Finite Group Databasebeta

← #625869

Subgroups — #625869

Order 512.

3004 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic, complemented
3C2×#2504C211yesmaximal, characteristic
4C2×#2503C211yesmaximal, characteristic, complemented
5C2×#2620C211yesmaximal, characteristic
6C2×#2317C211yesmaximal, characteristic, complemented
7#15808C218yesmaximal, complemented
8#15549C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic, complemented
10#15244C211yesmaximal, characteristic, complemented
11#15240C211yesmaximal, characteristic, complemented
12#15240C211yesmaximal, characteristic, complemented
13#14708C211yesmaximal, characteristic, complemented
14#14687C211yesmaximal, characteristic, complemented
15#14354C211yesmaximal, characteristic, complemented
16#14273C211yesmaximal, characteristic, complemented
17#13761C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2620—28no—
25#2504—28no—
26#2503—28no—
27#2500—28no—
28#2499—28no—
29C2×#400C2214yes—
30C2×#397C2214yes—
31C2×#397C2214yes—
32C2×#396C2214yes—
33C2×#394C2214yes—
34C2×#394C2214yes—
35#2317—28no—
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#389C2214yes—
47C2×#388C2214yes—
48C2×#387C2214yes—
49C2×#386C2214yes—
50C2×#385C2214yes—
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×#382C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#381C2214yes—
65C2×C4×C22⋊C4C2211yescharacteristic
66C2×C4×C22⋊C4C2211yescharacteristic
67C2×#376C2211yescharacteristic
68C2×C43C2211yescharacteristic
69#1445C2218yes—
70#1445C2218yes—
71#1444C2218yes—
72#1439C2218yes—
73#1439C2218yes—
74#1439C2218yes—
75#1437C2218yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2218yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117D418yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138#400—216no—
139#397—216no—
140#397—216no—
141#396—216no—
142#394—216no—
143#394—216no—
144#391—24no—
145#391—24no—
146#391—24no—
147#391—24no—
148#391—24no—
149#390—24no—
150#390—24no—
151#390—24no—
152#390—24no—
153#390—24no—
154#389—216no—
155#388—216no—
156#387—216no—
157#386—216no—
158#385—216no—
159#384—24no—
160#384—24no—
161#384—24no—
162#384—24no—
163#384—24no—
164#382—24no—
165#382—24no—
166#382—24no—
167#382—24no—
168#382—24no—
169#382—24no—
170#382—24no—
171#382—24no—
172#381—216no—
173C4×C22⋊C4D418yes—
174C4×C22⋊C4D418yes—
175#376—24no—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C43—24no—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—48no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—48no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241#124—416no—
242C22⋊Q8—416no—
243#121—416no—
244#117—232no—
245#117—416no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
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—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—416no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
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—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42#4114yes—
336C2×C42#4114yes—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—44no—
352#95—44no—
353#95C2×D418yes—
354#95C2×D418yes—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—28no—
369C24—28no—
370C24C22×D412yes—
371C24C22×D412yes—
372C24C22×D412yes—
373C24C2511yescharacteristic
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—88no—
391C2×D4—88no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
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—24no—
423C22×C4—24no—
424C22×C4—22no—
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—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4#127116yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C4⋊C4—216no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—28no—
506C42—28no—
507C42—28no—
508C42—28no—
509C42—416no—
510C42—48no—
511C42#123116yes—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—48no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—48no—
543C23—48no—
544C23—48no—
545C23C2×#12714yes—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4111yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+411yescharacteristic
553C23#58514yes—
554C23#58514yes—
555Q8—88no—
556D4—816no—
557D4—88no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—232no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—816no—
626C2×C4C2×#12318yes—
627C2×C4C2×#12314yes—
628C2×C4C2×#12714yes—
629C2×C4C2×#12714yes—
630C2×C4#53614yes—
631C2×C4#54714yes—
632C2×C4#54714yes—
633C2×C4#54714yes—
634C2×C4#54714yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22C22×#12311yescharacteristic
654C22C22×#12711yescharacteristic
655C22C2×#53611yescharacteristic
656C22#344912yes—
657C22#344912yes—
658C22C2×#54711yescharacteristic
659C22C2×#54711yescharacteristic
660C22C2×#54711yescharacteristic
661C22C2×#54711yescharacteristic
662C22#349314yes—
663C22#350012yes—
664C22#350012yes—
665C22#350012yes—
666C22#350012yes—
667C22#351214yes—
668C22#356514yes—
669C22#356514yes—
670C4—28no—
671C4—24no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4#250118yes—
685C2—44no—
686C2—162no—
687C2C2×#250111yescharacteristic
688C2#3532911yescharacteristic
689C2#3532911yescharacteristic
690C2#3561812yes—
691C2#3742312yes—
692C2#3787012yes—
693C2#3787112yes—
694C2#3948512yes—
695C2#3948512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2499C2semidirect
4. C2×#2503C2semidirect
6. C2×#2317C2semidirect
7. #15808C2semidirect
8. #15549C2semidirect
9. #15244C2semidirect
10. #15244C2semidirect
11. #15240C2semidirect
12. #15240C2semidirect
13. #14708C2semidirect
14. #14687C2semidirect
15. #14354C2semidirect
16. #14273C2semidirect
17. #13761C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#24992
C2×#25042
C2×#25032
C2×#26202
C2×#23172
#158082
#155492
#152442
#152402
#147082
#146872
#143542
#142732
#137612