Finite Group Databasebeta

← #612650

Subgroups — #612650

Order 512.

4542 conjugacy classes of subgroups.

The subgroup lattice has 730 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic, complemented
3C22×#380C211yesmaximal, characteristic
4C22×C4×C22⋊C4C211yesmaximal, characteristic
5#15236C212yesmaximal, complemented
6#15233C212yesmaximal, complemented
7#14841C214yesmaximal, complemented
8#14826C214yesmaximal, complemented
9#14702C212yesmaximal, complemented
10#14682C212yesmaximal, complemented
11#14139C211yesmaximal, characteristic, complemented
12#13972C214yesmaximal, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13881C211yesmaximal, characteristic, complemented
15#13869C214yesmaximal
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23C23×C22⋊C4C2212yes—
24C23×C42C2212yes—
25#2848—28no—
26#2503—28no—
27C2×#400C2214yescomplemented
28C2×#391C2212yescomplemented
29C2×#391C2212yescomplemented
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#382C2212yescomplemented
34C2×#382C2212yescomplemented
35C2×#381C2214yes—
36C2×#381C2214yes—
37C2×#380—28no—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#379C2212yes—
46C2×C4×C22⋊C4—28no—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C22×#95C2212yes—
53C22×#95C2212yes—
54#1445C2214yescomplemented
55#1443C2218yescomplemented
56#1442C2214yescomplemented
57#1442C2218yescomplemented
58#1441C2214yescomplemented
59#1440C2218yes—
60#1440C2218yes—
61#1436C2218yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65C4⋊#95C2212yescomplemented
66#1434C2218yescomplemented
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22⋊#95C2218yes—
70C22⋊#95C2218yes—
71C26C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C24×C4—24no—
74C24×C4—24no—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2314yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—216no—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C42—28no—
126C22×C42—28no—
127C22×C42—44no—
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131#400—216no—
132#391—28no—
133#391—44no—
134#385—28no—
135#385—28no—
136#385—28no—
137#382—28no—
138#382—44no—
139#381—216no—
140#381—216no—
141#380—24no—
142#380—24no—
143#380—24no—
144#380—28no—
145#380—28no—
146#380—28no—
147#380—28no—
148#379—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4D418yes—
154C2×#95—28no—
155C2×#95—28no—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2312yes—
172C25—22no—
173C25—24no—
174C25—22no—
175C25—24no—
176C25—24no—
177C25—42no—
178C25—42no—
179C25C2×D414yes—
180C25#4118yes—
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—232no—
199C23×C4—232no—
200C23×C4—28no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—416no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—432no—
228C23×C4—432no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4#4118yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22⋊Q8—416no—
253C22⋊Q8—416no—
254C22⋊Q8—416no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—416no—
282C2×C22⋊C4—264no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
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—216no—
300C2×C22⋊C4—216no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—432no—
308C2×C22⋊C4C2×D418yes—
309C2×C22⋊C4#4118yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—432no—
318C2×C42—44no—
319C2×C42C2×D418yes—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—216no—
328#95—216no—
329#95—216no—
330#95—216no—
331#95—48no—
332#95—48no—
333#95—48no—
334#95—48no—
335#95C2×D418yes—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—28no—
346C24—28no—
347C24—216no—
348C24—216no—
349C24—28no—
350C24—28no—
351C24—232no—
352C24—21no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—44no—
360C24—44no—
361C24—416no—
362C24—48no—
363C24—48no—
364C24—416no—
365C24—48no—
366C24—48no—
367C24—416no—
368C24#124116yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C22×D412yes—
372C24C22×D412yes—
373C24C2×#4114yes—
374C24C2×#4112yes—
375C24C2×#4114yes—
376C24C2×#4114yes—
377C24C2×#4114yes—
378C24C2×#4114yes—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—48no—
385C2×Q8—48no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—232no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—28no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—4128no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—416no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—416no—
478C22×C4—48no—
479C22×C4—816no—
480C22×C4#121116yes—
481C22×C4#123116yes—
482C22×C4C22×D412yes—
483C22×C4C22×D418yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4118yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4114yes—
494C22×C4H(5,𝔽2)14yes—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C22⋊C4—232no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C42—216no—
517C42—416no—
518C42—48no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—28no—
544C23—232no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—28no—
554C23—28no—
555C23—232no—
556C23—416no—
557C23—416no—
558C23—416no—
559C23—464no—
560C23—464no—
561C23—416no—
562C23—416no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23C2×#12114yes—
570C23C2×#12314yes—
571C23#52518yes—
572C23C2×#12414yes—
573C23C2×#12414yes—
574C23C2×#12414yes—
575C23#52918yes—
576C23#53214yes—
577C23#53614yes—
578C23#552116yes—
579C23C23×D411yescharacteristic
580C23C23×D411yescharacteristic
581C23C22×#4111yescharacteristic
582C23C22×#4112yes—
583C23C22×#4112yes—
584C23C22×#4112yes—
585C23C2×H(5,𝔽2)12yes—
586C23C2×2−1+412yes—
587C23#58512yes—
588Q8—88no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—232no—
598C2×C4—232no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—416no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—464no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—416no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—48no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—48no—
649C2×C4—48no—
650C2×C4—48no—
651C2×C4—816no—
652C2×C4—816no—
653C2×C4C2×#12114yes—
654C2×C4C2×#12114yes—
655C2×C4C2×#12314yes—
656C2×C4C2×#12314yes—
657C2×C4#53414yes—
658C2×C4#54618yes—
659C2×C4#55118yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—216no—
664C22—216no—
665C22—216no—
666C22—216no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—464no—
680C22—42no—
681C22—48no—
682C22—48no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22#250318yescomplemented
687C22#285018yes—
688C22C22×#12111yescharacteristic
689C22C22×#12311yescharacteristic
690C22C2×#52512yes—
691C22C22×#12411yescharacteristic
692C22C2×#52912yes—
693C22C2×#53211yescharacteristic
694C22C2×#53411yescharacteristic
695C22C2×#53611yescharacteristic
696C22#344911yescharacteristic
697C22C2×#54612yes—
698C22C2×#55112yes—
699C22C2×#55214yes—
700C22#347514yes—
701C22#348912yes—
702C22#349512yes—
703C22#350514yes—
704C22#352014yes—
705C4—24no—
706C4—24no—
707C4—48no—
708C4—48no—
709C4—42no—
710C4—84no—
711C4—82no—
712C4—82no—
713C4—84no—
714C4—82no—
715C4—82no—
716C4#241218yes—
717C2—28no—
718C2—28no—
719C2—44no—
720C2C2×#241211yescharacteristic
721C2C2×#250311yescharacteristic
722C2#3481011yescharacteristic
723C2#3501911yescharacteristic
724C2#3508511yescharacteristic
725C2#3530611yescharacteristic
726C2#3615614yes—
727C2#3616714yes—
728C2C2×#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2503C2semidirect
5. #15236C2semidirect
6. #15233C2semidirect
7. #14841C2semidirect
8. #14826C2semidirect
9. #14702C2semidirect
10. #14682C2semidirect
11. #14139C2semidirect
12. #13972C2semidirect
13. #13904C2semidirect
14. #13881C2semidirect
27. C2×#400C22semidirect
28. C2×#391C22semidirect
29. C2×#391C22semidirect
33. C2×#382C22semidirect
34. C2×#382C22semidirect
54. #1445C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. #1434C22semidirect
686. C22#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#25032
C22×#3802
C22×C4×C22⋊C42
#152362
#152332
#148412
#148262
#147022
#146822
#141392
#139722
#139042
#138812
#138692