Finite Group Databasebeta

← #612909

Subgroups — #612909

Order 512.

6858 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic, complemented
3C2×#2562C212yesmaximal, complemented
4#15181C212yesmaximal, complemented
5#15129C212yesmaximal, complemented
6#14860C214yesmaximal, complemented
7#14843C212yesmaximal, complemented
8#14724C212yesmaximal
9#14680C212yesmaximal, complemented
10#14680C212yesmaximal, complemented
11#14190C214yesmaximal, complemented
12#13971C214yesmaximal, complemented
13#13905C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C25×C4C2212yes—
17C22×#125C2211yescharacteristic, complemented
18C22×C4×D4—28no—
19C22×C4×D4—216no—
20C22×C4×D4C2212yes—
21C22×C4×D4C2212yes—
22C22×C4×D4C2212yes—
23C22×C4×D4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26C23×C42C2211yescharacteristic
27#2851—28no—
28#2562—216no—
29C2×#400C2214yescomplemented
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#389C2211yescharacteristic, complemented
33C2×#389C2211yescharacteristic, complemented
34C2×#388C2218yescomplemented
35C2×#387C2212yescomplemented
36C2×#387C2212yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#382C2211yescharacteristic, complemented
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2214yescomplemented
44C2×#380C2214yescomplemented
45C2×#380C2214yescomplemented
46C2×#380C2214yescomplemented
47C2×#379C2214yes—
48C2×#379C2214yes—
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C22⋊C4C2212yes—
51#1445C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2218yescomplemented
55#1439C2218yescomplemented
56#1438C2214yes—
57#1438C2214yes—
58#1437C2218yescomplemented
59C4⋊#95C2212yescomplemented
60#1434C2212yescomplemented
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2218yescomplemented
65C22⋊#95C2218yescomplemented
66C4×#95C2212yescomplemented
67C26C2312yes—
68C23×D4—28no—
69C23×D4—28no—
70C23×D4—28no—
71C23×D4—24no—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4—216no—
75C24×C4—28no—
76C24×C4—28no—
77C24×C4—28no—
78C24×C4—48no—
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#125—28no—
86C2×#125—24no—
87C2×C4×D4—2128no—
88C2×C4×D4—28no—
89C2×C4×D4—216no—
90C2×C4×D4—28no—
91C2×C4×D4—216no—
92C2×C4×D4—28no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4D4116yes—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—216no—
113C22×C22⋊C4C2318yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2318yes—
122C22×C42—28no—
123C22×C42D418yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129#400—216no—
130#394—216no—
131#394—216no—
132#389—24no—
133#389—24no—
134#388—232no—
135#387—28no—
136#387—28no—
137#385—28no—
138#385—28no—
139#382—24no—
140#380—216no—
141#380—216no—
142#380—24no—
143#380—24no—
144#380—28no—
145#380—48no—
146#380—48no—
147#379—216no—
148#379—48no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4—44no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C25—28no—
166C25—24no—
167C25—28no—
168C25—24no—
169C25—24no—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—44no—
174C25—44no—
175C25#4118yes—
176C25C2414yes—
177C25C2411yescharacteristic
178C22×D4—232no—
179C22×D4—2128no—
180C22×D4—232no—
181C22×D4—232no—
182C22×D4—28no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4C2×D418yes—
191C22×D4#4118yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—264no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—464no—
221C23×C4—464no—
222C23×C4—432no—
223C23×C4—432no—
224C23×C4C2×D414yes—
225C23×C4C2×D4116yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4118yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2418yes—
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#125—416no—
242C4×D4—264no—
243C4×D4—264no—
244C4×D4—264no—
245C4×D4—264no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—28no—
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—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D418yes—
271C2×C22⋊C4—264no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—216no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×D4116yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—232no—
302C2×C42—28no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D418yes—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—216no—
314#95—216no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—48no—
323#95—48no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—216no—
329C24—216no—
330C24—232no—
331C24—232no—
332C24—216no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—216no—
346C24—216no—
347C24—416no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—416no—
353C24—416no—
354C24—416no—
355C24—416no—
356C24—48no—
357C24—48no—
358C24—416no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24C22×D418yes—
364C24C22×D412yes—
365C24C2×#4118yes—
366C24C2×#4118yes—
367C24C2×#4114yes—
368C24C2×#4114yes—
369C24H(5,𝔽2)12yes—
370C242−1+412yes—
371C24C2511yescharacteristic
372C2×D4—216no—
373C2×D4—2256no—
374C2×D4—216no—
375C2×D4—232no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—232no—
379C2×D4—232no—
380C2×D4—232no—
381C2×D4—216no—
382C2×D4—216no—
383C22×C4—232no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—232no—
399C22×C4—216no—
400C22×C4—232no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—232no—
414C22×C4—232no—
415C22×C4—232no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—4256no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4#124116yes—
460C22×C4C22×D418yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4114yes—
473C22×C4H(5,𝔽2)14yes—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—416no—
485C22⋊C4—232no—
486C22⋊C4—232no—
487C22⋊C4—432no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—816no—
492C22⋊C4—816no—
493C22⋊C4—816no—
494C42—28no—
495C42—28no—
496C42—28no—
497C42—88no—
498C42#120116yes—
499C23—22no—
500C23—22no—
501C23—28no—
502C23—22no—
503C23—22no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—264no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—464no—
531C23—464no—
532C23—432no—
533C23—432no—
534C23—464no—
535C23—464no—
536C23—432no—
537C23—432no—
538C23—432no—
539C23—432no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23#525116yes—
557C23#52514yes—
558C23C2×#12414yes—
559C23#53714yes—
560C23#55218yes—
561C23#55318yes—
562C23C23×D411yescharacteristic
563C23C23×D412yes—
564C23C22×#4112yes—
565C23C22×#4112yes—
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+411yescharacteristic
570C23#58512yes—
571D4—264no—
572D4—264no—
573C2×C4—232no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—232no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—464no—
596C2×C4—432no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—464no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—48no—
625C2×C4—48no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4C2×#12014yes—
630C2×C4C2×#12014yes—
631C2×C4C2×#12014yes—
632C2×C4#52514yes—
633C2×C4C2×#12414yes—
634C2×C4C2×#12414yes—
635C2×C4#53514yes—
636C2×C4#546116yes—
637C22—24no—
638C22—24no—
639C22—216no—
640C22—216no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—216no—
646C22—216no—
647C22—216no—
648C22—4128no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—48no—
654C22—48no—
655C22—48no—
656C22—48no—
657C22—48no—
658C22—48no—
659C22—48no—
660C22—48no—
661C22#256218yescomplemented
662C22C22×#12011yescharacteristic
663C22C2×#52511yescharacteristic
664C22C2×#52514yes—
665C22C2×#52511yescharacteristic
666C22C22×#12411yescharacteristic
667C22C2×#53511yescharacteristic
668C22C2×#53711yescharacteristic
669C22#344911yescharacteristic
670C22C2×#54614yes—
671C22C2×#55212yes—
672C22C2×#55312yes—
673C22#347514yes—
674C22#348812yes—
675C22#348912yes—
676C22#350514yes—
677C22#351912yes—
678C22#352012yes—
679C4—24no—
680C4—24no—
681C4—24no—
682C4—24no—
683C4—48no—
684C4—42no—
685C4—82no—
686C4—82no—
687C4—88no—
688C4—82no—
689C4—82no—
690C4#243518yes—
691C2—28no—
692C2—416no—
693C2C2×#243511yescharacteristic
694C2C2×#256211yescharacteristic
695C2#3494611yescharacteristic
696C2#3510211yescharacteristic
697C2#3523611yescharacteristic
698C2#3525111yescharacteristic
699C2#3615614yes—
700C2#3615714yes—
701C2#3661411yescharacteristic
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×#2851C2semidirect
3. C2×#2562C2semidirect
4. #15181C2semidirect
5. #15129C2semidirect
6. #14860C2semidirect
7. #14843C2semidirect
9. #14680C2semidirect
10. #14680C2semidirect
11. #14190C2semidirect
12. #13971C2semidirect
13. #13905C2semidirect
14. C4×#380C2semidirect
17. C22×#125C22semidirect
29. C2×#400C22semidirect
32. C2×#389C22semidirect
33. C2×#389C22semidirect
34. C2×#388C22semidirect
35. C2×#387C22semidirect
36. C2×#387C22semidirect
39. C2×#382C22semidirect
43. C2×#380C22semidirect
44. C2×#380C22semidirect
45. C2×#380C22semidirect
46. C2×#380C22semidirect
49. C2×C4×C4⋊C4C22semidirect
51. #1445C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1440C22semidirect
55. #1439C22semidirect
58. #1437C22semidirect
59. C4⋊#95C22semidirect
60. #1434C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C4×#95C22semidirect
661. C22#2562semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×#28512
C2×#25622
#151812
#151292
#148602
#148432
#147242
#146802
#141902
#139712
#139052
C4×#3802