Finite Group Databasebeta

← #4387679

Subgroups — #4387679

Order 512.

3382 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16379C214yesmaximal, complemented
2#15989C214yesmaximal, complemented
3#15979C214yesmaximal, complemented
4#15687C214yesmaximal, complemented
5#15282C214yesmaximal, complemented
6#14985C214yesmaximal, complemented
7#14673C211yesmaximal, characteristic, complemented
8#14338C211yesmaximal, characteristic, complemented
9#14335C211yesmaximal, characteristic
10#14127C211yesmaximal, characteristic, complemented
11#13867C211yesmaximal, characteristic
12#13867C211yesmaximal, characteristic
13#13775C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17C2×#397C2214yescomplemented
18C2×#388C2214yescomplemented
19C2×#388C2214yescomplemented
20C2×#387C2214yescomplemented
21C2×#386C2212yes—
22C2×#386C2212yes—
23C2×#386C2212yes—
24C2×#386C2212yes—
25C2×#385C2212yescomplemented
26C2×#385C2212yescomplemented
27C2×#381C2212yes—
28C2×#380C2212yescomplemented
29C2×#379C2211yescharacteristic
30C2×#379C2211yescharacteristic
31C2×#379C2211yescharacteristic
32C2×#379C2211yescharacteristic
33C2×#379C2212yes—
34C2×#379C2212yes—
35C2×#379C2212yes—
36C2×C4×C22⋊C4C2212yescomplemented
37#1445C2214yescomplemented
38#1445C2214yescomplemented
39#1443C2214yescomplemented
40#1443C2214yescomplemented
41#1443C2214yescomplemented
42#1441C2214yescomplemented
43#1440C2214yescomplemented
44#1440C2214yescomplemented
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yescomplemented
60#1434C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yescomplemented
64C22⋊#95C2212yescomplemented
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C26C2311yescharacteristic
68C26C2311yescharacteristic
69C23×D4—24no—
70C23×D4—24no—
71C23×D4—24no—
72C23×D4—44no—
73C23×D4—44no—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—24no—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—44no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2314yes—
111#397—48no—
112#388—48no—
113#388—48no—
114#387—48no—
115#386—44no—
116#386—44no—
117#386—44no—
118#386—44no—
119#385—44no—
120#385—44no—
121#381—44no—
122#380—44no—
123#379—24no—
124#379—24no—
125#379—24no—
126#379—24no—
127#379—44no—
128#379—44no—
129#379—44no—
130C4×C22⋊C4—44no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25—22no—
150C25—24no—
151C25—22no—
152C25—22no—
153C25—42no—
154C25—42no—
155C25—42no—
156C25—42no—
157C25—42no—
158C25—42no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25—42no—
163C25—42no—
164C25#4114yes—
165C25#4114yes—
166C25#4114yes—
167C25C2412yes—
168C25C2411yescharacteristic
169C25C2411yescharacteristic
170C25C2411yescharacteristic
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—416no—
186C22×D4—432no—
187C22×D4—432no—
188C22×D4—416no—
189C22×D4—416no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—416no—
270C2×C22⋊C4—416no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C42—28no—
298C2×C42—44no—
299C2×C42—44no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—28no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—44no—
348C24—44no—
349C24—48no—
350C24—48no—
351C24—44no—
352C24—44no—
353C24—48no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24—44no—
362C24—48no—
363C24—48no—
364C24—44no—
365C24—82no—
366C24—82no—
367C24—82no—
368C24—82no—
369C24—84no—
370C24—84no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24—88no—
376C24—88no—
377C24—88no—
378C24—88no—
379C24#126116yes—
380C24C2×#4112yes—
381C24C2×#4112yes—
382C24C2×#4112yes—
383C24C2×#4112yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24C2×#4112yes—
387C24C2×#4112yes—
388C24C2×#4112yes—
389C24H(5,𝔽2)14yes—
390C24C2511yescharacteristic
391C2×D4—216no—
392C2×D4—216no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—4128no—
404C2×D4—48no—
405C2×D4—48no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4H(5,𝔽2)14yes—
480C22×C4H(5,𝔽2)14yes—
481C22×C4H(5,𝔽2)14yes—
482C4⋊C4—416no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—88no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C42—88no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—28no—
548C23—28no—
549C23—416no—
550C23—416no—
551C23—416no—
552C23—416no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—88no—
584C23—88no—
585C23—88no—
586C23—88no—
587C23—88no—
588C23—88no—
589C23—832no—
590C23—832no—
591C23—88no—
592C23—88no—
593C23—832no—
594C23—832no—
595C23—88no—
596C23—88no—
597C23C2×#12614yes—
598C23C2×#12614yes—
599C23C2×#12614yes—
600C23C22×#4111yescharacteristic
601C23C22×#4111yescharacteristic
602C23C22×#4111yescharacteristic
603C23C2×H(5,𝔽2)12yes—
604C23C2×H(5,𝔽2)12yes—
605C23C2×H(5,𝔽2)12yes—
606C23C2×H(5,𝔽2)12yes—
607C23#58512yes—
608C23#58512yes—
609D4—432no—
610D4—432no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—416no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
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—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—88no—
668C2×C4—88no—
669C2×C4—88no—
670C2×C4—88no—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—48no—
687C22—48no—
688C22—48no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—44no—
699C22—44no—
700C22—48no—
701C22—48no—
702C22—44no—
703C22—832no—
704C22—832no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22—82no—
710C22—82no—
711C22C22×#12611yescharacteristic
712C22#345912yes—
713C22#349011yescharacteristic
714C22#349011yescharacteristic
715C22#349012yes—
716C22#349012yes—
717C22#349011yescharacteristic
718C22#349011yescharacteristic
719C22#349111yescharacteristic
720C22#349111yescharacteristic
721C22#350512yes—
722C22#350512yes—
723C22#350512yes—
724C22#353414yes—
725C22#354014yes—
726C22#354214yes—
727C22#354314yes—
728C4—44no—
729C4—44no—
730C4—44no—
731C4—82no—
732C4—84no—
733C4—84no—
734C4—84no—
735C4—84no—
736C4—82no—
737C2—44no—
738C2—44no—
739C2—44no—
740C2—84no—
741C2#3499511yescharacteristic
742C2#3536911yescharacteristic
743C2#3537011yescharacteristic
744C2#4104914yes—
745C2#4638611yescharacteristic
746C2#4638711yescharacteristic
747C2#4699711yescharacteristic
748C2#4699811yescharacteristic
749C2#4799614yes—
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
1. #16379C2semidirect
2. #15989C2semidirect
3. #15979C2semidirect
4. #15687C2semidirect
5. #15282C2semidirect
6. #14985C2semidirect
7. #14673C2semidirect
8. #14338C2semidirect
10. #14127C2semidirect
13. #13775C2semidirect
17. C2×#397C22semidirect
18. C2×#388C22semidirect
19. C2×#388C22semidirect
20. C2×#387C22semidirect
25. C2×#385C22semidirect
26. C2×#385C22semidirect
28. C2×#380C22semidirect
36. C2×C4×C22⋊C4C22semidirect
37. #1445C22semidirect
38. #1445C22semidirect
39. #1443C22semidirect
40. #1443C22semidirect
41. #1443C22semidirect
42. #1441C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
59. #1437C22semidirect
60. #1434C22semidirect
63. C22⋊#95C22semidirect
64. C22⋊#95C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#163792
#159892
#159792
#156872
#152822
#149852
#146732
#143382
#143352
#141272
#138672
#137752