Finite Group Databasebeta

← #4435295

Subgroups — #4435295

Order 512.

2633 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C211yesmaximal, characteristic, complemented
2C2×#2503C211yesmaximal, characteristic, complemented
3C2×#2502C211yesmaximal, characteristic, complemented
4C2×#2623C211yesmaximal, characteristic
5C2×#2663C211yesmaximal, characteristic
6C2×#2663C211yesmaximal, characteristic
7#17064C212yesmaximal, complemented
8#16964C212yesmaximal, complemented
9#16813C212yesmaximal, complemented
10#16812C212yesmaximal, complemented
11#15549C214yesmaximal
12#15539C214yesmaximal, complemented
13#15173C212yesmaximal, complemented
14#15153C212yesmaximal, complemented
15#15065C212yesmaximal, complemented
16#15020C212yesmaximal, complemented
17#13807C211yesmaximal, characteristic, complemented
18C22×#123C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2817—28no—
23#2663—28no—
24#2663—28no—
25#2623—28no—
26#2503—28no—
27#2502—28no—
28C2×#400C2211yescharacteristic
29C2×#400C2211yescharacteristic
30C2×#400C2212yes—
31C2×#400C2212yes—
32C2×#400C2212yes—
33C2×#400C2212yes—
34C2×#400C2212yes—
35C2×#399C2211yescharacteristic
36C2×#399C2211yescharacteristic
37C2×#399C2212yes—
38C2×#398C2212yes—
39C2×#398C2212yes—
40C2×#398C2212yes—
41C2×#397C2212yes—
42C2×#397C2212yes—
43C2×#397C2212yes—
44C2×#397C2214yes—
45C2×#395C2211yescharacteristic
46C2×#395C2211yescharacteristic
47C2×#395C2212yes—
48C2×#395C2212yes—
49C2×#394C2212yes—
50C2×#394C2212yes—
51C2×#393C2211yescharacteristic
52C2×#393C2211yescharacteristic
53C2×#391C2212yes—
54C2×#391C2212yes—
55C2×#389C2212yes—
56C2×#389C2212yes—
57C2×#388C2212yes—
58C2×#388C2212yes—
59C2×#386C2212yes—
60C2×#385C2212yes—
61C2×#385C2214yes—
62C2×#384C2212yes—
63C2×#382C2212yes—
64C2×#381C2214yes—
65C2×#381C2211yescharacteristic
66C2×C4×C22⋊C4C2211yescharacteristic
67C2×C4×C22⋊C4C2211yescharacteristic
68#1445C2214yes—
69#1445C2214yes—
70#1445C2214yes—
71#1445C2214yes—
72#1445C2214yes—
73#1445C2214yes—
74#1444C2214yes—
75#1444C2214yes—
76#1444C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1437C2214yes—
81#1437C2214yes—
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×D4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2314yes—
136C22×C42C2312yes—
137C22×C42C2312yes—
138#400—28no—
139#400—28no—
140#400—28no—
141#400—24no—
142#400—24no—
143#400—28no—
144#400—28no—
145#399—28no—
146#399—24no—
147#399—24no—
148#398—28no—
149#398—28no—
150#398—28no—
151#397—216no—
152#397—28no—
153#397—28no—
154#397—28no—
155#395—28no—
156#395—28no—
157#395—24no—
158#395—24no—
159#394—28no—
160#394—28no—
161#393—24no—
162#393—24no—
163#391—28no—
164#391—28no—
165#389—28no—
166#389—28no—
167#388—28no—
168#388—28no—
169#386—28no—
170#385—28no—
171#385—216no—
172#384—28no—
173#382—28no—
174#381—216no—
175#381D418yes—
176C4×C22⋊C4D418yes—
177C4×C22⋊C4D418yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202C2×#95C2312yes—
203C2×#95C2311yescharacteristic
204C2×#95C2312yes—
205C2×#95C2311yescharacteristic
206C25C2411yescharacteristic
207C25C2411yescharacteristic
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—22no—
214C22×Q8—44no—
215C22×Q8—44no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—44no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244#123—416no—
245C22⋊Q8—416no—
246C22⋊Q8—416no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
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—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—416no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
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⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
328C2×C22⋊C4C2×D418yes—
329C2×C22⋊C4C2×D418yes—
330C2×C42—28no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42—44no—
334C2×C42—44no—
335C2×C42C2×D418yes—
336C2×C42C2×D418yes—
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—24no—
346#95—24no—
347#95—24no—
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—44no—
358#95—44no—
359#95C2×D414yes—
360#95C2×D414yes—
361#95C2×D414yes—
362#95C2×D414yes—
363#95C2×D418yes—
364#95C2×D418yes—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—21no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—24no—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C2511yescharacteristic
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×Q8—84no—
389C2×Q8—84no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—84no—
393C2×D4—84no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
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—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
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—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4#127116yes—
470C22×C4C22×D414yes—
471C22×C4C22×D418yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D414yes—
481C22×C42−1+414yes—
482C22×C42−1+414yes—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—84no—
494C4⋊C4—88no—
495C4⋊C4—84no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C42—48no—
513C42—48no—
514C42—88no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—41no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23C2×#12714yes—
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C2×2−1+412yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553C23#58512yes—
554C23#58514yes—
555Q8—88no—
556Q8—88no—
557D4—88no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—816no—
637C2×C4C2×#12714yes—
638C2×C4C2×#12714yes—
639C2×C4#53618yes—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—81no—
649C22—82no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22—81no—
656C22C22×#12711yescharacteristic
657C22C2×#53612yes—
658C22#344912yes—
659C22#348512yes—
660C22#348512yes—
661C22#348512yes—
662C22#349911yescharacteristic
663C22#349911yescharacteristic
664C22#349912yes—
665C22#350011yescharacteristic
666C22#350011yescharacteristic
667C22#350012yes—
668C22#355314yes—
669C22#355714yes—
670C22#356414yes—
671C22#356514yes—
672C4—24no—
673C4—24no—
674C4—28no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—48no—
679C4—81no—
680C4—82no—
681C4—84no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—81no—
686C4—82no—
687C2—44no—
688C2—161no—
689C2#3531711yescharacteristic
690C2#3532911yescharacteristic
691C2#3546311yescharacteristic
692C2#4137412yes—
693C2#4138312yes—
694C2#4231012yes—
695C2#4264712yes—
696C2#5086914yes—
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
1. C2×#2817C2semidirect
2. C2×#2503C2semidirect
3. C2×#2502C2semidirect
7. #17064C2semidirect
8. #16964C2semidirect
9. #16813C2semidirect
10. #16812C2semidirect
12. #15539C2semidirect
13. #15173C2semidirect
14. #15153C2semidirect
15. #15065C2semidirect
16. #15020C2semidirect
17. #13807C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#25032
C2×#25022
C2×#26232
C2×#26632
#170642
#169642
#168132
#168122
#155492
#155392
#151732
#151532
#150652
#150202
#138072