Finite Group Databasebeta

← #6500414

Subgroups — #6500414

Order 512.

2868 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3007C211yesmaximal, characteristic, complemented
2C2×#2975C211yesmaximal, characteristic, complemented
3#35084C211yesmaximal, characteristic, complemented
4#34736C211yesmaximal, characteristic
5#28113C214yesmaximal, complemented
6#25136C214yesmaximal, complemented
7#25134C214yesmaximal, complemented
8#24505C214yesmaximal, complemented
9#24389C214yesmaximal, complemented
10#24276C214yesmaximal, complemented
11#21943C211yesmaximal, characteristic, complemented
12#21768C211yesmaximal, characteristic, complemented
13#20600C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×#137C2212yescomplemented
16C2×#525C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#470C2214yescomplemented
20C2×#468C2214yescomplemented
21#3007—28no—
22C2×#452C2214yes—
23C2×#451C2212yescomplemented
24C2×#450C2212yescomplemented
25#2975—28no—
26C2×#436C2214yes—
27C2×#436C2214yescomplemented
28C2×#431C2212yescomplemented
29C2×#417C2212yescomplemented
30C2×#419C2212yescomplemented
31C2×#418C2212yescomplemented
32C2×#407C2211yescharacteristic
33C2×#407C2211yescharacteristic
34C2×#407C2212yescomplemented
35#2855C2212yescomplemented
36#2848C2212yescomplemented
37#2638—22no—
38#2583—21no—
39#2561—21no—
40#2486C2212yescomplemented
41#2483—21no—
42#2415C2214yes—
43#2409C2212yes—
44#2409C2212yescomplemented
45#2408C2212yes—
46#2407C2214yes—
47#2406—21no—
48#2405—21no—
49#2405—21no—
50#2404—21no—
51#2403—21no—
52#2388—22no—
53#2322—22no—
54C4×#120—22no—
55#2297C2212yes—
56#2280C2211yescharacteristic
57#2280C2211yescharacteristic
58#2067C2214yescomplemented
59#2066C2214yescomplemented
60#2056C2214yescomplemented
61#2055C2214yescomplemented
62#2036C2214yescomplemented
63#2034C2214yescomplemented
64#2001C2214yescomplemented
65#2000C2214yescomplemented
66#1969C2214yescomplemented
67#1968C2214yescomplemented
68#1934C2214yescomplemented
69#1933C2214yescomplemented
70#1895C2214yescomplemented
71#1894C2214yescomplemented
72#1894C2214yescomplemented
73#1873C2214yescomplemented
74#1854C2214yescomplemented
75#1854C2214yescomplemented
76#1824C2212yescomplemented
77#1809C2214yescomplemented
78#1788C2214yes—
79#1788C2214yescomplemented
80#1787C2212yescomplemented
81C22×#41—21no—
82C22×#41—21no—
83C22×#41—24no—
84C22×#41—24no—
85C22×#41—44no—
86C22×#41—44no—
87C22×#41C2312yes—
88C22×#41C2312yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2311yescharacteristic
91C23×D4—21no—
92C23×D4D411yescharacteristic
93C23×D4C2311yescharacteristic
94C24×C4—21no—
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C2×#137—28no—
98C2×#137—44no—
99C2×#137—44no—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C22×M16C2312yes—
103C22×M16C2314yes—
104C22×M16C2312yes—
105#525—44no—
106#525—44no—
107C2×#123—22no—
108C2×#123—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2312yes—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2314yes—
116C2×C22⋊Q8C2314yes—
117C2×#121C2314yes—
118C2×#120—22no—
119C2×#120—22no—
120C2×C4×D4—24no—
121C2×C4×D4—24no—
122C2×C4×D4—24no—
123C2×C4×D4D412yes—
124C2×C4×D4D412yes—
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×#117C2312yes—
128C2×#117C2312yes—
129C22×C4⋊C4—21no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4D411yescharacteristic
134C22×C22⋊C4D411yescharacteristic
135C22×C42C2311yescharacteristic
136#470—48no—
137#468—48no—
138#452—216no—
139#451—28no—
140#450—28no—
141#436—216no—
142#436—216no—
143#431—28no—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2314yes—
147#419—44no—
148#418—44no—
149#417—28no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#407—24no—
159#407—28no—
160#407D418yescomplemented
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2314yes—
164C2×C22⋊C8C2312yes—
165C2×C22⋊C8C2312yes—
166C2×C4×C8C2314yes—
167#398—22no—
168#397—22no—
169#397—24no—
170#395—22no—
171#395—22no—
172#394—24no—
173#394—22no—
174#394—24no—
175#393—22no—
176#393—22no—
177#392—24no—
178#392—22no—
179#391C2314yes—
180#391C2314yes—
181#390C2314yes—
182#389—24no—
183#389C2314yes—
184#388—24no—
185#387—24no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—24no—
191#385—24no—
192#385—22no—
193#385—22no—
194#385—24no—
195#385—22no—
196#385—22no—
197#381—22no—
198#381—21no—
199#381—21no—
200#381—22no—
201#380C2312yes—
202#380C2312yes—
203#380C2312yes—
204#380C2314yes—
205#380C2312yes—
206#379—22no—
207#379—21no—
208#379—21no—
209#379—22no—
210C4×C22⋊C4—21no—
211C4×C22⋊C4—21no—
212C4×C22⋊C4—21no—
213C4×C22⋊C4—21no—
214C4×C22⋊C4—24no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219C25—21no—
220C25C2×D411yescharacteristic
221C25C2×D411yescharacteristic
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—24no—
229C2×#41—24no—
230C2×#41—42no—
231C2×#41—42no—
232C2×#41—42no—
233C2×#41—42no—
234C2×#41—48no—
235C2×#41—416no—
236C2×#41—48no—
237C2×#41—416no—
238C2×#41—432no—
239C2×#41—44no—
240C2×#41—44no—
241C22×Q8—21no—
242C22×Q8—21no—
243C22×Q8—24no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—22no—
248C22×Q8—42no—
249C22×Q8—42no—
250C22×Q8—42no—
251C22×Q8C2412yes—
252C22×Q8C2412yes—
253C22×D4—24no—
254C22×D4—24no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—24no—
258C22×D4—22no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—44no—
262C22×D4—42no—
263C22×D4—48no—
264C22×D4—44no—
265C22×D4C2×D412yes—
266C22×D4C2×D412yes—
267C22×D4C2412yes—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—42no—
276C23×C4—42no—
277C23×C4—42no—
278C23×C4—42no—
279C23×C4—48no—
280C23×C4—44no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D412yes—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2×D414yes—
289C23×C4#4114yes—
290C23×C4C2412yes—
291C23×C4C2411yescharacteristic
292C23×C4C2411yescharacteristic
293C23×C4C2411yescharacteristic
294#137—432no—
295C2×Q16—28no—
296C2×Q16—28no—
297C2×Q16—44no—
298C2×SD16—28no—
299C2×SD16—28no—
300C2×SD16—44no—
301C2×M16—24no—
302C2×M16—28no—
303C2×M16—42no—
304C2×M16—42no—
305C2×M16—42no—
306C2×M16—42no—
307C2×M16—44no—
308C2×M16—44no—
309C2×M16C2×D418yes—
310C22×C8C2414yes—
311C22×C8C2414yes—
312#123—48no—
313#123—48no—
314C22⋊Q8—28no—
315C22⋊Q8—24no—
316C22⋊Q8—24no—
317C22⋊Q8—24no—
318C22⋊Q8—24no—
319C22⋊Q8—44no—
320C22⋊Q8—44no—
321C22⋊Q8—44no—
322C22⋊Q8—44no—
323C22⋊Q8—44no—
324#121—28no—
325#121—44no—
326#120—48no—
327#120—48no—
328C4×D4—28no—
329C4×D4—28no—
330C4×D4—24no—
331C4×D4—24no—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—416no—
335#117—42no—
336#117—42no—
337#117—42no—
338#117—42no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4C2×D412yes—
350C2×C4⋊C4C2×D412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C4⋊C4C2414yes—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4C2×D412yes—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42C2×D412yes—
380C2×C42C2412yes—
381#107—44no—
382#106—44no—
383C4⋊C8—28no—
384#103—28no—
385#103—28no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—28no—
390#102—28no—
391#102—44no—
392C22⋊C8—24no—
393C22⋊C8—24no—
394C22⋊C8—28no—
395C22⋊C8C2×D418yes—
396C22⋊C8C2×D418yes—
397C4×C8—28no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408C24—22no—
409C24—22no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—44no—
417C24—44no—
418C24—42no—
419C24—82no—
420C24#12114yes—
421C24C22×D412yes—
422C24C22×D411yescharacteristic
423C24C22×D411yescharacteristic
424C24C22×D411yescharacteristic
425#41—216no—
426#41—48no—
427#41—48no—
428#41—48no—
429#41—48no—
430#41—48no—
431#41—48no—
432#41—48no—
433#41—432no—
434#41—432no—
435C2×Q8—22no—
436C2×Q8—28no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—44no—
446C2×Q8—48no—
447C2×Q8—44no—
448C2×Q8—48no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8H(5,𝔽2)14yes—
455C2×D4—28no—
456C2×D4—28no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—48no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—416no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—44no—
475C2×D4—432no—
476C2×D4—44no—
477C2×D4—44no—
478C2×D4—48no—
479C2×D4—48no—
480C2×D4—42no—
481C2×D4—84no—
482C2×D4H(5,𝔽2)14yes—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—28no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—48no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—42no—
513C22×C4—42no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—432no—
518C22×C4—42no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—84no—
525C22×C4—82no—
526C22×C4—82no—
527C22×C4—82no—
528C22×C4#12114yes—
529C22×C4#12114yes—
530C22×C4#12114yes—
531C22×C4C22×D412yes—
532C22×C4C22×D412yes—
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D411yescharacteristic
535C22×C4C22×D411yescharacteristic
536C22×C4C22×D412yes—
537C22×C4C22×D412yes—
538C22×C4C22×D412yes—
539C22×C4C22×D412yes—
540C22×C4C2×#4112yes—
541C22×C4C2×#4112yes—
542C22×C4C2×#4112yes—
543C22×C4C2511yescharacteristic
544Q16—416no—
545SD16—416no—
546M16—48no—
547M16—88no—
548M16—84no—
549C2×C8—24no—
550C2×C8—24no—
551C2×C8—24no—
552C2×C8—24no—
553C2×C8—24no—
554C2×C8C22×D418yes—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—24no—
562C4⋊C4—24no—
563C4⋊C4—48no—
564C4⋊C4—42no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—48no—
568C4⋊C4—84no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—48no—
574C22⋊C4—48no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—48no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—48no—
583C22⋊C4—84no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C22⋊C4—84no—
587C42—24no—
588C42—22no—
589C42—22no—
590C42—44no—
591C42—82no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—22no—
599C23—22no—
600C23—22no—
601C23—24no—
602C23—22no—
603C23—24no—
604C23—24no—
605C23—22no—
606C23—22no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—48no—
611C23—44no—
612C23—44no—
613C23—42no—
614C23—42no—
615C23—44no—
616C23—82no—
617C23—82no—
618C23—84no—
619C23—84no—
620C23—88no—
621C23—88no—
622C23—816no—
623C23C2×#12112yes—
624C23C2×#12112yes—
625C23C2×#12112yes—
626C23C23×D411yescharacteristic
627Q8—24no—
628Q8—24no—
629Q8—48no—
630Q8—42no—
631Q8—42no—
632Q8—44no—
633Q8—42no—
634Q8—42no—
635Q8—84no—
636D4—24no—
637D4—24no—
638D4—48no—
639D4—48no—
640D4—48no—
641D4—416no—
642D4—416no—
643D4—42no—
644D4—42no—
645D4—816no—
646C2×C4—28no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—24no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—416no—
670C2×C4—416no—
671C2×C4—48no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—84no—
688C2×C4—82no—
689C2×C4—82no—
690C2×C4—84no—
691C2×C4—84no—
692C2×C4C2×#12112yes—
693C2×C4C2×#12112yes—
694C2×C4C2×#12112yes—
695C2×C4#57514yes—
696C2×C4C23×D411yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C2×H(5,𝔽2)12yes—
699C2×C4C2×H(5,𝔽2)12yes—
700C8—28no—
701C8—44no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—44no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—44no—
719C22—48no—
720C22—44no—
721C22—88no—
722C22—88no—
723C22—816no—
724C22—82no—
725C22—82no—
726C22—84no—
727C22C22×#12111yescharacteristic
728C22#348412yes—
729C22C2×#57512yes—
730C22#358612yes—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—24no—
735C4—42no—
736C4—44no—
737C4—42no—
738C4—82no—
739C4—82no—
740C4—84no—
741C4—84no—
742C4#348412yes—
743C4#349011yescharacteristic
744C4#349011yescharacteristic
745C2—42no—
746C2—42no—
747C2—42no—
748C2—82no—
749C2—84no—
750C2#3534811yescharacteristic
751C2#6044411yescharacteristic
752C2#6045511yescharacteristic
753C2#6158914yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#34736C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3007C2semidirect
2. C2×#2975C2semidirect
3. #35084C2semidirect
5. #28113C2semidirect
6. #25136C2semidirect
7. #25134C2semidirect
8. #24505C2semidirect
9. #24389C2semidirect
10. #24276C2semidirect
11. #21943C2semidirect
12. #21768C2semidirect
13. #20600C2semidirect
15. C22×#137C22semidirect
19. C2×#470C22semidirect
20. C2×#468C22semidirect
23. C2×#451C22semidirect
24. C2×#450C22semidirect
27. C2×#436C22semidirect
28. C2×#431C22semidirect
29. C2×#417C22semidirect
30. C2×#419C22semidirect
31. C2×#418C22semidirect
34. C2×#407C22semidirect
35. #2855C22semidirect
36. #2848C22semidirect
40. #2486C22semidirect
44. #2409C22semidirect
58. #2067C22semidirect
59. #2066C22semidirect
60. #2056C22semidirect
61. #2055C22semidirect
62. #2036C22semidirect
63. #2034C22semidirect
64. #2001C22semidirect
65. #2000C22semidirect
66. #1969C22semidirect
67. #1968C22semidirect
68. #1934C22semidirect
69. #1933C22semidirect
70. #1895C22semidirect
71. #1894C22semidirect
72. #1894C22semidirect
73. #1873C22semidirect
74. #1854C22semidirect
75. #1854C22semidirect
76. #1824C22semidirect
77. #1809C22semidirect
79. #1788C22semidirect
80. #1787C22semidirect
160. #407D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30072
C2×#29752
#350842
#347362
#281132
#251362
#251342
#245052
#243892
#242762
#219432
#217682
#206002