Finite Group Databasebeta

← #4867151

Subgroups — #4867151

Order 512.

3354 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2534C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic, complemented
3#17100C214yesmaximal, complemented
4#16807C212yesmaximal, complemented
5#16666C212yesmaximal, complemented
6#16665C212yesmaximal, complemented
7#16566C212yesmaximal, complemented
8#16365C212yesmaximal, complemented
9#16297C212yesmaximal, complemented
10#16286C214yesmaximal, complemented
11#15463C212yesmaximal, complemented
12#15266C212yesmaximal
13#14873C212yesmaximal, complemented
14#14717C212yesmaximal, complemented
15#14345C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic, complemented
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×#120C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2534—28no—
23#2409—28no—
24C2×#398C2214yescomplemented
25C2×#397C2214yescomplemented
26C2×#396C2212yes—
27C2×#395C2212yescomplemented
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33C2×#393C2214yes—
34C2×#391C2212yescomplemented
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2214yes—
39C2×#385C2212yescomplemented
40C2×#383C2211yescharacteristic, complemented
41C2×#382C2211yescharacteristic, complemented
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic, complemented
45C2×#379C2211yescharacteristic
46C2×#379C2214yes—
47#1446C2212yescomplemented
48#1446C2214yescomplemented
49#1446C2212yescomplemented
50#1445C2212yescomplemented
51#1445C2214yescomplemented
52#1444C2212yescomplemented
53#1444C2212yescomplemented
54#1443C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2212yescomplemented
72#1438C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yescomplemented
76C22⋊#95C2214yescomplemented
77C26C2311yescharacteristic
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—48no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—24no—
91C2×#123—44no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—48no—
94C2×#120—24no—
95C2×#120—44no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—24no—
108C22×C22⋊C4—42no—
109C22×C22⋊C4—42no—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#398—48no—
130#397—48no—
131#396—28no—
132#395—28no—
133#394—28no—
134#394—48no—
135#393—28no—
136#393—44no—
137#393—44no—
138#393—48no—
139#391—28no—
140#388—28no—
141#386—28no—
142#386—28no—
143#386—48no—
144#385—28no—
145#383—42no—
146#382—42no—
147#381—24no—
148#380—24no—
149#380—24no—
150#379—24no—
151#379—48no—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C25—22no—
176C25—24no—
177C25—24no—
178C25—41no—
179C25—41no—
180C25—42no—
181C25—42no—
182C25—44no—
183C25C2412yes—
184C25C2411yescharacteristic
185C25C2411yescharacteristic
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—28no—
189C22×Q8—28no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—416no—
197C22×Q8—432no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—28no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4C2×D414yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—41no—
223C23×C4—41no—
224C23×C4—44no—
225C23×C4—42no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240#123—416no—
241C22⋊Q8—432no—
242#120—416no—
243C4×D4—416no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
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—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—416no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42C2×D414yes—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—24no—
340C24—22no—
341C24—21no—
342C24—21no—
343C24—24no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—48no—
362C24—44no—
363C24—416no—
364C24—44no—
365C24—48no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—84no—
370C24—84no—
371C24—84no—
372C24—88no—
373C24—88no—
374C24—84no—
375C24—84no—
376C24—88no—
377C24C22×D412yes—
378C24C22×D412yes—
379C24H(5,𝔽2)14yes—
380C242−1+414yes—
381C242−1+414yes—
382C24C2511yescharacteristic
383C2×Q8—216no—
384C2×Q8—416no—
385C2×Q8—416no—
386C2×Q8—416no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—416no—
391C2×Q8—4128no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—44no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—416no—
401C2×D4—48no—
402C2×D4—84no—
403C2×D4—88no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4H(5,𝔽2)18yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C4⋊C4—44no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C42—28no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—24no—
531C23—28no—
532C23—24no—
533C23—28no—
534C23—24no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—41no—
541C23—44no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—416no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—42no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—816no—
569C23—88no—
570C23—816no—
571C23—816no—
572C23—832no—
573C23—816no—
574C23—88no—
575C23#53514yes—
576C23#53614yes—
577C23C23×D411yescharacteristic
578C23C2×H(5,𝔽2)11yescharacteristic
579C23C2×H(5,𝔽2)11yescharacteristic
580C23C2×H(5,𝔽2)12yes—
581C23C2×H(5,𝔽2)14yes—
582C23C2×2−1+412yes—
583C23C2×2−1+412yes—
584C23#58512yes—
585Q8—432no—
586Q8—432no—
587D4—416no—
588D4—816no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—416no—
605C2×C4—432no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
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—44no—
654C2×C4—44no—
655C2×C4—88no—
656C2×C4—88no—
657C2×C4#53414yes—
658C22—24no—
659C22—24no—
660C22—28no—
661C22—28no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—48no—
672C22—42no—
673C22—42no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—44no—
679C22—832no—
680C22—84no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22—82no—
685C22—84no—
686C22C2×#53411yescharacteristic
687C22C2×#53511yescharacteristic
688C22C2×#53611yescharacteristic
689C22#348412yes—
690C22#348912yes—
691C22#353212yes—
692C22#353212yes—
693C22#353612yes—
694C22#353612yes—
695C22#353614yes—
696C22#353811yescharacteristic
697C22#353814yes—
698C22#354011yescharacteristic
699C22#354112yes—
700C22#355814yes—
701C22#356114yes—
702C4—24no—
703C4—42no—
704C4—42no—
705C4—48no—
706C4—42no—
707C4—42no—
708C4—84no—
709C4—82no—
710C4—82no—
711C4—84no—
712C4—82no—
713C4—84no—
714C2—44no—
715C2—44no—
716C2—84no—
717C2#3671411yescharacteristic
718C2#3737411yescharacteristic
719C2#3745111yescharacteristic
720C2#4059312yes—
721C2#4059512yes—
722C2#5366014yes—
723C2#5892412yes—
724C2#5898712yes—
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
2. C2×#2409C2semidirect
3. #17100C2semidirect
4. #16807C2semidirect
5. #16666C2semidirect
6. #16665C2semidirect
7. #16566C2semidirect
8. #16365C2semidirect
9. #16297C2semidirect
10. #16286C2semidirect
11. #15463C2semidirect
13. #14873C2semidirect
14. #14717C2semidirect
15. #14345C2semidirect
16. C24×Q8C22semidirect
24. C2×#398C22semidirect
25. C2×#397C22semidirect
27. C2×#395C22semidirect
34. C2×#391C22semidirect
39. C2×#385C22semidirect
40. C2×#383C22semidirect
41. C2×#382C22semidirect
44. C2×#380C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1446C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1443C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
71. #1439C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25342
C2×#24092
#171002
#168072
#166662
#166652
#165662
#163652
#162972
#162862
#154632
#152662
#148732
#147172
#143452