Finite Group Databasebeta

← #770151

Subgroups — #770151

Order 512.

3670 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic, complemented
3C22×#393C211yesmaximal, characteristic
4C2×#2329C211yesmaximal, characteristic, complemented
5#15737C214yesmaximal, complemented
6#15633C214yesmaximal, complemented
7#15427C214yesmaximal, complemented
8#15322C214yesmaximal, complemented
9#14935C212yesmaximal, complemented
10#14726C212yesmaximal
11#13992C212yesmaximal, complemented
12#13991C212yesmaximal
13#13922C211yesmaximal, characteristic, complemented
14#13903C211yesmaximal, characteristic
15#13778C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yescomplemented
18C22×C22⋊Q8C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2561—28no—
23#2434—28no—
24C2×#397C2212yes—
25C2×#396C2214yes—
26C2×#395C2214yescomplemented
27C2×#393—28no—
28C2×#393C2214yes—
29C2×#393C2212yes—
30C2×#393C2214yes—
31#2329—28no—
32C2×#391C2212yescomplemented
33C2×#391C2212yescomplemented
34C2×#391C2212yescomplemented
35C2×#391C2212yescomplemented
36C2×#389C2212yescomplemented
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#387C2212yescomplemented
40C2×#387C2212yescomplemented
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2211yescharacteristic
54C22×#95C2211yescharacteristic
55C22×#95C2211yescharacteristic
56#1446C2214yescomplemented
57#1443C2214yescomplemented
58#1443C2214yescomplemented
59#1442C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1441C2214yescomplemented
63#1440C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yescomplemented
73#1436C2214yescomplemented
74#1436C2214yescomplemented
75#1434C2214yescomplemented
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—24no—
83C23×Q8—44no—
84C23×Q8—44no—
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8—44no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8D418yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4—44no—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#397—28no—
133#396—48no—
134#395—216no—
135#393—28no—
136#393—216no—
137#393—216no—
138#391—28no—
139#391—28no—
140#391—44no—
141#391—44no—
142#389—28no—
143#388—216no—
144#388—48no—
145#387—28no—
146#387—28no—
147#385—28no—
148#385—28no—
149#385—28no—
150#381—28no—
151#381—28no—
152#380—24no—
153#380—28no—
154#379—28no—
155#379—28no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—24no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95D418yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25—22no—
187C25—24no—
188C25—22no—
189C25—22no—
190C25—24no—
191C25—24no—
192C25—42no—
193C25—42no—
194C25—42no—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—416no—
216C22×Q8—416no—
217C22×Q8—432no—
218C22×Q8—432no—
219C22×Q8—416no—
220C22×Q8C2×D414yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2×D414yes—
248C23×C4#4114yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2414yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C4⋊Q8—432no—
263C22⋊Q8—232no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D414yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
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—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—416no—
318C2×C22⋊C4—416no—
319C2×C22⋊C4—416no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4#4118yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—44no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—44no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95C2×D418yes—
359#95C2×D418yes—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—28no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24—48no—
376C24—48no—
377C24—416no—
378C24—416no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—48no—
386C24—48no—
387C24—84no—
388C24—84no—
389C24—88no—
390C24—88no—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C2×#4112yes—
394C24C2×#4114yes—
395C242−1+414yes—
396C242−1+414yes—
397C24C2511yescharacteristic
398C2×Q8—216no—
399C2×Q8—216no—
400C2×Q8—28no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×Q8—48no—
407C2×Q8—48no—
408C2×Q8—48no—
409C2×Q8—48no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—4128no—
413C2×Q8—48no—
414C2×Q8—48no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—84no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
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—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
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—28no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
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—24no—
480C22×C4—24no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—88no—
503C22×C4—88no—
504C22×C4#121116yes—
505C22×C4C22×D414yes—
506C22×C4C22×D414yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C2×#4118yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4H(5,𝔽2)14yes—
517C22×C4H(5,𝔽2)14yes—
518C4⋊C4—28no—
519C4⋊C4—216no—
520C4⋊C4—416no—
521C4⋊C4—416no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—88no—
527C22⋊C4—216no—
528C22⋊C4—216no—
529C22⋊C4—416no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—88no—
538C22⋊C4—88no—
539C42—48no—
540C42—84no—
541C42—84no—
542C42—84no—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—416no—
561C23—416no—
562C23—416no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—832no—
582C23—832no—
583C23—88no—
584C23—88no—
585C23—88no—
586C23—88no—
587C23C2×#12114yes—
588C23#53214yes—
589C23#53614yes—
590C23#54818yes—
591C23#55618yes—
592C23C23×D411yescharacteristic
593C23C23×D411yescharacteristic
594C23C22×#4111yescharacteristic
595C23C22×#4112yes—
596C23C2×H(5,𝔽2)12yes—
597C23C2×H(5,𝔽2)12yes—
598C23C2×2−1+412yes—
599C23C2×2−1+412yes—
600C23#58512yes—
601Q8—432no—
602Q8—432no—
603Q8—88no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—416no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—88no—
678C2×C4—88no—
679C2×C4—88no—
680C2×C4C2×#12114yes—
681C2×C4C2×#12114yes—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—216no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—216no—
690C22—44no—
691C22—44no—
692C22—48no—
693C22—44no—
694C22—48no—
695C22—832no—
696C22—82no—
697C22—82no—
698C22—84no—
699C22—82no—
700C22—82no—
701C22—84no—
702C22#254518yes—
703C22C22×#12111yescharacteristic
704C22C2×#53211yescharacteristic
705C22C2×#53611yescharacteristic
706C22C2×#54812yes—
707C22C2×#55612yes—
708C22#348411yescharacteristic
709C22#348412yes—
710C22#348411yescharacteristic
711C22#348511yescharacteristic
712C22#348611yescharacteristic
713C22#348912yes—
714C22#349012yes—
715C22#349512yes—
716C22#351414yes—
717C22#352014yes—
718C22#353814yes—
719C22#356114yes—
720C4—24no—
721C4—24no—
722C4—44no—
723C4—44no—
724C4—44no—
725C4—44no—
726C4—48no—
727C4—44no—
728C4—44no—
729C4—82no—
730C4—82no—
731C4—84no—
732C4—82no—
733C2—28no—
734C2—84no—
735C2C2×#254511yescharacteristic
736C2#3534811yescharacteristic
737C2#3539711yescharacteristic
738C2#3546611yescharacteristic
739C2#3563311yescharacteristic
740C2#3670811yescharacteristic
741C2#3696611yescharacteristic
742C2#4310414yes—
743C2#4317614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2434C2semidirect
4. C2×#2329C2semidirect
5. #15737C2semidirect
6. #15633C2semidirect
7. #15427C2semidirect
8. #15322C2semidirect
9. #14935C2semidirect
11. #13992C2semidirect
13. #13922C2semidirect
15. #13778C2semidirect
17. C22×C4⋊Q8C22semidirect
26. C2×#395C22semidirect
32. C2×#391C22semidirect
33. C2×#391C22semidirect
34. C2×#391C22semidirect
35. C2×#391C22semidirect
36. C2×#389C22semidirect
39. C2×#387C22semidirect
40. C2×#387C22semidirect
56. #1446C22semidirect
57. #1443C22semidirect
58. #1443C22semidirect
59. #1442C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
74. #1436C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#24342
C22×#3932
C2×#23292
#157372
#156332
#154272
#153222
#149352
#147262
#139922
#139912
#139222
#139032
#137782