Finite Group Databasebeta

← #6470690

Subgroups — #6470690

Order 512.

2190 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3190C212yesmaximal, complemented
2#35032C211yesmaximal, characteristic
3C2×#2362C211yesmaximal, characteristic, complemented
4#33111C212yesmaximal, complemented
5#32906C211yesmaximal, characteristic, complemented
6#32903C211yesmaximal, characteristic, complemented
7#29756C211yesmaximal, characteristic, complemented
8#29754C211yesmaximal, characteristic, complemented
9#29719C212yesmaximal, complemented
10#25203C214yesmaximal, complemented
11#25201C214yesmaximal, complemented
12#25079C214yesmaximal, complemented
13#25077C214yesmaximal, complemented
14#21930C211yesmaximal, characteristic, complemented
15#20908C212yesmaximal, complemented
16C2×#525C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20#3190—216no—
21C2×#485C2214yes—
22C2×#483C2212yes—
23C2×#480C2212yes—
24C2×#467C2214yes—
25C2×#465C2214yes—
26C2×#462C2212yescomplemented
27C2×#460C2212yescomplemented
28C2×#452C2214yes—
29C2×#451C2212yes—
30C2×#448C2212yes—
31C2×#416C2214yes—
32C22×C22⋊C8C2212yes—
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2850C2212yes—
36#2617—21no—
37#2617—21no—
38#2602—22no—
39#2602—22no—
40#2500C2212yes—
41#2499C2212yes—
42#2432C2212yes—
43#2431C2212yes—
44#2362—22no—
45#2362—22no—
46#2362C2214yes—
47#2362C2214yes—
48#2325—21no—
49#2325—21no—
50C2×#391C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#388—21no—
54C2×#387—21no—
55C2×C4×C4⋊C4—21no—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58#2084C2212yescomplemented
59#2083C2212yescomplemented
60#2074C2214yescomplemented
61#2067C2214yescomplemented
62#2066C2214yescomplemented
63#2056C2214yescomplemented
64#2055C2214yescomplemented
65#2043C2212yescomplemented
66#2041C2212yescomplemented
67#2015C2214yescomplemented
68#1983C2214yescomplemented
69#1949C2214yescomplemented
70#1928C2214yescomplemented
71#1926C2214yescomplemented
72#1880C2214yescomplemented
73#1878C2214yescomplemented
74#1875C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1873C2214yes—
78#1872C2212yes—
79#1872C2212yes—
80#1850C2214yes—
81#1850C2214yes—
82#1803C2212yescomplemented
83#1803C2212yescomplemented
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1735C2214yes—
88C22×#41C2312yes—
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×Q16C2314yes—
92C22×SD16C2314yes—
93C23×C8C2314yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124C2312yes—
97#525—28no—
98#525—28no—
99C2×#123—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8D418yescomplemented
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2314yes—
106C2×#121—21no—
107C2×#121—21no—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#120—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—21no—
113C2×C4×Q8—44no—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4—22no—
117C2×C4×D4—22no—
118C2×#117—22no—
119C2×#117—44no—
120C2×#117C2312yes—
121C2×#117C2314yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#485—216no—
138#483—28no—
139#480—28no—
140#467—216no—
141#465—216no—
142#462—44no—
143#460—44no—
144#452—216no—
145#451—28no—
146#448—28no—
147C2×#107C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152#416—48no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8—28no—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×C22⋊C8C2314yes—
165#397—22no—
166#397—22no—
167#394—22no—
168#394—22no—
169#394—22no—
170#394—22no—
171#392—22no—
172#392—22no—
173#391—22no—
174#391C2314yes—
175#391C2312yes—
176#391C2312yes—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389C2312yes—
186#389C2312yes—
187#388—24no—
188#388—24no—
189#388—24no—
190#388—24no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—24no—
198#387—24no—
199#386—22no—
200#386—22no—
201#386—22no—
202#386—22no—
203#385—22no—
204#385—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382C2312yes—
208#380—21no—
209#380—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—24no—
212C4×C4⋊C4—24no—
213C4×C4⋊C4C2314yes—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C4×C22⋊C4C2312yes—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95—21no—
221C2×#95—21no—
222C2×#95D411yescharacteristic
223C2×#95D411yescharacteristic
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231C25C2×D411yescharacteristic
232C2×#41—24no—
233C2×#41—24no—
234C2×#41—44no—
235C2×#41—44no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—42no—
239C22×Q8—42no—
240C22×Q8C2×D414yes—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4C2412yes—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D414yes—
262C23×C4C2×D411yescharacteristic
263C23×C4#4114yes—
264C23×C4C2411yescharacteristic
265C23×C4C2412yes—
266C2×Q16—28no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—28no—
270C2×SD16—44no—
271C2×SD16—44no—
272C22×C8—28no—
273C22×C8—44no—
274C22×C8—44no—
275C22×C8C2414yes—
276C22×C8C2414yes—
277C4⋊Q8—42no—
278#127—42no—
279#124—44no—
280#123—48no—
281C22⋊Q8—216no—
282C22⋊Q8—28no—
283C22⋊Q8—24no—
284C22⋊Q8—24no—
285C22⋊Q8—24no—
286C22⋊Q8—24no—
287C22⋊Q8C2×D4116yes—
288#121—24no—
289#121—24no—
290#121—24no—
291#121—24no—
292#121—44no—
293#121—44no—
294#120—48no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—42no—
298C4×Q8—42no—
299C4×Q8—44no—
300C4×Q8—44no—
301C4×D4—48no—
302C4×D4—48no—
303#117—24no—
304#117—24no—
305#117—28no—
306#117—44no—
307#117—48no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D414yes—
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
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—42no—
352C2×C22⋊C4—44no—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—42no—
363C2×C42—42no—
364C2×C42—44no—
365C2×C42—44no—
366C2×C42—44no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D412yes—
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373#107—28no—
374#106—28no—
375C4⋊C8—44no—
376C4⋊C8—42no—
377C4⋊C8—42no—
378#103—28no—
379#103—28no—
380#103—28no—
381#103—28no—
382#103—44no—
383#102—28no—
384#102—28no—
385#102—44no—
386C22⋊C8—24no—
387C22⋊C8—24no—
388C22⋊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—22no—
405C24—22no—
406C24—22no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24#12114yes—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414#41—416no—
415#41—88no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—48no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—48no—
430C2×Q8C22×D414yes—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—42no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—42no—
442C2×D4—81no—
443C2×D4—81no—
444C2×D4—81no—
445C2×D4—81no—
446C2×D4—82no—
447C2×D4—82no—
448C2×D4—82no—
449C2×D4—82no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4#12114yes—
494C22×C4#12118yes—
495C22×C4#13514yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D412yes—
503C22×C4C22×D414yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C2×#4112yes—
507C22×C4C2×#4114yes—
508C22×C4C2511yescharacteristic
509Q16—88no—
510SD16—88no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—88no—
518C4⋊C4—24no—
519C4⋊C4—216no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—81no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4C22×D414yes—
542C4⋊C4H(5,𝔽2)14yes—
543C22⋊C4—216no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—48no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—22no—
561C42—22no—
562C42—24no—
563C42—42no—
564C42—41no—
565C42—41no—
566C42—48no—
567C42—44no—
568C42—44no—
569C42—48no—
570C42—44no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—28no—
582C23—44no—
583C23—44no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—82no—
592C23—82no—
593C23—82no—
594C23—82no—
595C23C2×#12114yes—
596C23C2×#12112yes—
597C23C2×#13512yes—
598C23#57512yes—
599C23C23×D411yescharacteristic
600Q8—24no—
601Q8—24no—
602Q8—48no—
603Q8—42no—
604Q8—42no—
605Q8—44no—
606Q8—44no—
607D4—44no—
608D4—44no—
609D4—88no—
610D4—88no—
611D4—88no—
612D4—162no—
613D4—162no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—24no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—44no—
642C2×C4—48no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—48no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—48no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—88no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—84no—
670C2×C4C2×#12114yes—
671C2×C4C2×#12112yes—
672C2×C4C2×#13512yes—
673C2×C4C2×#13512yes—
674C2×C4C23×D411yescharacteristic
675C2×C4C22×#4111yescharacteristic
676C2×C4C2×H(5,𝔽2)12yes—
677C2×C4#58512yes—
678C8—44no—
679C8—44no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—88no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22#303514yes—
702C22C22×#12111yescharacteristic
703C22#348411yescharacteristic
704C22#348511yescharacteristic
705C22C22×#13511yescharacteristic
706C22C2×#57511yescharacteristic
707C22#358611yescharacteristic
708C22#358611yescharacteristic
709C4—22no—
710C4—22no—
711C4—24no—
712C4—22no—
713C4—42no—
714C4—42no—
715C4—41no—
716C4—41no—
717C4—44no—
718C4—44no—
719C4—44no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4#345912yes—
725C4#349612yes—
726C2—24no—
727C2—44no—
728C2—162no—
729C2#3498411yescharacteristic
730C2C2×#303511yescharacteristic
731C2#6042411yescharacteristic
732C2#6156411yescharacteristic
733C2#6157511yescharacteristic
734C2#6170111yescharacteristic
735C2#6171111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3190C2semidirect
3. C2×#2362C2semidirect
4. #33111C2semidirect
5. #32906C2semidirect
6. #32903C2semidirect
7. #29756C2semidirect
8. #29754C2semidirect
9. #29719C2semidirect
10. #25203C2semidirect
11. #25201C2semidirect
12. #25079C2semidirect
13. #25077C2semidirect
14. #21930C2semidirect
15. #20908C2semidirect
26. C2×#462C22semidirect
27. C2×#460C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #2074C22semidirect
61. #2067C22semidirect
62. #2066C22semidirect
63. #2056C22semidirect
64. #2055C22semidirect
65. #2043C22semidirect
66. #2041C22semidirect
67. #2015C22semidirect
68. #1983C22semidirect
69. #1949C22semidirect
70. #1928C22semidirect
71. #1926C22semidirect
72. #1880C22semidirect
73. #1878C22semidirect
82. #1803C22semidirect
83. #1803C22semidirect
102. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31902
#350322
C2×#23622
#331112
#329062
#329032
#297562
#297542
#297192
#252032
#252012
#250792
#250772
#219302
#209082