Finite Group Databasebeta

← #4380498

Subgroups — #4380498

Order 512.

3362 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2599C211yesmaximal, characteristic
4C2×#2634C211yesmaximal, characteristic, complemented
5#16683C212yesmaximal, complemented
6#16560C212yesmaximal, complemented
7#16093C214yesmaximal, complemented
8#15985C214yesmaximal
9#15881C212yesmaximal, complemented
10#15849C212yesmaximal, complemented
11#15671C212yesmaximal
12#15670C212yesmaximal
13#14906C212yesmaximal
14#14843C212yesmaximal
15#14663C211yesmaximal, characteristic, complemented
16#14289C211yesmaximal, characteristic, complemented
17#13938C211yesmaximal, characteristic
18C2×#544—28no—
19C2×#544—28no—
20C2×#544—28no—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2211yescharacteristic
24C22×#125C2212yes—
25C22×#125C2211yescharacteristic
26C22×#125C2211yescharacteristic
27C22×C22⋊Q8C2212yes—
28C22×C22⋊Q8C2212yes—
29C22×C4×Q8C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33#2634—28no—
34#2599—28no—
35#2568—28no—
36C2×#400C2212yes—
37C2×#400C2212yes—
38C2×#400C2212yes—
39C2×#400C2212yes—
40C2×#398C2211yescharacteristic
41C2×#398C2212yes—
42C2×#397C2211yescharacteristic
43C2×#395C2212yes—
44C2×#395C2211yescharacteristic
45C2×#393C2211yescharacteristic
46C2×#391C2212yes—
47C2×#391C2214yes—
48C2×#388C2212yes—
49C2×#387C2214yes—
50C2×#387C2212yes—
51C2×#387C2212yes—
52C2×#385C2212yes—
53C2×#383C2211yescharacteristic
54C2×#383C2211yescharacteristic
55C2×#382C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57#1446C2214yes—
58#1446C2214yes—
59#1446C2214yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2212yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1437C2212yes—
80#1436C2212yes—
81#1436C2214yes—
82#1436C2212yes—
83#1436C2212yes—
84#1436C2214yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C22⋊#95C2214yes—
88C22⋊#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C23×Q8C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94#544—2128no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8D418yes—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—24no—
107C2×#125—28no—
108C2×#125—24no—
109C2×#125—24no—
110C2×#125—24no—
111C2×#125—24no—
112C2×#125D418yes—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8—28no—
115C2×C22⋊Q8—28no—
116C2×C22⋊Q8—28no—
117C2×C22⋊Q8—28no—
118C2×C22⋊Q8—28no—
119C2×C4×Q8—24no—
120C2×C4×Q8—24no—
121C2×C4×Q8—24no—
122C2×#117—24no—
123C2×#117—24no—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117—24no—
127C2×#117D418yes—
128C22×C4⋊C4—24no—
129C22×C4⋊C4—24no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2314yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C22⋊C4C2312yes—
151C22×C22⋊C4C2312yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2311yescharacteristic
154C22×C22⋊C4C2311yescharacteristic
155C22×C22⋊C4C2314yes—
156C22×C42C2312yes—
157C22×C42C2312yes—
158C22×C42C2311yescharacteristic
159C22×C42C2312yes—
160C22×C42C2311yescharacteristic
161#400—28no—
162#400—28no—
163#400—28no—
164#400—28no—
165#398—24no—
166#398—28no—
167#397—24no—
168#395—24no—
169#395—28no—
170#393—24no—
171#391—216no—
172#391—28no—
173#388—28no—
174#387—28no—
175#387—216no—
176#387—28no—
177#385—28no—
178#383—24no—
179#383—24no—
180#382—28no—
181C4×C4⋊C4—24no—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C25C2411yescharacteristic
213C22×Q8—22no—
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8—44no—
224C22×Q8—44no—
225C22×Q8—44no—
226C22×Q8—44no—
227C22×Q8C2×D418yes—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4C2×D414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C4⋊Q8—264no—
255C4⋊Q8—232no—
256C4⋊Q8—232no—
257#125—264no—
258#125—232no—
259#125—232no—
260C22⋊Q8—264no—
261C22⋊Q8—264no—
262C4×Q8—416no—
263#117—232no—
264#117—232no—
265C2×C4⋊C4—232no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2×D414yes—
316C2×C4⋊C4C2×D414yes—
317C2×C4⋊C4C2×D414yes—
318C2×C4⋊C4C2×D418yes—
319C2×C4⋊C4C2×D418yes—
320C2×C4⋊C4C2×D418yes—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4C2×D414yes—
337C2×C22⋊C4C2×D414yes—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—44no—
348C2×C42C2×D414yes—
349C2×C42C2×D414yes—
350C2×C42C2×D414yes—
351C2×C42C2×D414yes—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—44no—
378#95—44no—
379#95—44no—
380#95—44no—
381#95—44no—
382C24—21no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—24no—
388C24C22×D412yes—
389C242−1+414yes—
390C24C2511yescharacteristic
391C2×Q8—28no—
392C2×Q8—216no—
393C2×Q8—216no—
394C2×Q8—416no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—88no—
402C2×Q8—88no—
403C2×Q8—84no—
404C2×Q8—84no—
405C2×Q8—88no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—216no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—88no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—88no—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C42−1+414yes—
499C22×C42−1+414yes—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—48no—
514C4⋊C4—416no—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C22⋊C4—216no—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—88no—
525C42—216no—
526C42—28no—
527C42—48no—
528C42—48no—
529C42#120116yes—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23#53614yes—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C2×2−1+412yes—
550C23C2×2−1+412yes—
551C23C2×2−1+412yes—
552C23#58514yes—
553C23#58512yes—
554Q8—416no—
555Q8—816no—
556Q8—816no—
557C2×C4—232no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
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—816no—
639C2×C4C2×#12014yes—
640C2×C4C2×#12014yes—
641C2×C4C2×#12014yes—
642C2×C4#53614yes—
643C22—28no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—44no—
649C22—48no—
650C22—44no—
651C22C22×#12011yescharacteristic
652C22C2×#53611yescharacteristic
653C22C2×#53611yescharacteristic
654C22#348512yes—
655C22#348611yescharacteristic
656C22#348612yes—
657C22#348612yes—
658C22#348711yescharacteristic
659C22#348912yes—
660C22#348912yes—
661C22#350012yes—
662C22#350012yes—
663C22#355514yes—
664C22#356314yes—
665C22#356414yes—
666C22#356514yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—42no—
677C4—44no—
678C4—84no—
679C4—84no—
680C4—82no—
681C4—82no—
682C4—82no—
683C2—44no—
684C2#3524211yescharacteristic
685C2#3561611yescharacteristic
686C2#3565911yescharacteristic
687C2#4215012yes—
688C2#4291512yes—
689C2#4776914yes—
690C2#5205512yes—
691C2#5205612yes—
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
2. C2×#2568C2semidirect
4. C2×#2634C2semidirect
5. #16683C2semidirect
6. #16560C2semidirect
7. #16093C2semidirect
9. #15881C2semidirect
10. #15849C2semidirect
15. #14663C2semidirect
16. #14289C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#25682
C2×#25992
C2×#26342
#166832
#165602
#160932
#159852
#158812
#158492
#156712
#156702
#149062
#148432
#146632
#142892
#139382