Finite Group Databasebeta

← #763968

Subgroups — #763968

Order 512.

3748 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic, complemented
3C2×#2321C211yesmaximal, characteristic
4C2×#2315C211yesmaximal, characteristic, complemented
5#15585C214yesmaximal, complemented
6#15505C214yesmaximal, complemented
7#15439C214yesmaximal, complemented
8#15294C214yesmaximal, complemented
9#14729C212yesmaximal, complemented
10#14728C212yesmaximal, complemented
11#14149C211yesmaximal, characteristic, complemented
12#14134C212yesmaximal
13#13960C211yesmaximal, characteristic, complemented
14#13924C211yesmaximal, characteristic, complemented
15#13877C212yesmaximal, complemented
16C22×#120C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23#2487—28no—
24C2×#400C2212yescomplemented
25C2×#399—28no—
26C2×#399C2214yes—
27C2×#399C2214yes—
28C2×#399C2212yescomplemented
29C2×#398C2214yescomplemented
30C2×#396C2214yescomplemented
31#2321—28no—
32#2315—28no—
33C2×#390C2212yes—
34C2×#389C2212yescomplemented
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#386C2212yes—
38C2×#386C2211yescharacteristic
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#385C2211yescharacteristic
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2214yes—
50C2×#380C2211yescharacteristic
51C2×#380C2212yescomplemented
52C2×#379C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2211yescharacteristic
55#1445C2214yescomplemented
56#1444C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2212yescomplemented
61#1441C2214yescomplemented
62#1441C2214yescomplemented
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yescomplemented
67#1439C2212yescomplemented
68#1438C2214yes—
69#1436C2214yescomplemented
70C4⋊#95C2214yescomplemented
71#1434C2212yescomplemented
72C22⋊#95C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2214yes—
78C22⋊#95C2212yescomplemented
79C26C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#120—24no—
89C2×#120—24no—
90C2×#120Q818yes—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—28no—
115C22×C22⋊C4—48no—
116C22×C22⋊C4—44no—
117C22×C22⋊C4D418yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2312yes—
137#400—28no—
138#399—216no—
139#399—216no—
140#399—28no—
141#398—48no—
142#396—48no—
143#390—28no—
144#389—28no—
145#388—28no—
146#388—28no—
147#386—24no—
148#386—28no—
149#386—28no—
150#386—48no—
151#386—44no—
152#385—24no—
153#385—216no—
154#385—28no—
155#385—216no—
156#385—44no—
157#380—24no—
158#380—28no—
159#380—28no—
160#380—48no—
161#380D418yescomplemented
162#379—24no—
163C4×C22⋊C4—24no—
164C2×#95—24no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25—22no—
186C25—22no—
187C25—22no—
188C25—22no—
189C25—44no—
190C25—44no—
191C25C2×D414yes—
192C25C2×Q814yes—
193C25#4114yes—
194C25C2411yescharacteristic
195C25C2412yes—
196C25C2411yescharacteristic
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4C2×Q814yes—
208C22×D4C2×Q818yes—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—28no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250#120—232no—
251C4×D4—416no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—432no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—232no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—432no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—432no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×D418yes—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333C2×C22⋊C4C2×Q818yes—
334C2×C22⋊C4C2×Q814yes—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—44no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—28no—
371C24—28no—
372C24—28no—
373C24—28no—
374C24—28no—
375C24—28no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—24no—
381C24—21no—
382C24—21no—
383C24—24no—
384C24—48no—
385C24—48no—
386C24—416no—
387C24—416no—
388C24—84no—
389C24—84no—
390C24—84no—
391C24—84no—
392C24—88no—
393C24—88no—
394C24C22⋊Q8116yes—
395C24C22×D414yes—
396C24C22×D412yes—
397C24C22×D412yes—
398C24C22×Q812yes—
399C24C22×Q814yes—
400C24C22×Q812yes—
401C24C2×#4114yes—
402C24C2×#4112yes—
403C24C2×#4112yes—
404C24H(5,𝔽2)12yes—
405C242−1+412yes—
406C24C2511yescharacteristic
407C2×D4—28no—
408C2×D4—28no—
409C2×D4—28no—
410C2×D4—216no—
411C2×D4—416no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—88no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—216no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—416no—
489C22×C4—48no—
490C22×C4—44no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4—84no—
496C22×C4—88no—
497C22×C4—84no—
498C22×C4—88no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4—88no—
502C22×C4#121116yes—
503C22×C4C22×D418yes—
504C22×C4C22×D414yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×Q812yes—
510C22×C4C22×Q814yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)18yes—
514C4⋊C4—48no—
515C4⋊C4—416no—
516C4⋊C4—416no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—88no—
523C4⋊C4—88no—
524C4⋊C4—88no—
525C22⋊C4—216no—
526C22⋊C4—28no—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—416no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—48no—
536C22⋊C4—44no—
537C22⋊C4—84no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C42—44no—
543C42—84no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—28no—
550C23—28no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—28no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—232no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—28no—
573C23—432no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—44no—
594C23—44no—
595C23—816no—
596C23—88no—
597C23—88no—
598C23—816no—
599C23—816no—
600C23—816no—
601C23—816no—
602C23C2×#12114yes—
603C23C2×C22⋊Q818yes—
604C23C2×C22⋊Q814yes—
605C23#53714yes—
606C23#54614yes—
607C23D4×Q814yes—
608C23#55214yes—
609C23#55414yes—
610C23C23×D411yescharacteristic
611C23C23×D411yescharacteristic
612C23C23×Q811yescharacteristic
613C23C22×#4111yescharacteristic
614C23C22×#4111yescharacteristic
615C23C2×H(5,𝔽2)11yescharacteristic
616C23C2×H(5,𝔽2)14yes—
617C23C2×2−1+411yescharacteristic
618C23#58514yes—
619D4—416no—
620D4—816no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—28no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—28no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—416no—
642C2×C4—432no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—41no—
661C2×C4—41no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—88no—
688C2×C4—88no—
689C2×C4—832no—
690C2×C4—816no—
691C2×C4C2×#12114yes—
692C2×C4C2×#12114yes—
693C22—28no—
694C22—28no—
695C22—24no—
696C22—24no—
697C22—28no—
698C22—28no—
699C22—28no—
700C22—28no—
701C22—28no—
702C22—28no—
703C22—28no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—48no—
709C22—48no—
710C22—48no—
711C22—48no—
712C22—44no—
713C22—42no—
714C22—44no—
715C22—44no—
716C22—832no—
717C22—82no—
718C22—81no—
719C22—81no—
720C22—82no—
721C22—82no—
722C22#241718yes—
723C22C22×#12111yescharacteristic
724C22C22×C22⋊Q811yescharacteristic
725C22C2×#53711yescharacteristic
726C22C2×#54611yescharacteristic
727C22C2×D4×Q811yescharacteristic
728C22C2×#55211yescharacteristic
729C22C2×#55411yescharacteristic
730C22#348412yes—
731C22#348512yes—
732C22#348512yes—
733C22#349112yes—
734C22#349812yes—
735C22#350114yes—
736C22#350514yes—
737C22#350812yes—
738C22#353414yes—
739C22#353514yes—
740C4—24no—
741C4—24no—
742C4—44no—
743C4—48no—
744C4—42no—
745C4—44no—
746C4—48no—
747C4—82no—
748C4—81no—
749C4—81no—
750C4—84no—
751C4—84no—
752C2—28no—
753C2—48no—
754C2—82no—
755C2C2×#241711yescharacteristic
756C2#3534912yes—
757C2#3544711yescharacteristic
758C2#3552811yescharacteristic
759C2#3670912yes—
760C2#4349112yes—
761C2#4349412yes—
762C2#4350912yes—
763C2#4351212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2487C2semidirect
2. C22×#399C2semidirect
4. C2×#2315C2semidirect
5. #15585C2semidirect
6. #15505C2semidirect
7. #15439C2semidirect
8. #15294C2semidirect
9. #14729C2semidirect
10. #14728C2semidirect
11. #14149C2semidirect
13. #13960C2semidirect
14. #13924C2semidirect
15. #13877C2semidirect
24. C2×#400C22semidirect
28. C2×#399C22semidirect
29. C2×#398C22semidirect
30. C2×#396C22semidirect
34. C2×#389C22semidirect
51. C2×#380C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
69. #1436C22semidirect
70. C4⋊#95C22semidirect
71. #1434C22semidirect
78. C22⋊#95C22semidirect
161. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C22×#3992
C2×#23212
C2×#23152
#155852
#155052
#154392
#152942
#147292
#147282
#141492
#141342
#139602
#139242
#138772