Finite Group Databasebeta

← #6487671

Subgroups — #6487671

Order 512.

2338 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3105C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic, complemented
3C2×#2349C211yesmaximal, characteristic, complemented
4#32908C212yesmaximal, complemented
5#32905C211yesmaximal, characteristic, complemented
6#32902C211yesmaximal, characteristic
7#29761C212yesmaximal, complemented
8#29757C211yesmaximal, characteristic, complemented
9#29755C211yesmaximal, characteristic, complemented
10#24822C214yesmaximal, complemented
11#24820C214yesmaximal, complemented
12#24737C214yesmaximal, complemented
13#24735C214yesmaximal, complemented
14#21951C211yesmaximal, characteristic, complemented
15#21861C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3105—216no—
21C2×#484C2212yes—
22C2×#481C2212yes—
23C2×#478C2214yes—
24C2×#463C2214yes—
25C2×#463C2214yes—
26C2×#461C2212yes—
27C2×#459C2212yes—
28C2×#450C2212yes—
29C2×#447C2212yes—
30C2×C4×SD16C2214yes—
31C2×C4×D8C2214yes—
32C2×#431C2212yes—
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2850C2212yes—
36#2618—21no—
37#2618—21no—
38#2601—22no—
39#2601—22no—
40#2566C2212yescomplemented
41#2565C2212yescomplemented
42#2545C2212yescomplemented
43#2544C2212yescomplemented
44#2349—22no—
45#2349—22no—
46#2349C2214yes—
47#2349C2214yes—
48#2325—21no—
49#2325—21no—
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#385—21no—
53C2×#385—21no—
54C2×#382C2211yescharacteristic
55C2×#380—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C22×#95—21no—
58#2085C2214yescomplemented
59#2084C2212yescomplemented
60#2083C2212yescomplemented
61#2068C2214yescomplemented
62#2067C2214yescomplemented
63#2066C2214yescomplemented
64#2065C2214yescomplemented
65#2043C2212yescomplemented
66#2042C2214yescomplemented
67#2041C2212yescomplemented
68#1991C2214yescomplemented
69#1958C2214yescomplemented
70#1939C2214yescomplemented
71#1939C2214yescomplemented
72#1879C2214yes—
73#1879C2214yes—
74#1877C2214yes—
75#1877C2214yes—
76#1874C2212yes—
77#1874C2212yes—
78#1872C2212yes—
79#1872C2212yes—
80#1805C2214yescomplemented
81#1804C2214yes—
82#1804C2214yescomplemented
83#1802C2212yes—
84#1802C2212yes—
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1797C2212yes—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C24×C4—21no—
91C24×C4—21no—
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96#529—28no—
97#529—28no—
98C2×#126—22no—
99C2×#125C2312yes—
100C2×#124C2312yes—
101C2×#124C2312yes—
102C2×#123—22no—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4—44no—
115C2×C4×D4—44no—
116C2×C4×D4D418yescomplemented
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2314yes—
121C2×C4×D4C2314yes—
122C2×#117—22no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#484—28no—
137#481—28no—
138#478—216no—
139#463—216no—
140#463—48no—
141#461—44no—
142#459—44no—
143#450—28no—
144#447—28no—
145C4×SD16—216no—
146C4×D8—216no—
147#431—28no—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C4×C8C2314yes—
165#398—22no—
166#398—22no—
167#396—22no—
168#396—22no—
169#396—22no—
170#396—22no—
171#394—22no—
172#394—22no—
173#391C2312yes—
174#391C2312yes—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—22no—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2314yes—
186#389C2312yes—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#387—22no—
192#387—22no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386—22no—
197#385—24no—
198#385—24no—
199#385—24no—
200#385—24no—
201#385—22no—
202#385—22no—
203#384C2311yescharacteristic
204#384C2311yescharacteristic
205#382—22no—
206#382C2312yes—
207#382C2314yes—
208#380—21no—
209#380—21no—
210#380—24no—
211#380—24no—
212C4×C4⋊C4D412yes—
213C4×C4⋊C4D412yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—24no—
221C2×#95—24no—
222C2×#95D411yescharacteristic
223C2×#95D411yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2312yes—
229#336C2312yes—
230#336C2314yes—
231C25—21no—
232C22×Q8C2412yes—
233C22×D4—24no—
234C22×D4—24no—
235C22×D4—22no—
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—22no—
240C22×D4C2×D414yes—
241C22×D4C2412yes—
242C22×D4C2412yes—
243C23×C4—21no—
244C23×C4—28no—
245C23×C4—24no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—82no—
257C23×C4—82no—
258C23×C4—82no—
259C23×C4—82no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×SD16—28no—
268C2×SD16—28no—
269C2×SD16—28no—
270C2×D8—28no—
271C2×D8—28no—
272C2×D8—28no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—42no—
276#127—42no—
277#126—48no—
278#125—24no—
279#125—24no—
280#124—24no—
281#124—24no—
282#124—44no—
283#123—48no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286#121—24no—
287#121—24no—
288#121—44no—
289#121—44no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—28no—
295C4×D4—216no—
296C4×D4—24no—
297C4×D4—28no—
298C4×D4—24no—
299C4×D4—44no—
300C4×D4—42no—
301C4×D4—42no—
302C4×D4—44no—
303C4×D4—44no—
304C4×D4—48no—
305C4×D4—48no—
306C4×D4C2×D4116yes—
307#117—48no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4—82no—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D414yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4—44no—
359C2×C42—22no—
360C2×C42—42no—
361C2×C42—42no—
362C2×C42C2×D412yes—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D414yes—
367C2×C42#4114yes—
368C2×C42C2412yes—
369C2×C42C2411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2412yes—
372#107—28no—
373#106—28no—
374C4⋊C8—28no—
375C4⋊C8—42no—
376C4⋊C8—42no—
377C4⋊C8—44no—
378#103—28no—
379#103—44no—
380#103—44no—
381#102—28no—
382#102—28no—
383#102—28no—
384#102—44no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388C4×C8—28no—
389#95—24no—
390#95—24no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95C2×D412yes—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—22no—
408C24—22no—
409C24—42no—
410C24—42no—
411C24—42no—
412C24—42no—
413C24C22×D411yescharacteristic
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—216no—
421C2×D4—216no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—22no—
429C2×D4—24no—
430C2×D4—42no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—81no—
436C2×D4—81no—
437C2×D4—81no—
438C2×D4—81no—
439C2×D4C22×D414yes—
440C2×D4H(5,𝔽2)14yes—
441C2×D42−1+414yes—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—216no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—82no—
488C22×C4—84no—
489C22×C4—816no—
490C22×C4—816no—
491C22×C4—816no—
492C22×C4#12114yes—
493C22×C4#12114yes—
494C22×C4#13514yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2511yescharacteristic
507SD16—416no—
508D8—416no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—81no—
537C4⋊C4—81no—
538C4⋊C4—84no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—81no—
542C4⋊C4—81no—
543C4⋊C4C22×D414yes—
544C22⋊C4—216no—
545C22⋊C4—48no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C42—24no—
560C42—22no—
561C42—22no—
562C42—41no—
563C42—41no—
564C42—42no—
565C42—44no—
566C42#12118yes—
567C42C22×D414yes—
568C42C2×#4114yes—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—216no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—48no—
578C23—48no—
579C23—48no—
580C23—48no—
581C23—48no—
582C23—42no—
583C23—42no—
584C23—48no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23C2×#12112yes—
595C23C2×#13512yes—
596C23#57512yes—
597C23C23×D411yescharacteristic
598Q8—44no—
599Q8—44no—
600D4—216no—
601D4—24no—
602D4—24no—
603D4—24no—
604D4—44no—
605D4—44no—
606D4—88no—
607D4—162no—
608D4—162no—
609D4#53718yes—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—28no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—816no—
658C2×C4—88no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4—82no—
662C2×C4—82no—
663C2×C4—816no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4C2×#12114yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#12114yes—
669C2×C4C2×#13512yes—
670C2×C4C2×#13512yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)12yes—
674C2×C4C2×2−1+412yes—
675C8—44no—
676C8—44no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—28no—
684C22—432no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22#303514yes—
699C22C22×#12111yescharacteristic
700C22#348411yescharacteristic
701C22#348611yescharacteristic
702C22C22×#13511yescharacteristic
703C22C2×#57511yescharacteristic
704C22#358511yescharacteristic
705C22#358511yescharacteristic
706C4—24no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—24no—
711C4—44no—
712C4—42no—
713C4—42no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4C2×#53712yes—
722C4#349212yes—
723C2—24no—
724C2—48no—
725C2—162no—
726C2#3522311yescharacteristic
727C2C2×#303511yescharacteristic
728C2#6042311yescharacteristic
729C2#6156111yescharacteristic
730C2#6158111yescharacteristic
731C2#6181911yescharacteristic
732C2#6183111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32902C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3105C2semidirect
2. #35103C2semidirect
3. C2×#2349C2semidirect
4. #32908C2semidirect
5. #32905C2semidirect
7. #29761C2semidirect
8. #29757C2semidirect
9. #29755C2semidirect
10. #24822C2semidirect
11. #24820C2semidirect
12. #24737C2semidirect
13. #24735C2semidirect
14. #21951C2semidirect
15. #21861C2semidirect
40. #2566C22semidirect
41. #2565C22semidirect
42. #2545C22semidirect
43. #2544C22semidirect
58. #2085C22semidirect
59. #2084C22semidirect
60. #2083C22semidirect
61. #2068C22semidirect
62. #2067C22semidirect
63. #2066C22semidirect
64. #2065C22semidirect
65. #2043C22semidirect
66. #2042C22semidirect
67. #2041C22semidirect
68. #1991C22semidirect
69. #1958C22semidirect
70. #1939C22semidirect
71. #1939C22semidirect
80. #1805C22semidirect
82. #1804C22semidirect
116. C2×C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31052
#351032
C2×#23492
#329082
#329052
#329022
#297612
#297572
#297552
#248222
#248202
#247372
#247352
#219512
#218612