Finite Group Databasebeta

← #618453

Subgroups — #618453

Order 512.

5614 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#124C211yesmaximal, characteristic
2C2×#2483C211yesmaximal, characteristic, complemented
3C2×#2405C211yesmaximal, characteristic
4C2×#2315C211yesmaximal, characteristic, complemented
5#16032C214yesmaximal, complemented
6#15311C218yesmaximal, complemented
7#15262C214yesmaximal, complemented
8#14918C212yesmaximal, complemented
9#14874C212yesmaximal, complemented
10#14705C212yesmaximal, complemented
11#14341C211yesmaximal, characteristic, complemented
12#14262C212yesmaximal, complemented
13#13869C211yesmaximal, characteristic, complemented
14#13867C211yesmaximal, characteristic
15C24×Q8C2211yescharacteristic
16C24×D4C2211yescharacteristic
17C22×#124—28no—
18C22×#124—28no—
19C22×#124—28no—
20C22×#124C2214yes—
21C22×#124C2214yes—
22C22×#123C2212yes—
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C22⋊C4C2211yescharacteristic
29C23×C42C2211yescharacteristic
30#2483—28no—
31#2405—28no—
32C2×#397C2214yescomplemented
33C2×#395C2218yescomplemented
34C2×#394C2214yes—
35#2315—28no—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yescomplemented
43C2×#385C2212yescomplemented
44C2×#385C2212yescomplemented
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yescomplemented
48C2×#380C2214yescomplemented
49C2×#379C2211yescharacteristic
50C2×#379C2211yescharacteristic
51C2×#379C2212yes—
52C2×#379C2212yes—
53#1446C2218yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1440C2218yescomplemented
58#1439C2218yescomplemented
59#1439C2218yescomplemented
60#1439C2218yescomplemented
61#1438C2214yes—
62#1436C2218yescomplemented
63#1436C2218yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—48no—
75C23×Q8C2314yes—
76C23×D4—24no—
77C23×D4—24no—
78C23×D4—48no—
79C23×D4D418yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#124—2128no—
91C2×#124—216no—
92C2×#124—216no—
93C2×#124—216no—
94C2×#124—216no—
95C2×#124—216no—
96C2×#124—216no—
97C2×#123—28no—
98C2×#123—48no—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—28no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4D418yes—
119C22×C22⋊C4D418yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2318yes—
139C22×C42—24no—
140C22×C42—24no—
141C22×C42—24no—
142C22×C42C2314yes—
143#397—216no—
144#395—416no—
145#394—216no—
146#386—28no—
147#386—48no—
148#385—28no—
149#385—28no—
150#385—28no—
151#385—28no—
152#385—28no—
153#385—28no—
154#385—28no—
155#381—28no—
156#381—28no—
157#381—28no—
158#380—216no—
159#379—28no—
160#379—24no—
161#379—24no—
162#379—28no—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2318yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C25—24no—
175C25—24no—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—44no—
181C25—44no—
182C25—44no—
183C25—44no—
184C25C2×D414yes—
185C25C2×D414yes—
186C25C2×D414yes—
187C25C2×D414yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C25C2411yescharacteristic
191C25C2411yescharacteristic
192C22×Q8—232no—
193C22×Q8—28no—
194C22×Q8—28no—
195C22×Q8—28no—
196C22×Q8—216no—
197C22×Q8—48no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8—432no—
201C22×D4—232no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—432no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—432no—
212C22×D4—432no—
213C22×D4C2×D418yes—
214C22×D4C2×D418yes—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—28no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—28no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2×D414yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258#124—2128no—
259#124—2128no—
260#123—432no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—48no—
268C2×C4⋊C4—48no—
269C2×C22⋊C4—232no—
270C2×C22⋊C4—232no—
271C2×C22⋊C4—232no—
272C2×C22⋊C4—232no—
273C2×C22⋊C4—232no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
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—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—216no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—416no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4C2×D418yes—
321C2×C22⋊C4C2×D418yes—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4#4118yes—
325C2×C42—232no—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—216no—
336#95—216no—
337#95—48no—
338#95—48no—
339#95—48no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—28no—
351C24—28no—
352C24—216no—
353C24—216no—
354C24—28no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—28no—
362C24—216no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—48no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—44no—
376C24—44no—
377C24—48no—
378C24—816no—
379C24—816no—
380C24—816no—
381C24—816no—
382C24#120116yes—
383C24C22×D414yes—
384C24C22×D412yes—
385C24C22×D412yes—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C22×D412yes—
389C24C22×D412yes—
390C24C22×D412yes—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C2×#4114yes—
394C242−1+414yes—
395C24C2511yescharacteristic
396C2×Q8—232no—
397C2×Q8—432no—
398C2×Q8—432no—
399C2×Q8—4128no—
400C2×Q8—416no—
401C2×Q8—416no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—216no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—4128no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—216no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—216no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—28no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—816no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4—84no—
481C22×C4—84no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4C22×D414yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C2×#4118yes—
490C22×C4H(5,𝔽2)18yes—
491C4⋊C4—416no—
492C4⋊C4—88no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—216no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—816no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—88no—
514C42—232no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—216no—
522C23—216no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—232no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—416no—
550C23—416no—
551C23—416no—
552C23—416no—
553C23—48no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23—416no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—48no—
566C23—48no—
567C23—48no—
568C23—48no—
569C23—832no—
570C23—832no—
571C23—816no—
572C23—816no—
573C23—832no—
574C23—816no—
575C23—816no—
576C23—832no—
577C23C2×#12014yes—
578C23C2×#12014yes—
579C23C2×#12014yes—
580C23#52514yes—
581C23#53614yes—
582C23#54718yes—
583C23#54718yes—
584C23C23×D411yescharacteristic
585C23C23×D411yescharacteristic
586C23C23×D411yescharacteristic
587C23C22×#4112yes—
588C23C2×H(5,𝔽2)14yes—
589C23C2×2−1+412yes—
590C23#58514yes—
591Q8—464no—
592D4—432no—
593D4—432no—
594C2×C4—232no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—216no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
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—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—48no—
642C2×C4—48no—
643C2×C4—816no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4#53418yes—
647C22—216no—
648C22—24no—
649C22—28no—
650C22—28no—
651C22—216no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—48no—
660C22—48no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—48no—
667C22—48no—
668C22—48no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—832no—
673C22—832no—
674C22—84no—
675C22—84no—
676C22—88no—
677C22#240618yes—
678C22C22×#12011yescharacteristic
679C22C2×#52511yescharacteristic
680C22C2×#53412yes—
681C22C2×#53611yescharacteristic
682C22#344912yes—
683C22C2×#54712yes—
684C22C2×#54712yes—
685C22#348412yes—
686C22#348412yes—
687C22#348512yes—
688C22#348512yes—
689C22#351418yes—
690C22#356118yes—
691C4—28no—
692C4—44no—
693C4—48no—
694C4—44no—
695C4—44no—
696C4—44no—
697C4—84no—
698C4—82no—
699C4—84no—
700C4—82no—
701C2—28no—
702C2—44no—
703C2—44no—
704C2—84no—
705C2C2×#240611yescharacteristic
706C2#3498011yescharacteristic
707C2#3508811yescharacteristic
708C2#3545112yes—
709C2#3696612yes—
710C2#3804414yes—
711C2#3891814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2483C2semidirect
4. C2×#2315C2semidirect
5. #16032C2semidirect
6. #15311C2semidirect
7. #15262C2semidirect
8. #14918C2semidirect
9. #14874C2semidirect
10. #14705C2semidirect
11. #14341C2semidirect
12. #14262C2semidirect
13. #13869C2semidirect
32. C2×#397C22semidirect
33. C2×#395C22semidirect
42. C2×#385C22semidirect
43. C2×#385C22semidirect
44. C2×#385C22semidirect
47. C2×#381C22semidirect
48. C2×#380C22semidirect
53. #1446C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
68. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1242
C2×#24832
C2×#24052
C2×#23152
#160322
#153112
#152622
#149182
#148742
#147052
#143412
#142622
#138692
#138672