Finite Group Databasebeta

← #4416782

Subgroups — #4416782

Order 512.

2244 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2821C211yesmaximal, characteristic
2C2×#2821C211yesmaximal, characteristic
3C2×#2862C211yesmaximal, characteristic, complemented
4C2×C4×#126C211yesmaximal, characteristic
5#16806C212yesmaximal, complemented
6#16805C212yesmaximal
7#16804C212yesmaximal, complemented
8#16803C212yesmaximal
9#16705C212yesmaximal, complemented
10#16701C212yesmaximal
11#15164C212yesmaximal, complemented
12#15161C212yesmaximal
13#15097C212yesmaximal, complemented
14#15085C212yesmaximal, complemented
15#15050C212yesmaximal
16#15016C212yesmaximal
17#14688C211yesmaximal, characteristic, complemented
18#14663C211yesmaximal, characteristic, complemented
19#14663C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2211yescharacteristic
23#2862—28no—
24#2821—28no—
25#2821—28no—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#398C2212yes—
31C2×#398C2212yes—
32C2×#397C2212yes—
33C2×#397C2212yes—
34C2×#395C2212yes—
35C2×#395C2212yes—
36C2×#393C2212yes—
37C2×#393C2212yes—
38C4×#126—28no—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#388C2211yescharacteristic
44C2×#388C2211yescharacteristic
45C2×#388C2211yescharacteristic
46C2×#387C2211yescharacteristic
47C2×#387C2211yescharacteristic
48C2×#387C2211yescharacteristic
49C2×#383C2211yescharacteristic
50C2×#383C2211yescharacteristic
51C2×#383C2211yescharacteristic
52C2×#382C2212yes—
53C2×#382C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60C2×C43C2211yescharacteristic
61#1446C2212yes—
62#1446C2212yes—
63#1446C2212yes—
64#1446C2212yes—
65#1445C2212yes—
66#1445C2212yes—
67#1445C2212yes—
68#1445C2212yes—
69#1445C2212yes—
70#1445C2212yes—
71#1445C2212yes—
72#1445C2212yes—
73#1443C2214yes—
74#1443C2214yes—
75#1442C2214yes—
76#1442C2214yes—
77#1441C2214yes—
78#1441C2214yes—
79#1441C2214yes—
80#1441C2214yes—
81#1441C2214yes—
82#1440C2214yes—
83#1440C2214yes—
84#1439C2214yes—
85#1436C2214yes—
86#1436C2214yes—
87C4⋊#95C2212yes—
88C4⋊#95C2212yes—
89C4⋊#95C2212yes—
90#1434C2214yes—
91#1434C2212yes—
92#1434C2212yes—
93#1434C2212yes—
94C22⋊#95C2214yes—
95C23×Q8C2312yes—
96C23×Q8C2312yes—
97C24×C4C2311yescharacteristic
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×#126—24no—
105C2×#126—24no—
106C2×#126D418yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#400—28no—
139#400—28no—
140#399—28no—
141#399—28no—
142#398—28no—
143#398—28no—
144#397—28no—
145#397—28no—
146#395—28no—
147#395—28no—
148#393—28no—
149#393—28no—
150#391—28no—
151#391—28no—
152#391—28no—
153#391—28no—
154#388—24no—
155#388—24no—
156#388—24no—
157#387—24no—
158#387—24no—
159#387—24no—
160#383—24no—
161#383—24no—
162#383—24no—
163#382—28no—
164#382—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4—24no—
168C4×C22⋊C4—24no—
169C4×C22⋊C4—24no—
170C4×C22⋊C4—24no—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C43D418yes—
209C25C2411yescharacteristic
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×Q8—44no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C4⋊Q8—416no—
243C4⋊Q8—416no—
244#126—232no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
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—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42#4114yes—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24C22×D412yes—
369C24C2511yescharacteristic
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—88no—
377C2×Q8—88no—
378C2×Q8—88no—
379C2×Q8—88no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—88no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C42−1+414yes—
473C22×C42−1+414yes—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—28no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—28no—
496C42—28no—
497C42—44no—
498C42—44no—
499C42—44no—
500C42—44no—
501C42—44no—
502C42—44no—
503C42—44no—
504C42#121116yes—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C23×D411yescharacteristic
523C23C23×D411yescharacteristic
524C23C22×#4111yescharacteristic
525C23C2×2−1+412yes—
526C23C2×2−1+412yes—
527C23#58512yes—
528C23#58512yes—
529C23#58512yes—
530C23#58512yes—
531Q8—816no—
532Q8—816no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4C2×#12114yes—
620C2×C4C2×#12114yes—
621C2×C4C2×#12114yes—
622C2×C4#53614yes—
623C2×C4#53614yes—
624C22—48no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—84no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C22×#12111yescharacteristic
634C22C2×#53611yescharacteristic
635C22C2×#53611yescharacteristic
636C22#344911yescharacteristic
637C22#344911yescharacteristic
638C22#344911yescharacteristic
639C22#344911yescharacteristic
640C22#349912yes—
641C22#350012yes—
642C22#350012yes—
643C22#350912yes—
644C22#351012yes—
645C22#351012yes—
646C22#353114yes—
647C22#355714yes—
648C22#356414yes—
649C22#356414yes—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—24no—
655C4—42no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—84no—
661C4—84no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—84no—
667C4—84no—
668C2—82no—
669C2#3531611yescharacteristic
670C2#3532711yescharacteristic
671C2#3532711yescharacteristic
672C2#3669911yescharacteristic
673C2#3754811yescharacteristic
674C2#3755111yescharacteristic
675C2#3755711yescharacteristic
676C2#4614614yes—
677C2#4615614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2862C2semidirect
5. #16806C2semidirect
7. #16804C2semidirect
9. #16705C2semidirect
11. #15164C2semidirect
13. #15097C2semidirect
14. #15085C2semidirect
17. #14688C2semidirect
18. #14663C2semidirect
19. #14663C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#28622
C2×C4×#1262
#168062
#168052
#168042
#168032
#167052
#167012
#151642
#151612
#150972
#150852
#150502
#150162
#146882
#146632