Finite Group Databasebeta

← #4421458

Subgroups — #4421458

Order 512.

3348 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic, complemented
2C2×#2599C211yesmaximal, characteristic
3C2×#2599C211yesmaximal, characteristic
4C2×#2628C211yesmaximal, characteristic
5#16683C212yesmaximal, complemented
6#16682C212yesmaximal
7#16603C212yesmaximal, complemented
8#16602C212yesmaximal
9#16104C214yesmaximal, complemented
10#15996C214yesmaximal
11#14882C212yesmaximal
12#14879C212yesmaximal, complemented
13#14876C212yesmaximal, complemented
14#14875C212yesmaximal
15#14663C211yesmaximal, characteristic, complemented
16#14346C211yesmaximal, characteristic, complemented
17#14346C211yesmaximal, characteristic, complemented
18C2×#557—28no—
19C2×#557—28no—
20C2×#557—28no—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2212yes—
24C22×#125C2214yes—
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2212yes—
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×#117C2211yescharacteristic
30#2628—28no—
31#2599—28no—
32#2599—28no—
33C2×#400C2212yes—
34C2×#399C2212yes—
35C2×#398C2211yescharacteristic
36C2×#398C2211yescharacteristic
37C2×#397C2211yescharacteristic
38C2×#397C2211yescharacteristic
39C2×#395C2211yescharacteristic
40C2×#395C2212yes—
41C2×#395C2211yescharacteristic
42C2×#393C2211yescharacteristic
43C2×#393C2212yes—
44C2×#393C2211yescharacteristic
45C2×#389C2212yes—
46C2×#388C2212yes—
47C2×#388C2212yes—
48C2×#387C2212yes—
49C2×#387C2212yes—
50C2×#387C2212yes—
51C2×#387C2212yes—
52C2×#385C2212yes—
53C2×#385C2212yes—
54C2×#383C2211yescharacteristic
55C2×#383C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#380C2212yes—
58#1446C2212yes—
59#1446C2212yes—
60#1446C2212yes—
61#1446C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1445C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1442C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87#1434C2212yes—
88#1434C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C23×Q8C2311yescharacteristic
92C24×C4C2311yescharacteristic
93#557—2128no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8D418yes—
100C2×#125—216no—
101C2×#125—28no—
102C2×#125—216no—
103C2×#125—216no—
104C2×#125—28no—
105C2×#125—28no—
106C2×C4×Q8—28no—
107C2×C4×Q8—28no—
108C2×C4×Q8—28no—
109C2×C4×Q8—28no—
110C2×C4×Q8—28no—
111C2×C4×Q8—28no—
112C2×C4×Q8—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—24no—
116C2×C4×Q8D418yes—
117C2×C4×Q8D418yes—
118C2×#117—24no—
119C2×#117—24no—
120C2×#117—24no—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2311yescharacteristic
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2312yes—
145C22×C42C2311yescharacteristic
146C22×C42C2314yes—
147C22×C42C2312yes—
148C22×C42C2312yes—
149C22×C42C2311yescharacteristic
150#400—28no—
151#399—28no—
152#398—24no—
153#398—24no—
154#397—24no—
155#397—24no—
156#395—24no—
157#395—24no—
158#395—28no—
159#393—24no—
160#393—24no—
161#393—28no—
162#389—28no—
163#388—28no—
164#388—28no—
165#387—28no—
166#387—28no—
167#387—28no—
168#387—28no—
169#385—28no—
170#385—28no—
171#383—24no—
172#383—24no—
173#382—24no—
174#380—28no—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C25C2411yescharacteristic
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—22no—
214C22×Q8—28no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8—44no—
218C22×Q8C2×D414yes—
219C22×Q8C2×D418yes—
220C22×Q8C2×D414yes—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—41no—
227C23×C4—41no—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C4⋊Q8—264no—
245C4⋊Q8—232no—
246#125—2128no—
247#125—264no—
248C4×Q8—264no—
249C4×Q8—264no—
250C4×Q8—232no—
251C4×Q8—232no—
252#117—416no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
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—24no—
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—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2×D418yes—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×D418yes—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
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—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—28no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D418yes—
332C2×C42C2×D418yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—24no—
372C24—21no—
373C24—24no—
374C24—21no—
375C24—21no—
376C242−1+412yes—
377C24C2511yescharacteristic
378C2×Q8—216no—
379C2×Q8—216no—
380C2×Q8—28no—
381C2×Q8—216no—
382C2×Q8—28no—
383C2×Q8—216no—
384C2×Q8—48no—
385C2×Q8—416no—
386C2×Q8—416no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×Q8—48no—
390C2×Q8—48no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
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×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C42−1+412yes—
485C22×C42−1+418yes—
486C4⋊C4—232no—
487C4⋊C4—232no—
488C4⋊C4—232no—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—216no—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—88no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—88no—
510C42—232no—
511C42—216no—
512C42—216no—
513C42—44no—
514C42#120116yes—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—28no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23C23×D411yescharacteristic
533C23C23×D411yescharacteristic
534C23C23×D411yescharacteristic
535C23C2×2−1+414yes—
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23#58514yes—
539C23#58511yescharacteristic
540C23#58511yescharacteristic
541Q8—232no—
542Q8—416no—
543Q8—432no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—416no—
576C2×C4—41no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
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—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4C2×#12014yes—
630C2×C4C2×#12014yes—
631C2×C4C2×#12014yes—
632C2×C4#53618yes—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—81no—
638C22—82no—
639C22—82no—
640C22—81no—
641C22—82no—
642C22C22×#12011yescharacteristic
643C22C2×#53612yes—
644C22#344912yes—
645C22#348512yes—
646C22#348611yescharacteristic
647C22#348612yes—
648C22#348612yes—
649C22#348711yescharacteristic
650C22#348712yes—
651C22#348911yescharacteristic
652C22#348911yescharacteristic
653C22#348911yescharacteristic
654C22#348911yescharacteristic
655C22#355514yes—
656C22#355714yes—
657C22#356314yes—
658C22#356514yes—
659C4—24no—
660C4—24no—
661C4—24no—
662C4—28no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—48no—
668C4—81no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—81no—
673C4—82no—
674C4—82no—
675C2—82no—
676C2#3514911yescharacteristic
677C2#3524211yescharacteristic
678C2#3565911yescharacteristic
679C2#4217112yes—
680C2#4217412yes—
681C2#4288412yes—
682C2#4288812yes—
683C2#5205512yes—
684C2#5205812yes—
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
1. C22×#557C2semidirect
5. #16683C2semidirect
7. #16603C2semidirect
9. #16104C2semidirect
12. #14879C2semidirect
13. #14876C2semidirect
15. #14663C2semidirect
16. #14346C2semidirect
17. #14346C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#25992
C2×#26282
#166832
#166822
#166032
#166022
#161042
#159962
#148822
#148792
#148762
#148752
#146632
#143462