Finite Group Databasebeta

← #6481760

Subgroups — #6481760

Order 512.

2152 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3114C212yesmaximal, complemented
2#35103C211yesmaximal, characteristic, complemented
3C2×#2354C211yesmaximal, characteristic, complemented
4#33111C212yesmaximal, complemented
5#33108C211yesmaximal, characteristic, complemented
6#33106C211yesmaximal, characteristic, complemented
7#29718C212yesmaximal, complemented
8#29714C211yesmaximal, characteristic, complemented
9#29712C211yesmaximal, characteristic
10#24690C214yesmaximal, complemented
11#24688C214yesmaximal, complemented
12#24640C214yesmaximal, complemented
13#24638C214yesmaximal, complemented
14#21955C211yesmaximal, characteristic, complemented
15#21867C212yesmaximal, complemented
16C2×#529C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117—21no—
19#3114—216no—
20C2×#483C2212yes—
21C2×#480C2212yes—
22C2×#478C2214yes—
23C2×#463C2214yes—
24C2×#450C2212yes—
25C2×#447C2212yes—
26C2×#442C2214yes—
27C2×#440C2214yes—
28C2×#432C2212yes—
29#2857C2211yescharacteristic
30#2856C2211yescharacteristic
31#2850C2212yes—
32#2618—21no—
33#2618—21no—
34#2601—22no—
35#2601—22no—
36#2566C2212yescomplemented
37#2565C2212yescomplemented
38#2545C2212yescomplemented
39#2544C2212yescomplemented
40#2354—22no—
41#2354—22no—
42#2354C2214yes—
43#2354C2214yes—
44#2325—21no—
45#2325—21no—
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#388—21no—
49C2×#388—21no—
50C2×#384C2211yescharacteristic
51C2×#380—21no—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55#2073C2214yes—
56#2071C2212yes—
57#2071C2212yes—
58#2055C2214yes—
59#2055C2214yes—
60#2054C2214yes—
61#2054C2214yes—
62#2013C2212yes—
63#2012C2214yes—
64#2011C2212yes—
65#1992C2214yescomplemented
66#1983C2214yescomplemented
67#1982C2212yescomplemented
68#1981C2212yescomplemented
69#1959C2214yescomplemented
70#1949C2214yescomplemented
71#1948C2212yescomplemented
72#1947C2212yescomplemented
73#1940C2214yescomplemented
74#1940C2214yescomplemented
75#1928C2214yescomplemented
76#1927C2214yescomplemented
77#1926C2214yescomplemented
78#1925C2214yescomplemented
79#1874C2212yes—
80#1874C2212yes—
81#1872C2212yes—
82#1872C2212yes—
83#1803C2214yescomplemented
84#1802C2214yescomplemented
85#1797C2211yescharacteristic
86#1797C2211yescharacteristic
87#1797C2212yes—
88C23×D4C2311yescharacteristic
89C24×C4—21no—
90C22×SD16C2314yes—
91C22×D8C2314yes—
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94#529—28no—
95#529—28no—
96C2×#126—22no—
97C2×#125C2312yes—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—24no—
109C2×C4×D4D418yescomplemented
110C2×C4×D4C2312yes—
111C2×C4×D4C2314yes—
112C2×#117—24no—
113C2×#117—24no—
114C2×#117—22no—
115C2×#117—48no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4D411yescharacteristic
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42—21no—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#483—28no—
134#480—28no—
135#478—216no—
136#463—216no—
137#450—28no—
138#447—28no—
139#442—216no—
140#440—216no—
141#432—28no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2312yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×#97C2314yes—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#396—22no—
164#396—22no—
165#394—22no—
166#394—22no—
167#391C2312yes—
168#391C2312yes—
169#390—21no—
170#390—21no—
171#390—22no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2314yes—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—24no—
186#388—24no—
187#388—24no—
188#388—24no—
189#387—22no—
190#387—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#386—22no—
195#385—22no—
196#385—22no—
197#384—22no—
198#384C2311yescharacteristic
199#384C2311yescharacteristic
200#384C2314yes—
201#382C2312yes—
202#380—21no—
203#380—21no—
204#380—24no—
205#380—24no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C2×#95—21no—
217C2×#95—21no—
218C2×#95D411yescharacteristic
219C2×#95D411yescharacteristic
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2312yes—
225#336C2312yes—
226#336C2314yes—
227C25—21no—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—42no—
235C22×D4—42no—
236C22×D4C2×D414yes—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—28no—
241C23×C4—24no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×SD16—28no—
255C2×SD16—44no—
256C2×SD16—44no—
257C2×D8—28no—
258C2×D8—44no—
259C2×D8—44no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—42no—
263#127—42no—
264#126—48no—
265#125—24no—
266#125—24no—
267#124—24no—
268#124—24no—
269#124—44no—
270#123—48no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273#121—24no—
274#121—24no—
275#121—44no—
276#121—44no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×D4—216no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—48no—
284C4×D4C2×D4116yes—
285#117—416no—
286#117—416no—
287#117—48no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—82no—
342C2×C22⋊C4—82no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—42no—
348C2×C42—42no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D412yes—
355C2×C42C2×D414yes—
356C2×C42#4114yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2412yes—
361#107—28no—
362#106—28no—
363C4⋊C8—28no—
364C4⋊C8—44no—
365C4⋊C8—42no—
366C4⋊C8—42no—
367#103—28no—
368#103—44no—
369#103—44no—
370#102—28no—
371#102—28no—
372#102—28no—
373#102—44no—
374#102—44no—
375C22⋊C8—24no—
376C22⋊C8—24no—
377#97—28no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—24no—
392C24—22no—
393C24—22no—
394C24—42no—
395C24—42no—
396C24—42no—
397C24—42no—
398C24—42no—
399C24—42no—
400C24C22×D411yescharacteristic
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—48no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—42no—
422C2×D4—81no—
423C2×D4—81no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4C22×D414yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—48no—
453C22×C4—432no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4#12114yes—
467C22×C4#12114yes—
468C22×C4AGL(1,ℤ/8)14yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C2×#4112yes—
480C22×C4C2511yescharacteristic
481SD16—416no—
482D8—416no—
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—84no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4C22×D414yes—
517C22⋊C4—216no—
518C22⋊C4—48no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—41no—
539C42—41no—
540C42—42no—
541C42—48no—
542C42—44no—
543C42—44no—
544C42—44no—
545C42#12118yes—
546C42C22×D414yes—
547C42C2×#4114yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—24no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23C2×#12112yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23#57512yes—
576C23C23×D411yescharacteristic
577Q8—44no—
578Q8—44no—
579D4—28no—
580D4—48no—
581D4—44no—
582D4—44no—
583D4—44no—
584D4—88no—
585D4—162no—
586D4—162no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—28no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—432no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—42no—
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—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—88no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4C2×#12114yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12114yes—
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4C2×AGL(1,ℤ/8)12yes—
645C2×C4C23×D411yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4#58512yes—
648C2×C4#58512yes—
649C8—44no—
650C8—44no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—816no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22#303614yes—
673C22C22×#12111yescharacteristic
674C22#348511yescharacteristic
675C22#348711yescharacteristic
676C22C22×AGL(1,ℤ/8)11yescharacteristic
677C22C2×#57511yescharacteristic
678C22#358511yescharacteristic
679C22#358611yescharacteristic
680C4—24no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—24no—
685C4—42no—
686C4—42no—
687C4—41no—
688C4—41no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4#344912yes—
696C4#349312yes—
697C2—24no—
698C2—84no—
699C2—162no—
700C2#3515011yescharacteristic
701C2C2×#303611yescharacteristic
702C2#6042511yescharacteristic
703C2#6170611yescharacteristic
704C2#6171011yescharacteristic
705C2#6189011yescharacteristic
706C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#29712C2
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×#3114C2semidirect
2. #35103C2semidirect
3. C2×#2354C2semidirect
4. #33111C2semidirect
5. #33108C2semidirect
6. #33106C2semidirect
7. #29718C2semidirect
8. #29714C2semidirect
10. #24690C2semidirect
11. #24688C2semidirect
12. #24640C2semidirect
13. #24638C2semidirect
14. #21955C2semidirect
15. #21867C2semidirect
36. #2566C22semidirect
37. #2565C22semidirect
38. #2545C22semidirect
39. #2544C22semidirect
65. #1992C22semidirect
66. #1983C22semidirect
67. #1982C22semidirect
68. #1981C22semidirect
69. #1959C22semidirect
70. #1949C22semidirect
71. #1948C22semidirect
72. #1947C22semidirect
73. #1940C22semidirect
74. #1940C22semidirect
75. #1928C22semidirect
76. #1927C22semidirect
77. #1926C22semidirect
78. #1925C22semidirect
83. #1803C22semidirect
84. #1802C22semidirect
109. C2×C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31142
#351032
C2×#23542
#331112
#331082
#331062
#297182
#297142
#297122
#246902
#246882
#246402
#246382
#219552
#218672