Finite Group Databasebeta

← #921051

Subgroups — #921051

Order 512.

3830 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic, complemented
2C22×#394C211yesmaximal, characteristic
3#17125C214yesmaximal, complemented
4#16652C212yesmaximal, complemented
5#16541C214yesmaximal, complemented
6#16514C212yesmaximal, complemented
7#15833C212yesmaximal, complemented
8#15643C212yesmaximal, complemented
9#14726C212yesmaximal, complemented
10#14724C212yesmaximal
11#14717C212yesmaximal, complemented
12#14265C212yesmaximal, complemented
13#13991C212yesmaximal, complemented
14#13971C212yesmaximal, complemented
15#13923C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21#2502—28no—
22C2×#398C2212yescomplemented
23C2×#398C2212yescomplemented
24C2×#396C2214yescomplemented
25C2×#394—28no—
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#394C2212yes—
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#393C2214yescomplemented
33C2×#391C2211yescharacteristic, complemented
34C2×#391C2211yescharacteristic, complemented
35C2×#388C2214yescomplemented
36C2×#388C2214yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#384C2211yescharacteristic, complemented
41C2×#384C2211yescharacteristic, complemented
42C2×#380C2211yescharacteristic
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×#379C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2212yescomplemented
55#1441C2214yescomplemented
56#1440C2214yescomplemented
57#1440C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1438C2214yes—
67#1438C2214yes—
68#1437C2214yescomplemented
69#1437C2214yescomplemented
70#1436C2212yescomplemented
71#1436C2212yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4—24no—
80C23×D4—42no—
81C23×D4—42no—
82C23×D4—48no—
83C23×D4C2312yes—
84C23×D4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—42no—
91C2×C22⋊Q8—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—48no—
110C22×C22⋊C4—48no—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#398—28no—
131#398—28no—
132#396—48no—
133#394—28no—
134#394—28no—
135#394—28no—
136#394—28no—
137#394—48no—
138#394—48no—
139#393—48no—
140#391—24no—
141#391—24no—
142#388—48no—
143#388—48no—
144#385—28no—
145#385—28no—
146#385—28no—
147#384—24no—
148#384—24no—
149#380—42no—
150#379—28no—
151#379—28no—
152#379—44no—
153#379—44no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—42no—
156C2×#95—24no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C25—24no—
178C25—28no—
179C25—24no—
180C25—24no—
181C25—24no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—44no—
186C25—44no—
187C25—44no—
188C25C2414yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—41no—
193C22×Q8—41no—
194C22×Q8—42no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—416no—
206C22×D4—432no—
207C22×D4—432no—
208C22×D4—432no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C22⋊Q8—416no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
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—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
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—432no—
307C2×C22⋊C4—432no—
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#4118yes—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—44no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—216no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—416no—
368C24—416no—
369C24—416no—
370C24—416no—
371C24—416no—
372C24—416no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—48no—
377C24—48no—
378C24—48no—
379C24—48no—
380C24—48no—
381C24—48no—
382C24—48no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—88no—
388C24—88no—
389C24C22×D412yes—
390C24C2×#4114yes—
391C24H(5,𝔽2)18yes—
392C24H(5,𝔽2)12yes—
393C24H(5,𝔽2)12yes—
394C242−1+414yes—
395C24C2511yescharacteristic
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×D4—416no—
402C2×D4—416no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—416no—
409C2×D4—4128no—
410C2×D4—48no—
411C2×D4—48no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4H(5,𝔽2)14yes—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4—88no—
508C4⋊C4—84no—
509C22⋊C4—216no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C42—44no—
526C42—44no—
527C42—84no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—432no—
547C23—432no—
548C23—432no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—832no—
578C23—832no—
579C23—832no—
580C23—832no—
581C23—816no—
582C23—816no—
583C23—816no—
584C23—816no—
585C23#53414yes—
586C23#54218yes—
587C23#54818yes—
588C23#55514yes—
589C23#56314yes—
590C23C23×D411yescharacteristic
591C23C22×#4112yes—
592C23C2×H(5,𝔽2)11yescharacteristic
593C23C2×H(5,𝔽2)11yescharacteristic
594C23C2×H(5,𝔽2)12yes—
595C23C2×H(5,𝔽2)14yes—
596C23C2×2−1+412yes—
597C23#58512yes—
598Q8—88no—
599D4—432no—
600D4—432no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—416no—
621C2×C4—416no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
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—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
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—44no—
653C2×C4—44no—
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—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—88no—
670C2×C4—816no—
671C2×C4#53514yes—
672C22—24no—
673C22—216no—
674C22—24no—
675C22—28no—
676C22—28no—
677C22—28no—
678C22—28no—
679C22—48no—
680C22—48no—
681C22—48no—
682C22—48no—
683C22—48no—
684C22—48no—
685C22—48no—
686C22—864no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22—84no—
691C22—84no—
692C22—84no—
693C22—84no—
694C22—84no—
695C22#273118yes—
696C22C2×#53411yescharacteristic
697C22C2×#53511yescharacteristic
698C22#344911yescharacteristic
699C22C2×#54212yes—
700C22C2×#54812yes—
701C22C2×#55511yescharacteristic
702C22#348412yes—
703C22#351314yes—
704C22#351914yes—
705C22C2×#56311yescharacteristic
706C22#353212yes—
707C22#353212yes—
708C22#353412yes—
709C22#353412yes—
710C22#353514yes—
711C22#353614yes—
712C4—24no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—44no—
720C4—82no—
721C4—84no—
722C4—82no—
723C4—84no—
724C4—84no—
725C2—28no—
726C2—88no—
727C2#3670211yescharacteristic
728C2#3744011yescharacteristic
729C2#3911612yes—
730C2#3912012yes—
731C2#4078514yes—
732C2#4319812yes—
733C2C2×#273111yescharacteristic
734C2#5241112yes—
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×#2502C2semidirect
3. #17125C2semidirect
4. #16652C2semidirect
5. #16541C2semidirect
6. #16514C2semidirect
7. #15833C2semidirect
8. #15643C2semidirect
9. #14726C2semidirect
11. #14717C2semidirect
12. #14265C2semidirect
13. #13991C2semidirect
14. #13971C2semidirect
15. #13923C2semidirect
22. C2×#398C22semidirect
23. C2×#398C22semidirect
24. C2×#396C22semidirect
32. C2×#393C22semidirect
33. C2×#391C22semidirect
34. C2×#391C22semidirect
35. C2×#388C22semidirect
36. C2×#388C22semidirect
40. C2×#384C22semidirect
41. C2×#384C22semidirect
50. #1445C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
68. #1437C22semidirect
69. #1437C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C22×#3942
#171252
#166522
#165412
#165142
#158332
#156432
#147262
#147242
#147172
#142652
#139912
#139712
#139232