Finite Group Databasebeta

← #6470698

Subgroups — #6470698

Order 512.

2244 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3196C212yesmaximal, complemented
2#35032C211yesmaximal, characteristic
3C2×#2360C211yesmaximal, characteristic, complemented
4#33110C212yesmaximal, complemented
5#32906C211yesmaximal, characteristic, complemented
6#32903C211yesmaximal, characteristic, complemented
7#29757C211yesmaximal, characteristic, complemented
8#29755C211yesmaximal, characteristic, complemented
9#29719C212yesmaximal, complemented
10#25222C214yesmaximal, complemented
11#25220C214yesmaximal, complemented
12#25098C214yesmaximal, complemented
13#25096C214yesmaximal, complemented
14#21934C211yesmaximal, characteristic, complemented
15#20909C212yesmaximal, complemented
16C2×#525C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#3196—216no—
21C2×#484C2212yes—
22C2×#481C2212yes—
23C2×#485C2214yes—
24C2×#470C2214yes—
25C2×#468C2214yes—
26C2×#462C2212yescomplemented
27C2×#460C2212yescomplemented
28C2×#452C2214yes—
29C2×#451C2212yes—
30C2×#448C2212yes—
31C2×#417C2214yes—
32C2×#407C2212yes—
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#2360—22no—
45#2360—22no—
46#2360C2214yes—
47#2360C2214yes—
48#2325—21no—
49#2325—21no—
50C2×#391C2211yescharacteristic
51C2×#389—21no—
52C2×#389C2211yescharacteristic
53C2×#387—21no—
54C2×#385—21no—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4—21no—
57C2×C4×C22⋊C4C2211yescharacteristic
58#2084C2212yescomplemented
59#2083C2212yescomplemented
60#2074C2214yescomplemented
61#2068C2214yescomplemented
62#2065C2214yescomplemented
63#2056C2214yescomplemented
64#2055C2214yescomplemented
65#2043C2212yescomplemented
66#2041C2212yescomplemented
67#2015C2214yescomplemented
68#1983C2214yescomplemented
69#1949C2214yescomplemented
70#1927C2214yescomplemented
71#1925C2214yescomplemented
72#1880C2214yescomplemented
73#1878C2214yescomplemented
74#1874C2214yes—
75#1874C2212yes—
76#1874C2212yes—
77#1872C2214yes—
78#1872C2212yes—
79#1872C2212yes—
80#1851C2214yes—
81#1851C2214yes—
82#1803C2212yescomplemented
83#1803C2212yescomplemented
84#1797C2212yes—
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1736C2214yes—
88C22×#41C2312yes—
89C23×Q8C2311yescharacteristic
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94C22×M16C2314yes—
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#124C2312yes—
98#525—28no—
99#525—28no—
100C2×#123—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8D418yescomplemented
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×#121—21no—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#120—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—44no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—44no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2314yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C22⋊C4—21no—
131C22×C22⋊C4D411yescharacteristic
132C22×C22⋊C4D411yescharacteristic
133C22×C42—21no—
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#485—216no—
138#484—28no—
139#481—28no—
140#470—216no—
141#468—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#417—48no—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161#407—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#389—24no—
186#389—24no—
187#389C2312yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#387—22no—
194#387—22no—
195#387—24no—
196#387—24no—
197#386—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#385—24no—
202#385—24no—
203#385—22no—
204#385—22no—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#382C2312yes—
208#380—21no—
209#380—21no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—24no—
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—24no—
254C23×C4—22no—
255C23×C4—42no—
256C23×C4—42no—
257C23×C4—44no—
258C23×C4—44no—
259C23×C4—82no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D414yes—
266C23×C4C2×D411yescharacteristic
267C23×C4#4114yes—
268C23×C4C2411yescharacteristic
269C23×C4C2412yes—
270C2×Q16—28no—
271C2×Q16—44no—
272C2×Q16—44no—
273C2×SD16—28no—
274C2×SD16—44no—
275C2×SD16—44no—
276C2×M16—28no—
277C2×M16—44no—
278C2×M16—44no—
279C22×C8C2414yes—
280C22×C8C2414yes—
281C4⋊Q8—42no—
282#127—42no—
283#124—44no—
284#123—48no—
285C22⋊Q8—216no—
286C22⋊Q8—28no—
287C22⋊Q8—24no—
288C22⋊Q8—24no—
289C22⋊Q8—24no—
290C22⋊Q8—24no—
291C22⋊Q8C2×D4116yes—
292#121—24no—
293#121—24no—
294#121—24no—
295#121—24no—
296#121—44no—
297#121—44no—
298#120—48no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—42no—
302C4×Q8—42no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×D4—48no—
306C4×D4—48no—
307#117—24no—
308#117—24no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—48no—
333C2×C4⋊C4—44no—
334C2×C4⋊C4—44no—
335C2×C4⋊C4—82no—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—24no—
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—42no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42—44no—
368C2×C42—82no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2412yes—
375#107—28no—
376#106—28no—
377C4⋊C8—44no—
378C4⋊C8—42no—
379C4⋊C8—42no—
380#103—28no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—44no—
385#102—28no—
386#102—28no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8—24no—
390C22⋊C8—28no—
391#95—22no—
392#95—22no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405#95C2×D412yes—
406C24—22no—
407C24—22no—
408C24—22no—
409C24—41no—
410C24—41no—
411C24—41no—
412C24—41no—
413C24#12114yes—
414C24C22×D412yes—
415C24C22×D411yescharacteristic
416#41—416no—
417#41—88no—
418C2×Q8—24no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—24no—
428C2×Q8—48no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—48no—
432C2×Q8C22×D414yes—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—42no—
438C2×D4—44no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—81no—
445C2×D4—81no—
446C2×D4—81no—
447C2×D4—81no—
448C2×D4—82no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4—82no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
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—22no—
468C22×C4—22no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—41no—
476C22×C4—42no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—41no—
483C22×C4—48no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—48no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—84no—
498C22×C4—816no—
499C22×C4#12114yes—
500C22×C4#12118yes—
501C22×C4#13714yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D412yes—
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2×#4112yes—
513C22×C4C2×#4114yes—
514C22×C4C2511yescharacteristic
515Q16—88no—
516SD16—88no—
517M16—88no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C4⋊C4—24no—
525C4⋊C4—216no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—81no—
543C4⋊C4—81no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—81no—
547C4⋊C4—81no—
548C4⋊C4C22×D414yes—
549C4⋊C4H(5,𝔽2)14yes—
550C22⋊C4—216no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—42no—
554C22⋊C4—42no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C22⋊C4—82no—
566C42—22no—
567C42—22no—
568C42—24no—
569C42—42no—
570C42—41no—
571C42—41no—
572C42—44no—
573C42—44no—
574C42—44no—
575C42—84no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—28no—
585C23—22no—
586C23—22no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23C2×#12114yes—
601C23C2×#12112yes—
602C23C2×#13712yes—
603C23#57512yes—
604C23C23×D411yescharacteristic
605Q8—24no—
606Q8—24no—
607Q8—48no—
608Q8—42no—
609Q8—42no—
610Q8—44no—
611Q8—44no—
612D4—44no—
613D4—44no—
614D4—88no—
615D4—88no—
616D4—88no—
617D4—162no—
618D4—162no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—44no—
649C2×C4—48no—
650C2×C4—44no—
651C2×C4—48no—
652C2×C4—48no—
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—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—88no—
672C2×C4—82no—
673C2×C4—82no—
674C2×C4—816no—
675C2×C4—84no—
676C2×C4C2×#12114yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#13712yes—
679C2×C4C2×#13712yes—
680C2×C4C23×D411yescharacteristic
681C2×C4C22×#4111yescharacteristic
682C2×C4C2×H(5,𝔽2)12yes—
683C2×C4#58512yes—
684C8—44no—
685C8—44no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—416no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—42no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22#303714yes—
708C22C22×#12111yescharacteristic
709C22#348411yescharacteristic
710C22#348511yescharacteristic
711C22C22×#13711yescharacteristic
712C22C2×#57511yescharacteristic
713C22#358511yescharacteristic
714C22#358611yescharacteristic
715C4—22no—
716C4—22no—
717C4—24no—
718C4—22no—
719C4—44no—
720C4—42no—
721C4—42no—
722C4—41no—
723C4—41no—
724C4—44no—
725C4—44no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4#345912yes—
731C4#349612yes—
732C2—24no—
733C2—44no—
734C2—162no—
735C2#3498411yescharacteristic
736C2C2×#303711yescharacteristic
737C2#6042511yescharacteristic
738C2#6157411yescharacteristic
739C2#6158111yescharacteristic
740C2#6171111yescharacteristic
741C2#6171911yescharacteristic
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×#3196C2semidirect
3. C2×#2360C2semidirect
4. #33110C2semidirect
5. #32906C2semidirect
6. #32903C2semidirect
7. #29757C2semidirect
8. #29755C2semidirect
9. #29719C2semidirect
10. #25222C2semidirect
11. #25220C2semidirect
12. #25098C2semidirect
13. #25096C2semidirect
14. #21934C2semidirect
15. #20909C2semidirect
26. C2×#462C22semidirect
27. C2×#460C22semidirect
58. #2084C22semidirect
59. #2083C22semidirect
60. #2074C22semidirect
61. #2068C22semidirect
62. #2065C22semidirect
63. #2056C22semidirect
64. #2055C22semidirect
65. #2043C22semidirect
66. #2041C22semidirect
67. #2015C22semidirect
68. #1983C22semidirect
69. #1949C22semidirect
70. #1927C22semidirect
71. #1925C22semidirect
72. #1880C22semidirect
73. #1878C22semidirect
82. #1803C22semidirect
83. #1803C22semidirect
103. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31962
#350322
C2×#23602
#331102
#329062
#329032
#297572
#297552
#297192
#252222
#252202
#250982
#250962
#219342
#209092