Finite Group Databasebeta

← #4462509

Subgroups — #4462509

Order 512.

3690 conjugacy classes of subgroups.

The subgroup lattice has 772 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic, complemented
2C22×#394C211yesmaximal, characteristic
3#17151C214yesmaximal, complemented
4#17107C214yesmaximal, complemented
5#17103C214yesmaximal, complemented
6#17092C214yesmaximal, complemented
7#14936C211yesmaximal, characteristic, complemented
8#14934C211yesmaximal, characteristic, complemented
9#14918C212yesmaximal, complemented
10#14918C212yesmaximal, complemented
11#14786C212yesmaximal, complemented
12#14734C211yesmaximal, characteristic, complemented
13#14724C212yesmaximal
14#14717C211yesmaximal, characteristic, complemented
15#14341C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2626—28no—
24C2×#399C2212yescomplemented
25C2×#399C2212yescomplemented
26C2×#398C2214yescomplemented
27C2×#396C2212yescomplemented
28C2×#396C2212yescomplemented
29C2×#394—28no—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#393C2212yescomplemented
35C2×#393C2212yescomplemented
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C22×#95C2211yescharacteristic
47#1446C2212yescomplemented
48#1445C2214yescomplemented
49#1445C2212yescomplemented
50#1445C2214yescomplemented
51#1444C2212yescomplemented
52#1444C2214yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2214yescomplemented
57#1440C2214yescomplemented
58#1440C2214yescomplemented
59#1440C2214yescomplemented
60#1440C2214yescomplemented
61#1440C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2212yescomplemented
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1439C2212yescomplemented
70#1438C2214yes—
71#1438C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C26C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—24no—
81C23×D4—44no—
82C23×D4—44no—
83C23×D4—44no—
84C23×D4C2314yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4D418yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131#399—28no—
132#399—44no—
133#398—48no—
134#396—28no—
135#396—28no—
136#394—216no—
137#394—216no—
138#394—48no—
139#394—48no—
140#393—28no—
141#393—44no—
142#385—28no—
143#385—44no—
144#385—44no—
145#381—44no—
146#381—44no—
147#380—24no—
148#379—28no—
149#379—44no—
150#379—44no—
151#379—44no—
152C2×#95—24no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—22no—
175C25—24no—
176C25—24no—
177C25—24no—
178C25—22no—
179C25—22no—
180C25—24no—
181C25—24no—
182C25—22no—
183C25—22no—
184C25—42no—
185C25—42no—
186C25—42no—
187C25—42no—
188C25—42no—
189C25—42no—
190C25—42no—
191C25C2×D414yes—
192C25C2412yes—
193C25C2412yes—
194C25C2411yescharacteristic
195C22×Q8—22no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—416no—
207C22×D4—416no—
208C22×D4—416no—
209C22×D4—464no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C22⋊Q8—416no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—416no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—416no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4C2×D418yes—
326C2×C22⋊C4C2×D418yes—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—42no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—42no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—28no—
353C24—28no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—28no—
375C24—28no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—416no—
381C24—416no—
382C24—48no—
383C24—48no—
384C24—44no—
385C24—416no—
386C24—44no—
387C24—44no—
388C24—416no—
389C24—44no—
390C24—44no—
391C24—44no—
392C24—48no—
393C24—48no—
394C24—48no—
395C24—48no—
396C24—88no—
397C24—88no—
398C24—84no—
399C24—84no—
400C24—88no—
401C24—88no—
402C24—84no—
403C24—84no—
404C24—84no—
405C24—84no—
406C24C22×D414yes—
407C24C22×D412yes—
408C24H(5,𝔽2)14yes—
409C24H(5,𝔽2)14yes—
410C24H(5,𝔽2)14yes—
411C242−1+412yes—
412C242−1+412yes—
413C24C2511yescharacteristic
414C2×Q8—44no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—88no—
418C2×D4—416no—
419C2×D4—416no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—416no—
423C2×D4—416no—
424C2×D4—4128no—
425C2×D4—416no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—416no—
492C22×C4—416no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—88no—
504C22×C4—88no—
505C22×C4C22×D412yes—
506C22×C4C22×D414yes—
507C22×C4C22×D412yes—
508C22×C4H(5,𝔽2)14yes—
509C4⋊C4—416no—
510C4⋊C4—48no—
511C4⋊C4—44no—
512C4⋊C4—82no—
513C4⋊C4—84no—
514C4⋊C4—82no—
515C4⋊C4—84no—
516C4⋊C4—88no—
517C4⋊C4—88no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—88no—
533C22⋊C4—88no—
534C22⋊C4—88no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C22⋊C4—88no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—24no—
561C23—24no—
562C23—416no—
563C23—416no—
564C23—416no—
565C23—416no—
566C23—416no—
567C23—416no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—42no—
591C23—42no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—88no—
603C23—88no—
604C23—88no—
605C23—88no—
606C23—832no—
607C23—832no—
608C23—88no—
609C23—88no—
610C23—88no—
611C23—832no—
612C23—832no—
613C23—88no—
614C23#53414yes—
615C23#53514yes—
616C23#53614yes—
617C23#56218yes—
618C23#56218yes—
619C23C23×D411yescharacteristic
620C23C2×H(5,𝔽2)12yes—
621C23C2×H(5,𝔽2)12yes—
622C23C2×H(5,𝔽2)12yes—
623C23C2×H(5,𝔽2)12yes—
624C23C2×2−1+411yescharacteristic
625C23C2×2−1+411yescharacteristic
626C23#58512yes—
627C23#58512yes—
628Q8—88no—
629D4—464no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—28no—
641C2×C4—416no—
642C2×C4—416no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—88no—
695C2×C4—88no—
696C2×C4—88no—
697C22—24no—
698C22—24no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—216no—
703C22—216no—
704C22—24no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—48no—
714C22—48no—
715C22—44no—
716C22—44no—
717C22—832no—
718C22—832no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—84no—
724C22—84no—
725C22—84no—
726C22—84no—
727C22—84no—
728C22—84no—
729C22#281618yes—
730C22C2×#53411yescharacteristic
731C22C2×#53511yescharacteristic
732C22C2×#53611yescharacteristic
733C22#348411yescharacteristic
734C22#348411yescharacteristic
735C22#348511yescharacteristic
736C22#348511yescharacteristic
737C22C2×#56212yes—
738C22C2×#56212yes—
739C22#353214yes—
740C22#353314yes—
741C22#353414yes—
742C22#353512yes—
743C22#353512yes—
744C22#354214yes—
745C22#356112yes—
746C22#356112yes—
747C4—44no—
748C4—44no—
749C4—42no—
750C4—42no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—84no—
757C4—84no—
758C4—82no—
759C2—28no—
760C2—84no—
761C2—84no—
762C2C2×#281611yescharacteristic
763C2#3670311yescharacteristic
764C2#3670411yescharacteristic
765C2#3672011yescharacteristic
766C2#3695611yescharacteristic
767C2#3696611yescharacteristic
768C2#3711711yescharacteristic
769C2#5234614yes—
770C2#5237314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2626C2semidirect
3. #17151C2semidirect
4. #17107C2semidirect
5. #17103C2semidirect
6. #17092C2semidirect
7. #14936C2semidirect
8. #14934C2semidirect
9. #14918C2semidirect
10. #14918C2semidirect
11. #14786C2semidirect
12. #14734C2semidirect
14. #14717C2semidirect
15. #14341C2semidirect
24. C2×#399C22semidirect
25. C2×#399C22semidirect
26. C2×#398C22semidirect
27. C2×#396C22semidirect
28. C2×#396C22semidirect
34. C2×#393C22semidirect
35. C2×#393C22semidirect
47. #1446C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
58. #1440C22semidirect
59. #1440C22semidirect
60. #1440C22semidirect
61. #1440C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
69. #1439C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C22×#3942
#171512
#171072
#171032
#170922
#149362
#149342
#149182
#147862
#147342
#147242
#147172
#143412