Finite Group Databasebeta

← #760432

Subgroups — #760432

Order 512.

2754 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C4×C22⋊Q8C211yesmaximal, characteristic
4C2×#2283C211yesmaximal, characteristic
5#16394C212yesmaximal
6#16262C212yesmaximal
7#15179C212yesmaximal
8#15132C212yesmaximal
9#15076C214yesmaximal, complemented
10#14908C212yesmaximal, complemented
11#14901C212yesmaximal, complemented
12#14202C214yesmaximal
13#14150C211yesmaximal, characteristic, complemented
14#14045C212yesmaximal, complemented
15#14021C212yesmaximal, complemented
16#13942C211yesmaximal, characteristic, complemented
17#13925C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2581—28no—
24C2×#400C2212yes—
25C2×#397C2214yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C4×C22⋊Q8—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#387C2212yes—
38C2×#387C2214yes—
39C2×#387C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42#2283—28no—
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2212yes—
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2212yes—
58C2×#376C2212yes—
59#1446C2212yes—
60#1445C2212yes—
61#1444C2212yes—
62#1444C2214yes—
63#1443C2214yes—
64#1443C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1440C2214yes—
74#1440C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42—24no—
125C22×C42—24no—
126C22×C42D418yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400—28no—
137#397—216no—
138#395—28no—
139#391—28no—
140#391—28no—
141#391—24no—
142#391—24no—
143#389—28no—
144#389—28no—
145#389—24no—
146#389—24no—
147#387—28no—
148#387—28no—
149#387—216no—
150#385—28no—
151#385—28no—
152#383—28no—
153#382—24no—
154#382D418yes—
155#381—28no—
156#380—24no—
157#380—24no—
158#380—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—28no—
165C4×C22⋊C4—28no—
166C4×C22⋊C4—28no—
167#376—28no—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—42no—
210C22×Q8#4114yes—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C22⋊Q8—232no—
245C22⋊Q8—232no—
246C4×Q8—416no—
247C4×Q8—416no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4#4114yes—
293C2×C22⋊C4—24no—
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—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4#4118yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—232no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42#4114yes—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
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—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D418yes—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24C22×D412yes—
374C24C2×#4114yes—
375C24C2511yescharacteristic
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
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—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4C22×D412yes—
482C22×C4C22×D418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C42−1+414yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—216no—
508C4⋊C4—416no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—88no—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C42—28no—
525C42—28no—
526C42—28no—
527C42—416no—
528C42—48no—
529C42—48no—
530C42—48no—
531C42—48no—
532C42#121116yes—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23#52514yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4111yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4112yes—
562C23C2×2−1+411yescharacteristic
563C23C2×2−1+411yescharacteristic
564C23C2×2−1+412yes—
565C23#58512yes—
566C23#58512yes—
567Q8—416no—
568Q8—416no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—416no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4C2×#12114yes—
661C2×C4C2×#12114yes—
662C2×C4C2×#12114yes—
663C2×C4#52514yes—
664C2×C4#53614yes—
665C2×C4#53614yes—
666C2×C4#54714yes—
667C2×C4#54814yes—
668C2×C4#55214yes—
669C2×C4#55614yes—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—42no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—42no—
678C22—44no—
679C22—44no—
680C22—42no—
681C22C22×#12111yescharacteristic
682C22C2×#52511yescharacteristic
683C22C2×#52511yescharacteristic
684C22C2×#53611yescharacteristic
685C22C2×#53611yescharacteristic
686C22#344911yescharacteristic
687C22#344911yescharacteristic
688C22C2×#54711yescharacteristic
689C22C2×#54811yescharacteristic
690C22D4×#4112yes—
691C22C2×#55211yescharacteristic
692C22C2×#55611yescharacteristic
693C22#347512yes—
694C22#348212yes—
695C22#348212yes—
696C22#348912yes—
697C22#349512yes—
698C22#350712yes—
699C22#350712yes—
700C22#352112yes—
701C22#352412yes—
702C22#355512yes—
703C22#355512yes—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—44no—
708C4—44no—
709C4—42no—
710C4—44no—
711C4—42no—
712C4—44no—
713C4—48no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—84no—
718C4—82no—
719C4#241518yes—
720C2—44no—
721C2C2×#241511yescharacteristic
722C2#3473711yescharacteristic
723C2#3497111yescharacteristic
724C2#3511111yescharacteristic
725C2#3521011yescharacteristic
726C2#3530611yescharacteristic
727C2#3763511yescharacteristic
728C2#5906912yes—
729C2#5908212yes—
730C2#5924612yes—
731C2#5926112yes—
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. C2×#2581C2semidirect
9. #15076C2semidirect
10. #14908C2semidirect
11. #14901C2semidirect
13. #14150C2semidirect
14. #14045C2semidirect
15. #14021C2semidirect
16. #13942C2semidirect
17. #13925C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×C4×C22⋊Q82
C2×#22832
#163942
#162622
#151792
#151322
#150762
#149082
#149012
#142022
#141502
#140452
#140212
#139422
#139252