Finite Group Databasebeta

← #4435578

Subgroups — #4435578

Order 512.

2709 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2821C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic, complemented
3C2×#2603C211yesmaximal, characteristic
4C2×#2642C211yesmaximal, characteristic, complemented
5C2×#2664C211yesmaximal, characteristic, complemented
6C2×#2633C211yesmaximal, characteristic, complemented
7#16964C212yesmaximal, complemented
8#16814C212yesmaximal, complemented
9#16812C212yesmaximal, complemented
10#16809C212yesmaximal
11#15526C214yesmaximal, complemented
12#15513C214yesmaximal, complemented
13#15170C212yesmaximal, complemented
14#15167C212yesmaximal, complemented
15#15065C212yesmaximal, complemented
16#15020C212yesmaximal, complemented
17#13806C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2821—28no—
23#2664—28no—
24#2642—28no—
25#2633—28no—
26#2603—28no—
27#2563—28no—
28C2×#400C2211yescharacteristic
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#400C2212yes—
32C2×#400C2214yes—
33C2×#399C2212yes—
34C2×#399C2211yescharacteristic
35C2×#399C2212yes—
36C2×#398C2212yes—
37C2×#398C2212yes—
38C2×#398C2211yescharacteristic
39C2×#398C2211yescharacteristic
40C2×#398C2212yes—
41C2×#397C2212yes—
42C2×#397C2211yescharacteristic
43C2×#397C2211yescharacteristic
44C2×#397C2212yes—
45C2×#396C2212yes—
46C2×#395C2212yes—
47C2×#394C2212yes—
48C2×#394C2211yescharacteristic
49C2×#393C2212yes—
50C2×#393C2212yes—
51C2×#393C2212yes—
52C2×#392C2211yescharacteristic
53C2×#391C2212yes—
54C2×#391C2212yes—
55C2×#390C2212yes—
56C2×#390C2212yes—
57C2×#389C2212yes—
58C2×#388C2212yes—
59C2×#388C2212yes—
60C2×#386C2212yes—
61C2×#385C2214yes—
62C2×#385C2212yes—
63C2×#383C2211yescharacteristic
64C2×#382C2212yes—
65C2×#380C2214yes—
66C2×C4×C4⋊C4C2211yescharacteristic
67C2×#376C2211yescharacteristic
68#1445C2214yes—
69#1445C2214yes—
70#1445C2214yes—
71#1445C2214yes—
72#1444C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1438C2214yes—
79#1438C2214yes—
80#1437C2214yes—
81#1437C2214yes—
82#1437C2214yes—
83#1436C2214yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C23×Q8C2312yes—
90C23×D4C2311yescharacteristic
91C23×D4C2312yes—
92C24×C4C2312yes—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×#126—24no—
97C2×#126—24no—
98C2×#126—24no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2314yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#400—28no—
141#400—216no—
142#400—28no—
143#400—28no—
144#400—24no—
145#399—28no—
146#399—28no—
147#399—24no—
148#398—28no—
149#398—28no—
150#398—24no—
151#398—28no—
152#398—24no—
153#397—28no—
154#397—24no—
155#397—28no—
156#397—24no—
157#396—28no—
158#395—28no—
159#394—24no—
160#394—28no—
161#393—28no—
162#393—28no—
163#393—28no—
164#392—24no—
165#391—28no—
166#391—28no—
167#390—28no—
168#390—28no—
169#389—28no—
170#388—28no—
171#388—28no—
172#386—28no—
173#385—28no—
174#385—216no—
175#383D418yescomplemented
176#382—28no—
177#380—216no—
178C4×C4⋊C4D418yescomplemented
179#376D418yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C2×#95C2312yes—
202C2×#95C2311yescharacteristic
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C25C2411yescharacteristic
207C25C2412yes—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—24no—
219C22×D4—42no—
220C22×D4—42no—
221C22×D4—44no—
222C22×D4—44no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C4⋊Q8—416no—
244#126—416no—
245#125—416no—
246C4×D4—416no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4C2×D418yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42—44no—
339C2×C42C2×D418yes—
340C2×C42C2×D418yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D414yes—
364#95C2×D414yes—
365#95C2×D414yes—
366#95C2×D414yes—
367#95C2×D418yes—
368#95C2×D418yes—
369#95C2×D418yes—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—22no—
387C24—22no—
388C242−1+414yes—
389C24C2511yescharacteristic
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—48no—
393C2×Q8—88no—
394C2×Q8—88no—
395C2×D4—28no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—44no—
402C2×D4—84no—
403C2×D4—88no—
404C2×D4—88no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—416no—
471C22×C4—48no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—84no—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D414yes—
478C22×C4C22×D418yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C4⋊C4—84no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C42—48no—
521C42—48no—
522C42—88no—
523C42#127116yes—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—42no—
553C23—42no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C23×D411yescharacteristic
566C23C2×2−1+412yes—
567C23#58512yes—
568C23#58514yes—
569C23#58512yes—
570C23#58512yes—
571Q8—816no—
572D4—416no—
573D4—816no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
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—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—816no—
651C2×C4C2×#12714yes—
652C2×C4C2×#12714yes—
653C2×C4C2×#12714yes—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—44no—
658C22—48no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22C22×#12711yescharacteristic
674C22#344912yes—
675C22#344912yes—
676C22#348511yescharacteristic
677C22#348611yescharacteristic
678C22#348912yes—
679C22#348912yes—
680C22#350011yescharacteristic
681C22#350012yes—
682C22#350011yescharacteristic
683C22#350012yes—
684C22#350012yes—
685C22#355314yes—
686C22#356414yes—
687C22#356614yes—
688C22#356614yes—
689C4—24no—
690C4—24no—
691C4—24no—
692C4—44no—
693C4—44no—
694C4—44no—
695C4—44no—
696C4—48no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—82no—
701C4—84no—
702C4—81no—
703C4—82no—
704C2—84no—
705C2—161no—
706C2#3532911yescharacteristic
707C2#3532911yescharacteristic
708C2#3563411yescharacteristic
709C2#4138512yes—
710C2#4230412yes—
711C2#4238912yes—
712C2#4239112yes—
713C2#5087114yes—
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×#2821C2semidirect
2. C2×#2563C2semidirect
4. C2×#2642C2semidirect
5. C2×#2664C2semidirect
6. C2×#2633C2semidirect
7. #16964C2semidirect
8. #16814C2semidirect
9. #16812C2semidirect
11. #15526C2semidirect
12. #15513C2semidirect
13. #15170C2semidirect
14. #15167C2semidirect
15. #15065C2semidirect
16. #15020C2semidirect
17. #13806C2semidirect
175. #383D4semidirect
178. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#25632
C2×#26032
C2×#26422
C2×#26642
C2×#26332
#169642
#168142
#168122
#168092
#155262
#155132
#151702
#151672
#150652
#150202
#138062