Finite Group Databasebeta

← #646660

Subgroups — #646660

Order 512.

3078 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2633C211yesmaximal, characteristic, complemented
3C2×C4×#121C211yesmaximal, characteristic
4C22×#383C211yesmaximal, characteristic, complemented
5#16679C212yesmaximal, complemented
6#16677C212yesmaximal, complemented
7#16676C212yesmaximal, complemented
8#16664C212yesmaximal, complemented
9#14925C214yesmaximal, complemented
10#14836C212yesmaximal, complemented
11#14766C212yesmaximal, complemented
12#14747C212yesmaximal
13#14735C212yesmaximal, complemented
14#14345C211yesmaximal, characteristic, complemented
15#14298C211yesmaximal, characteristic, complemented
16#14289C211yesmaximal, characteristic, complemented
17#13975C214yesmaximal, complemented
18C22×#125C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C42C2211yescharacteristic
23#2851—28no—
24#2633—28no—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#394C2212yes—
29C4×#121—28no—
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#386C2214yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#383—28no—
44C2×#383C2211yescharacteristic
45C2×#383C2212yes—
46C2×#383C2212yes—
47C2×#383C2212yes—
48C2×#383C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2214yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C22×#95C2212yes—
57C22×#95C2212yes—
58C22×#95C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1445C2212yes—
63#1444C2212yes—
64#1444C2214yes—
65#1444C2212yes—
66#1443C2214yes—
67#1442C2214yes—
68#1442C2212yes—
69#1442C2214yes—
70#1442C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2214yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1439C2214yes—
82#1438C2212yes—
83#1436C2214yes—
84#1436C2214yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×D4C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2314yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#121—28no—
98C2×#121—24no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—48no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42D418yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#399—28no—
138#398—28no—
139#397—28no—
140#394—28no—
141#390—24no—
142#389—24no—
143#389—24no—
144#389—24no—
145#388—28no—
146#387—28no—
147#387—28no—
148#386—216no—
149#386—28no—
150#386—48no—
151#385—28no—
152#385—28no—
153#385—216no—
154#383—24no—
155#383—28no—
156#383—28no—
157#383—28no—
158#383D418yescomplemented
159#382—24no—
160#380—24no—
161#380—24no—
162#380—24no—
163#380—48no—
164C4×C4⋊C4—24no—
165C4×C22⋊C4—28no—
166C2×#95—28no—
167C2×#95—28no—
168C2×#95—28no—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C25C2412yes—
198C25C2411yescharacteristic
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—42no—
208C22×D4#4118yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251#125—416no—
252#121—232no—
253C4×D4—432no—
254C4×D4—416no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
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—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
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×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—232no—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42#4114yes—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×D418yes—
361#95C2×D418yes—
362#95C2×D418yes—
363C24—21no—
364C24—21no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24C22×D412yes—
379C24C2×#4114yes—
380C24C2×#4114yes—
381C24H(5,𝔽2)14yes—
382C242−1+414yes—
383C24C2511yescharacteristic
384C2×D4—28no—
385C2×D4—216no—
386C2×D4—232no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—416no—
393C2×D4—44no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—416no—
477C22×C4—416no—
478C22×C4—416no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—416no—
482C22×C4—48no—
483C22×C4—88no—
484C22×C4—88no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4C22×D412yes—
489C22×C4C22×D418yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4114yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C4⋊C4—28no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—88no—
515C4⋊C4—88no—
516C22⋊C4—216no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C42—28no—
528C42—28no—
529C42—28no—
530C42—48no—
531C42#121116yes—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—24no—
546C23—28no—
547C23—24no—
548C23—24no—
549C23—28no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23#52514yes—
569C23#53514yes—
570C23#53614yes—
571C23#539116yes—
572C23C23×D411yescharacteristic
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+412yes—
578C23#58512yes—
579C23#58512yes—
580C23#58512yes—
581D4—416no—
582D4—432no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—816no—
665C2×C4—832no—
666C2×C4C2×#12114yes—
667C2×C4C2×#12114yes—
668C2×C4C2×#12114yes—
669C22—216no—
670C22—24no—
671C22—216no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—48no—
678C22—44no—
679C22—44no—
680C22—48no—
681C22—44no—
682C22—84no—
683C22—84no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#263318yescomplemented
689C22C22×#12111yescharacteristic
690C22C2×#52511yescharacteristic
691C22C2×#53511yescharacteristic
692C22C2×#53611yescharacteristic
693C22#344911yescharacteristic
694C22#344911yescharacteristic
695C22#344911yescharacteristic
696C22C2×#53914yes—
697C22D4×#4112yes—
698C22#347512yes—
699C22#348812yes—
700C22#348912yes—
701C22#349412yes—
702C22#349512yes—
703C22#352114yes—
704C22#356014yes—
705C22#356414yes—
706C4—24no—
707C4—24no—
708C4—24no—
709C4—44no—
710C4—48no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—82no—
715C4—84no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C2—28no—
722C2—84no—
723C2C2×#263311yescharacteristic
724C2#3473711yescharacteristic
725C2#3514411yescharacteristic
726C2#3525211yescharacteristic
727C2#3530611yescharacteristic
728C2#3755611yescharacteristic
729C2#3755711yescharacteristic
730C2#3772414yes—
731C2#3791314yes—
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
1. C2×#2851C2semidirect
2. C2×#2633C2semidirect
4. C22×#383C2semidirect
5. #16679C2semidirect
6. #16677C2semidirect
7. #16676C2semidirect
8. #16664C2semidirect
9. #14925C2semidirect
10. #14836C2semidirect
11. #14766C2semidirect
13. #14735C2semidirect
14. #14345C2semidirect
15. #14298C2semidirect
16. #14289C2semidirect
17. #13975C2semidirect
158. #383D4semidirect
688. C22#2633semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#26332
C2×C4×#1212
C22×#3832
#166792
#166772
#166762
#166642
#149252
#148362
#147662
#147472
#147352
#143452
#142982
#142892
#139752