Finite Group Databasebeta

← #763823

Subgroups — #763823

Order 512.

3772 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic, complemented
2C2×#2412C212yesmaximal
3C22×#397C211yesmaximal, characteristic, complemented
4C2×#2321C211yesmaximal, characteristic
5C2×#2315C211yesmaximal, characteristic, complemented
6#15497C214yesmaximal, complemented
7#15472C214yesmaximal, complemented
8#15390C214yesmaximal, complemented
9#15290C214yesmaximal, complemented
10#14935C212yesmaximal, complemented
11#14927C212yesmaximal, complemented
12#14149C211yesmaximal, characteristic, complemented
13#13960C211yesmaximal, characteristic, complemented
14#13904C211yesmaximal, characteristic, complemented
15#13869C212yesmaximal, complemented
16C22×#123C2212yes—
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23#2486—28no—
24#2412—216no—
25C2×#398C2212yescomplemented
26C2×#397—28no—
27C2×#397C2214yes—
28C2×#397C2214yes—
29C2×#397C2214yes—
30C2×#397C2212yescomplemented
31C2×#394C2214yes—
32#2321—28no—
33#2315—28no—
34C2×#390C2212yes—
35C2×#389C2212yescomplemented
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#386C2212yes—
39C2×#386C2211yescharacteristic
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#385C2214yes—
43C2×#385C2214yes—
44C2×#385C2211yescharacteristic
45C2×#385C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#380C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2214yes—
52C2×#380C2211yescharacteristic
53C2×#380C2212yescomplemented
54C2×#379C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2211yescharacteristic
57C22×#95C2212yes—
58#1445C2214yescomplemented
59#1443C2214yescomplemented
60#1442C2214yescomplemented
61#1441C2214yescomplemented
62#1441C2214yescomplemented
63#1441C2214yescomplemented
64#1441C2214yescomplemented
65#1441C2214yescomplemented
66#1441C2212yescomplemented
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yescomplemented
70#1439C2214yescomplemented
71#1439C2212yescomplemented
72#1436C2214yescomplemented
73C4⋊#95C2214yescomplemented
74#1434C2212yescomplemented
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2214yes—
79C22⋊#95C2212yescomplemented
80C26C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—44no—
92C2×#120—24no—
93C2×#120—24no—
94C2×#120D418yes—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
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—28no—
116C22×C22⋊C4—28no—
117C22×C22⋊C4—44no—
118C22×C22⋊C4D418yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138#398—28no—
139#397—216no—
140#397—216no—
141#397—216no—
142#397—28no—
143#394—216no—
144#390—28no—
145#389—28no—
146#388—28no—
147#388—28no—
148#386—28no—
149#386—216no—
150#386—24no—
151#386—28no—
152#385—24no—
153#385—216no—
154#385—28no—
155#385—216no—
156#381—28no—
157#381—28no—
158#381—44no—
159#380—24no—
160#380—28no—
161#380—28no—
162#380—216no—
163#380D418yescomplemented
164#379—24no—
165C4×C22⋊C4—24no—
166C2×#95—28no—
167C2×#95—24no—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C25—22no—
191C25—22no—
192C25—22no—
193C25—22no—
194C25—44no—
195C25—44no—
196C25C2×D414yes—
197C25C2×D414yes—
198C25#4114yes—
199C25C2411yescharacteristic
200C25C2412yes—
201C25C2411yescharacteristic
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4C2×D414yes—
213C22×D4C2×D418yes—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—28no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4#4114yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2414yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257#123—432no—
258#120—232no—
259C4×D4—416no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D418yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—232no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—432no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—432no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333C2×C22⋊C4C2×D418yes—
334C2×C22⋊C4C2×D418yes—
335C2×C22⋊C4C2×D414yes—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—44no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—28no—
355#95—24no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363C24—24no—
364C24—24no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—28no—
374C24—28no—
375C24—28no—
376C24—28no—
377C24—28no—
378C24—28no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24—21no—
385C24—24no—
386C24—24no—
387C24—48no—
388C24—48no—
389C24—416no—
390C24—416no—
391C24—84no—
392C24—84no—
393C24—84no—
394C24—84no—
395C24—88no—
396C24—88no—
397C24#121116yes—
398C24C22×D412yes—
399C24C22×D414yes—
400C24C22×D414yes—
401C24C22×D412yes—
402C24C22×D412yes—
403C24C22×D412yes—
404C24C2×#4114yes—
405C24C2×#4112yes—
406C24C2×#4112yes—
407C24H(5,𝔽2)12yes—
408C242−1+412yes—
409C24C2511yescharacteristic
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—28no—
413C2×D4—216no—
414C2×D4—416no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4—88no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
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—28no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—416no—
484C22×C4—48no—
485C22×C4—416no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—44no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—88no—
500C22×C4—88no—
501C22×C4—88no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4—88no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4—88no—
508C22×C4#121116yes—
509C22×C4C22×D414yes—
510C22×C4C22×D414yes—
511C22×C4C22×D412yes—
512C22×C4C22×D418yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C4⋊C4—48no—
520C4⋊C4—416no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—44no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—88no—
527C4⋊C4—88no—
528C22⋊C4—216no—
529C22⋊C4—28no—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—216no—
533C22⋊C4—416no—
534C22⋊C4—416no—
535C22⋊C4—416no—
536C22⋊C4—416no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—48no—
540C22⋊C4—44no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—88no—
545C42—44no—
546C42—84no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—24no—
552C23—28no—
553C23—24no—
554C23—24no—
555C23—28no—
556C23—28no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—24no—
568C23—24no—
569C23—28no—
570C23—28no—
571C23—232no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—28no—
576C23—432no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—88no—
599C23—816no—
600C23—88no—
601C23—816no—
602C23—816no—
603C23—816no—
604C23—816no—
605C23C2×#12118yes—
606C23C2×#12114yes—
607C23C2×#12114yes—
608C23#53714yes—
609C23D4214yes—
610C23#54714yes—
611C23#54814yes—
612C23#55314yes—
613C23C23×D411yescharacteristic
614C23C23×D411yescharacteristic
615C23C23×D411yescharacteristic
616C23C22×#4111yescharacteristic
617C23C22×#4111yescharacteristic
618C23C2×H(5,𝔽2)11yescharacteristic
619C23C2×2−1+411yescharacteristic
620C23#58514yes—
621C23#58514yes—
622D4—416no—
623D4—816no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—416no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—41no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—88no—
690C2×C4—88no—
691C2×C4—832no—
692C2×C4—816no—
693C2×C4—816no—
694C2×C4C2×#12114yes—
695C2×C4C2×#12114yes—
696C22—24no—
697C22—28no—
698C22—24no—
699C22—28no—
700C22—28no—
701C22—28no—
702C22—28no—
703C22—28no—
704C22—28no—
705C22—28no—
706C22—28no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—48no—
712C22—48no—
713C22—48no—
714C22—48no—
715C22—42no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—832no—
720C22—81no—
721C22—82no—
722C22—81no—
723C22—82no—
724C22—82no—
725C22#241618yes—
726C22C22×#12111yescharacteristic
727C22C22×#12111yescharacteristic
728C22C2×#53711yescharacteristic
729C22C2×D4211yescharacteristic
730C22C2×#54711yescharacteristic
731C22C2×#54811yescharacteristic
732C22C2×#55311yescharacteristic
733C22#348512yes—
734C22#348512yes—
735C22#348512yes—
736C22#348514yes—
737C22#349114yes—
738C22#349112yes—
739C22#350012yes—
740C22#351012yes—
741C22#356014yes—
742C22#356414yes—
743C4—24no—
744C4—24no—
745C4—44no—
746C4—42no—
747C4—44no—
748C4—48no—
749C4—81no—
750C4—82no—
751C4—84no—
752C4—81no—
753C4—84no—
754C4—84no—
755C2—28no—
756C2—48no—
757C2—82no—
758C2C2×#241611yescharacteristic
759C2#3544711yescharacteristic
760C2#3544712yes—
761C2#3561711yescharacteristic
762C2#3696012yes—
763C2#4349212yes—
764C2#4349312yes—
765C2#4351012yes—
766C2#4351112yes—
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×#2486C2semidirect
3. C22×#397C2semidirect
5. C2×#2315C2semidirect
6. #15497C2semidirect
7. #15472C2semidirect
8. #15390C2semidirect
9. #15290C2semidirect
10. #14935C2semidirect
11. #14927C2semidirect
12. #14149C2semidirect
13. #13960C2semidirect
14. #13904C2semidirect
15. #13869C2semidirect
25. C2×#398C22semidirect
30. C2×#397C22semidirect
35. C2×#389C22semidirect
53. C2×#380C22semidirect
58. #1445C22semidirect
59. #1443C22semidirect
60. #1442C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
63. #1441C22semidirect
64. #1441C22semidirect
65. #1441C22semidirect
66. #1441C22semidirect
69. #1440C22semidirect
70. #1439C22semidirect
71. #1439C22semidirect
72. #1436C22semidirect
73. C4⋊#95C22semidirect
74. #1434C22semidirect
79. C22⋊#95C22semidirect
163. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#24122
C22×#3972
C2×#23212
C2×#23152
#154972
#154722
#153902
#152902
#149352
#149272
#141492
#139602
#139042
#138692