Finite Group Databasebeta

← #626420

Subgroups — #626420

Order 512.

3026 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×#2566C211yesmaximal, characteristic
5C2×#2642C211yesmaximal, characteristic, complemented
6C2×#2326C211yesmaximal, characteristic
7#15808C218yesmaximal, complemented
8#15513C218yesmaximal, complemented
9#15243C211yesmaximal, characteristic
10#15243C211yesmaximal, characteristic
11#15240C211yesmaximal, characteristic, complemented
12#15199C211yesmaximal, characteristic, complemented
13#14714C211yesmaximal, characteristic, complemented
14#14708C211yesmaximal, characteristic
15#14274C211yesmaximal, characteristic, complemented
16#14273C211yesmaximal, characteristic
17#13759C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#124C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C22×#124C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27#2642—28no—
28#2566—28no—
29#2565—28no—
30#2547—28no—
31#2546—28no—
32C2×#400C2214yes—
33C2×#400C2214yes—
34C2×#398C2214yes—
35C2×#398C2214yes—
36C2×#397C2214yes—
37C2×#396C2214yes—
38#2326—28no—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#389C2214yes—
54C2×#388C2214yes—
55C2×#387C2214yes—
56C2×#385C2214yes—
57C2×#385C2214yes—
58C2×#384C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#380C2214yes—
64C2×C4×C4⋊C4C2211yescharacteristic
65C2×C4×C4⋊C4C2211yescharacteristic
66C2×#376C2211yescharacteristic
67C2×#376C2211yescharacteristic
68C2×C43C2211yescharacteristic
69#1445C2218yes—
70#1445C2218yes—
71#1444C2218yes—
72#1439C2218yes—
73#1439C2218yes—
74#1439C2218yes—
75#1437C2218yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2218yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—42no—
94C2×C4⋊Q8—42no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×#124—24no—
99C2×#124—24no—
100C2×#124—24no—
101C2×#124—24no—
102C2×#124—24no—
103C2×#124—24no—
104C2×#124—24no—
105C2×#124—42no—
106C2×#124—42no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2314yes—
144C22×C42C2311yescharacteristic
145#400—216no—
146#400—216no—
147#398—216no—
148#398—216no—
149#397—216no—
150#396—216no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—24no—
156#391—24no—
157#391—24no—
158#391—24no—
159#391—24no—
160#391—24no—
161#391—24no—
162#390—24no—
163#390—24no—
164#390—24no—
165#389—216no—
166#388—216no—
167#387—216no—
168#385—216no—
169#385—216no—
170#384—24no—
171#384—24no—
172#384—24no—
173#382—24no—
174#382—24no—
175#380—216no—
176C4×C4⋊C4D418yes—
177C4×C4⋊C4D418yes—
178#376—42no—
179#376D418yes—
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×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C43—42no—
201C25C2412yes—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8#4114yes—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—22no—
227C22×D4—22no—
228C22×D4—22no—
229C22×D4#4114yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—48no—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C4⋊Q8—416no—
251C4⋊Q8—416no—
252C4⋊Q8—416no—
253#125—416no—
254#124—232no—
255#124—416no—
256#124—416no—
257C4×Q8—416no—
258C4×D4—416no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
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—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—28no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—48no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—48no—
300C2×C4⋊C4C2×D418yes—
301C2×C4⋊C4C2×D418yes—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—216no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—48no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—44no—
362#95—44no—
363#95C2×D418yes—
364#95C2×D418yes—
365#95C2×D418yes—
366#95C2×D418yes—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—28no—
373C24—22no—
374C24—28no—
375C24C2×#4114yes—
376C24C2511yescharacteristic
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—44no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—44no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
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—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+418yes—
490C22×C42−1+412yes—
491C4⋊C4—216no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C22⋊C4—232no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—48no—
511C22⋊C4—416no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—28no—
515C42—28no—
516C42—28no—
517C42—28no—
518C42—28no—
519C42—28no—
520C42—416no—
521C42—48no—
522C42#127116yes—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—216no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23C23×D411yescharacteristic
541C23C23×D411yescharacteristic
542C23C23×D411yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+414yes—
547C23C2×2−1+411yescharacteristic
548C23#58514yes—
549Q8—416no—
550Q8—88no—
551Q8—88no—
552D4—416no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
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—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
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—44no—
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—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
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×C4#52514yes—
627C2×C4C2×#12714yes—
628C2×C4C2×#12714yes—
629C2×C4C2×#12714yes—
630C2×C4#53618yes—
631C2×C4#53714yes—
632C2×C4#54614yes—
633C2×C4#54614yes—
634C2×C4#54814yes—
635C2×C4#54814yes—
636C22—48no—
637C22—48no—
638C22—44no—
639C22—48no—
640C22—82no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—84no—
645C22C2×#52511yescharacteristic
646C22C22×#12711yescharacteristic
647C22C2×#53612yes—
648C22C2×#53711yescharacteristic
649C22#344912yes—
650C22C2×#54611yescharacteristic
651C22C2×#54611yescharacteristic
652C22C2×#54811yescharacteristic
653C22C2×#54811yescharacteristic
654C22#349912yes—
655C22#349912yes—
656C22#350012yes—
657C22#350012yes—
658C22#351514yes—
659C22#351614yes—
660C22#354114yes—
661C22#355514yes—
662C4—24no—
663C4—28no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4#254018yes—
678C2—84no—
679C2C2×#254011yescharacteristic
680C2#3531711yescharacteristic
681C2#3532911yescharacteristic
682C2#3562212yes—
683C2#3738912yes—
684C2#3808512yes—
685C2#3808812yes—
686C2#3916212yes—
687C2#3917312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2546C2semidirect
3. C2×#2565C2semidirect
5. C2×#2642C2semidirect
7. #15808C2semidirect
8. #15513C2semidirect
11. #15240C2semidirect
12. #15199C2semidirect
13. #14714C2semidirect
15. #14274C2semidirect
17. #13759C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25462
C2×#25652
C2×#25662
C2×#26422
C2×#23262
#158082
#155132
#152432
#152402
#151992
#147142
#147082
#142742
#142732
#137592