736C2#5205411yescharacteristic
737C2#5907011yescharacteristic
738C2#5917312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#559C2semidirect
2. C2×#2406C2semidirect
4. #16032C2semidirect
5. #15950C2semidirect
6. #15314C2semidirect
7. #15262C2semidirect
8. #14881C2semidirect
9. #14721C2semidirect
10. #14505C2semidirect
11. #14419C2semidirect
12. #14172C2semidirect
13. #14015C2semidirect
103. C2×#123D4semidirect
351. #95#41semidirect

Maximal subgroups

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

subgroupindex
C22×#5592
C2×#24062
C2×#24312
#160322
#159502
#153142
#152622
#148812
#147212
#145052
#144192
#141722
#140152