Finite Group Databasebeta

← #4422765

Subgroups — #4422765

Order 512.

2902 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2598C211yesmaximal, characteristic
3C2×#2628C211yesmaximal, characteristic, complemented
4C2×#2343C211yesmaximal, characteristic, complemented
5#17063C212yesmaximal, complemented
6#16802C212yesmaximal, complemented
7#16564C212yesmaximal, complemented
8#16392C212yesmaximal, complemented
9#15354C214yesmaximal, complemented
10#15323C214yesmaximal, complemented
11#15135C212yesmaximal, complemented
12#14977C212yesmaximal
13#14910C212yesmaximal, complemented
14#14836C212yesmaximal, complemented
15#14663C211yesmaximal, characteristic, complemented
16#14356C211yesmaximal, characteristic, complemented
17#14345C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2214yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2628—28no—
23#2598—28no—
24#2542—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#399C2212yes—
28C2×#399C2211yescharacteristic
29C2×#399C2212yes—
30C2×#399C2212yes—
31C2×#395C2212yes—
32C2×#394C2211yescharacteristic
33C2×#393C2212yes—
34C2×#392C2211yescharacteristic
35#2343—28no—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2214yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#384C2212yes—
49C2×#383C2211yescharacteristic
50C2×#383C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#381C2211yescharacteristic
54C2×#380C2212yes—
55C2×#380C2214yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58C2×#376C2211yescharacteristic
59C22×#95C2211yescharacteristic
60#1446C2214yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2214yes—
65#1445C2212yes—
66#1445C2212yes—
67#1445C2214yes—
68#1444C2212yes—
69#1444C2214yes—
70#1442C2212yes—
71#1442C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1440C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1439C2214yes—
82#1438C2214yes—
83#1438C2212yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C22⋊Q8—216no—
97C2×C22⋊Q8—48no—
98C2×C22⋊Q8—48no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×#117—24no—
103C2×#117—24no—
104C2×#117D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#400—24no—
136#400—28no—
137#399—24no—
138#399—28no—
139#399—28no—
140#399—28no—
141#395—28no—
142#394—24no—
143#393—28no—
144#392—24no—
145#390—28no—
146#389—28no—
147#389—48no—
148#388—28no—
149#388—48no—
150#387—28no—
151#387—28no—
152#386—28no—
153#386—28no—
154#386—48no—
155#385—28no—
156#385—28no—
157#384—28no—
158#383—24no—
159#383—24no—
160#382—24no—
161#382—24no—
162#381—24no—
163#380—28no—
164#380—48no—
165C4×C4⋊C4D418yescomplemented
166C4×C22⋊C4—28no—
167#376—24no—
168C2×#95—24no—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C25C2412yes—
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—28no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—44no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—42no—
229C23×C4C2×D414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C4⋊Q8—416no—
246C22⋊Q8—464no—
247C4×D4—416no—
248#117—232no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
319C2×C22⋊C4C2×D414yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—44no—
325C2×C42C2×D414yes—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42#4114yes—
330#95—22no—
331#95—24no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D414yes—
358#95C2×D414yes—
359C24—21no—
360C24—21no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—22no—
374C24—22no—
375C24C22×D412yes—
376C242−1+414yes—
377C242−1+414yes—
378C24C2511yescharacteristic
379C2×Q8—416no—
380C2×Q8—48no—
381C2×Q8—88no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×D4—28no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—44no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—84no—
393C2×D4—88no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
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—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)18yes—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—48no—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—416no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—28no—
513C42—48no—
514C42—48no—
515C42#121116yes—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23#53614yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)14yes—
559C23C2×2−1+412yes—
560C23C2×2−1+412yes—
561C23#58512yes—
562C23#58512yes—
563Q8—816no—
564D4—416no—
565D4—816no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—416no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
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—816no—
641C2×C4—816no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12114yes—
644C2×C4C2×#12114yes—
645C2×C4#53518yes—
646C22—28no—
647C22—24no—
648C22—42no—
649C22—44no—
650C22—48no—
651C22—44no—
652C22—42no—
653C22—44no—
654C22—44no—
655C22—48no—
656C22—48no—
657C22—44no—
658C22—44no—
659C22—84no—
660C22—84no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C22×#12111yescharacteristic
666C22C2×#53512yes—
667C22C2×#53611yescharacteristic
668C22#344911yescharacteristic
669C22#348511yescharacteristic
670C22#348611yescharacteristic
671C22#348912yes—
672C22#348912yes—
673C22#349512yes—
674C22#349512yes—
675C22#349812yes—
676C22#350812yes—
677C22#353314yes—
678C22#353514yes—
679C22#353814yes—
680C22#354114yes—
681C4—24no—
682C4—28no—
683C4—24no—
684C4—24no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—84no—
693C4—82no—
694C4—82no—
695C4—84no—
696C2—44no—
697C2—84no—
698C2#3527211yescharacteristic
699C2#3530611yescharacteristic
700C2#3563311yescharacteristic
701C2#4140912yes—
702C2#4195312yes—
703C2#4195912yes—
704C2#4280812yes—
705C2#5105714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2542C2semidirect
3. C2×#2628C2semidirect
4. C2×#2343C2semidirect
5. #17063C2semidirect
6. #16802C2semidirect
7. #16564C2semidirect
8. #16392C2semidirect
9. #15354C2semidirect
10. #15323C2semidirect
11. #15135C2semidirect
13. #14910C2semidirect
14. #14836C2semidirect
15. #14663C2semidirect
16. #14356C2semidirect
17. #14345C2semidirect
165. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#25982
C2×#26282
C2×#23432
#170632
#168022
#165642
#163922
#153542
#153232
#151352
#149772
#149102
#148362
#146632
#143562
#143452