Finite Group Databasebeta

← #6355361

Subgroups — #6355361

Order 512.

1926 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×#3226C211yesmaximal, characteristic, complemented
2#60415C211yesmaximal, characteristic
3C2×#3039C211yesmaximal, characteristic, complemented
4#60277C211yesmaximal, characteristic, complemented
5#19452C218yesmaximal, complemented
6#19440C214yesmaximal, complemented
7#19428C214yesmaximal, complemented
8#19301C214yesmaximal, complemented
9#19296C214yesmaximal, complemented
10#17813C211yesmaximal, characteristic, complemented
11#17704C211yesmaximal, characteristic, complemented
12C2×#1546C211yesmaximal, characteristic, complemented
13C2×#575—21no—
14C22×#137—21no—
15C2×#532C2211yescharacteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18#3226—22no—
19#3226—22no—
20#3226C2214yes—
21#3226C2214yescomplemented
22C2×#500C2212yescomplemented
23C2×#486C2212yescomplemented
24C2×#480—22no—
25C2×#474—22no—
26#3040—22no—
27#3039—22no—
28#3039—22no—
29#3039C2212yes—
30#3039C2214yes—
31#3039C2214yescomplemented
32#3038C2212yes—
33#3037—21no—
34#3037C2212yes—
35#3036—21no—
36#3036C2212yes—
37#3035—22no—
38#3035C2214yes—
39C2×#462—21no—
40C2×#462C2212yescomplemented
41C2×#461—21no—
42C2×#461C2212yescomplemented
43#2989—22no—
44#2989C2214yescomplemented
45#2983C2212yescomplemented
46#2975C2212yescomplemented
47#2973—22no—
48#2965—22no—
49#2965C2214yes—
50#2920—22no—
51C2×#431C2211yescharacteristic
52C2×#431C2211yescharacteristic
53C2×#423C2211yescharacteristic
54C2×#423C2211yescharacteristic
55#2896—21no—
56C2×#419—21no—
57#1702C22116yescomplemented
58#1701C22116yescomplemented
59#1694C22116yescomplemented
60#1693C22116yescomplemented
61#1570C2214yescomplemented
62#1570C2214yescomplemented
63#1566C2214yescomplemented
64#1547C2214yescomplemented
65#1546C2214yescomplemented
66#1546C2214yescomplemented
67#1546C2212yescomplemented
68#1545C2212yescomplemented
69#1540C2214yes—
70C2×#332C2214yescomplemented
71C22×#41—21no—
72C22×#41D411yescharacteristic
73C23×Q8—21no—
74#575—48no—
75C2×#137—24no—
76C2×#137—24no—
77C2×#137—22no—
78C2×#137—22no—
79C2×#137—48no—
80C2×AGL(1,ℤ/8)—22no—
81C2×AGL(1,ℤ/8)—22no—
82C2×#135—24no—
83C22×Q16—22no—
84C22×SD16—22no—
85C22×M16—21no—
86C22×M16—21no—
87#532—21no—
88#532—21no—
89#532—21no—
90#532—21no—
91#532—21no—
92#532—21no—
93#532C2312yes—
94#532C2312yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127C2311yescharacteristic
109C2×#124—21no—
110C2×#121—21no—
111C2×#121—21no—
112C4×#41—21no—
113C4×#41—21no—
114C4×#41—21no—
115C4×#41—21no—
116C4×#41D412yes—
117C4×#41D412yes—
118C4×#41C2312yes—
119C4×#41C2312yes—
120C2×C4×Q8D411yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4D411yescharacteristic
127C2×C4×D4C2311yescharacteristic
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134#500—24no—
135#500C2318yes—
136#486—24no—
137#486C2318yes—
138#480—28no—
139#480—28no—
140#474—28no—
141#474—28no—
142#464—24no—
143#464—24no—
144#463—24no—
145#463—24no—
146#462—24no—
147#462—24no—
148#462—24no—
149#462—22no—
150#462C2314yes—
151#462C2318yes—
152#461—24no—
153#461—24no—
154#461—24no—
155#461—22no—
156#461C2314yes—
157#461C2318yes—
158#460—22no—
159#460C2314yes—
160#459—22no—
161#459C2314yes—
162#446—24no—
163#446C2318yes—
164#436—24no—
165#436C2318yes—
166#435—28no—
167#435—28no—
168#432—24no—
169#432—24no—
170#431—21no—
171#431—21no—
172#431—24no—
173#431D412yes—
174#431D412yes—
175#431C2312yes—
176#431C2312yes—
177#431C2312yes—
178#431C2312yes—
179C2×#107—24no—
180#423—21no—
181#423—21no—
182#423—21no—
183#423—21no—
184#423—24no—
185#423C2312yes—
186#423C2312yes—
187#423C2312yes—
188#423C2312yes—
189C2×C4⋊C8—21no—
190C2×C4⋊C8—21no—
191C2×C4⋊C8D412yes—
192C2×C4⋊C8C2312yes—
193C2×C4⋊C8C2312yes—
194C2×C4⋊C8C2312yes—
195#419—24no—
196#419—24no—
197#419—22no—
198#418—22no—
199#417—22no—
200#417—22no—
201#416—24no—
202C2×#103—22no—
203C2×#103—22no—
204C2×#102—24no—
205#407—24no—
206C2×C22⋊C8—21no—
207C2×C22⋊C8D412yes—
208C4×M16—24no—
209C2×C4×C8—21no—
210C2×C4×C8C2312yes—
211#334C23116yes—
212#332C23116yes—
213#332C2318yes—
214#331C2318yes—
215C2×#41—21no—
216C2×#41—21no—
217C2×#41—24no—
218C2×#41—24no—
219C2×#41—22no—
220C2×#41—22no—
221C2×#41—41no—
222C2×#41—41no—
223C2×#41—44no—
224C2×#41—44no—
225C2×#41C2×D412yes—
226C2×#41C2×D412yes—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—41no—
232C22×Q8—41no—
233C22×Q8—41no—
234C22×Q8—44no—
235C22×Q8—44no—
236C22×Q8C2×D411yescharacteristic
237C22×D4—21no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—41no—
242C22×D4C2×D411yescharacteristic
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—41no—
247C23×C4—41no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4#4111yescharacteristic
251#137—48no—
252#137—416no—
253#137—416no—
254#137—48no—
255AGL(1,ℤ/8)—48no—
256AGL(1,ℤ/8)—48no—
257#135—416no—
258C2×Q16—28no—
259C2×Q16—24no—
260C2×Q16—48no—
261C2×SD16—24no—
262C2×SD16—28no—
263C2×SD16—24no—
264C2×SD16—48no—
265C2×D8—24no—
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—28no—
271C2×M16—44no—
272C2×M16—44no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8C2×D414yes—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4⋊Q8—22no—
280C4⋊Q8—21no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—21no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—21no—
289C4⋊Q8—41no—
290C4⋊Q8—41no—
291C4⋊Q8—41no—
292C4⋊Q8—41no—
293C4⋊Q8—41no—
294C4⋊Q8—41no—
295C4⋊Q8C2414yes—
296C4⋊Q8C2412yes—
297#127—21no—
298#127—21no—
299#127—21no—
300#127C2412yes—
301#124—22no—
302#124—22no—
303#124—41no—
304#124—41no—
305#121—22no—
306#121—22no—
307#121—22no—
308#121—22no—
309#121—41no—
310#121—41no—
311#121—41no—
312#121—41no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—21no—
316C4×Q8—21no—
317C4×Q8C2×D412yes—
318C4×Q8C2×D412yes—
319C4×Q8C2×D412yes—
320C4×Q8C2412yes—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—22no—
328C4×D4—22no—
329C4×D4—21no—
330C4×D4—21no—
331C4×D4—21no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—21no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—41no—
338C4×D4—41no—
339C4×D4—41no—
340C4×D4—41no—
341C4×D4C2×D412yes—
342C4×D4C2×D412yes—
343C4×D4C2×D412yes—
344C4×D4C2412yes—
345#117—21no—
346#117—21no—
347#117—22no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117C2×D412yes—
352#117C2×D412yes—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—41no—
364C2×C4⋊C4—41no—
365C2×C4⋊C4—41no—
366C2×C4⋊C4—41no—
367C2×C4⋊C4—41no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—41no—
375C2×C22⋊C4—41no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42#4111yescharacteristic
390C2×C42#4111yescharacteristic
391C2×C42#4111yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395#107—216no—
396C4⋊C8—28no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399C4⋊C8—22no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404C4⋊C8C2×D414yes—
405C4⋊C8C2×D414yes—
406C4⋊C8C2414yes—
407C4⋊C8C2414yes—
408C4⋊C8C2414yes—
409#103—28no—
410#103—28no—
411#103—24no—
412#103—24no—
413#102—216no—
414#102—24no—
415#102—24no—
416C22⋊C8—22no—
417C22⋊C8—22no—
418C22⋊C8—28no—
419C22⋊C8C2×D414yes—
420C22⋊C8C2×D414yes—
421#97—28no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8C2×D414yes—
425C4×C8C2414yes—
426C24—21no—
427C24—41no—
428C24#12111yescharacteristic
429#41—24no—
430#41—24no—
431#41—24no—
432#41—24no—
433#41—44no—
434#41—44no—
435#41—44no—
436#41—44no—
437#41—44no—
438#41—44no—
439#41—84no—
440#41—84no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—24no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—42no—
454C2×Q8—44no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—84no—
459C2×Q8—84no—
460C2×Q8—84no—
461C2×Q8C22×D412yes—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—41no—
475C2×D4—41no—
476C2×D4—41no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—82no—
480C2×D4—84no—
481C2×D4C22×D412yes—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—41no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4#12111yescharacteristic
505C22×C4#12111yescharacteristic
506C22×C4#12111yescharacteristic
507C22×C4#12111yescharacteristic
508C22×C4#12111yescharacteristic
509C22×C4#12111yescharacteristic
510C22×C4#12111yescharacteristic
511C22×C4#12111yescharacteristic
512C22×C4#12111yescharacteristic
513C22×C4AGL(1,ℤ/8)14yes—
514C22×C4AGL(1,ℤ/8)14yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C2×#4111yescharacteristic
522C22×C4C2×#4111yescharacteristic
523C22×C4C2×#4111yescharacteristic
524Q16—416no—
525Q16—48no—
526SD16—416no—
527SD16—48no—
528SD16—48no—
529D8—48no—
530M16—44no—
531M16—44no—
532M16—44no—
533M16—44no—
534M16—88no—
535C2×C8—24no—
536C2×C8—24no—
537C2×C8—24no—
538C2×C8—44no—
539C2×C8—44no—
540C2×C8—44no—
541C2×C8—82no—
542C2×C8C22×D418yes—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—24no—
552C4⋊C4—22no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—41no—
560C4⋊C4—41no—
561C4⋊C4C22×D412yes—
562C4⋊C4C22×D412yes—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42#12111yescharacteristic
580C42#12111yescharacteristic
581C42#12111yescharacteristic
582C42#12111yescharacteristic
583C42#12111yescharacteristic
584C42#12111yescharacteristic
585C42C22×D411yescharacteristic
586C42C22×D411yescharacteristic
587C42C22×D411yescharacteristic
588C42C22×D411yescharacteristic
589C42C22×D411yescharacteristic
590C42C22×D411yescharacteristic
591C42C2×#4111yescharacteristic
592C42C2×#4111yescharacteristic
593C42C2×#4111yescharacteristic
594C42C2511yescharacteristic
595C23—22no—
596C23—41no—
597C23—41no—
598C23—42no—
599C23—42no—
600C23—81no—
601C23—81no—
602C23—82no—
603C23—82no—
604C23—82no—
605C23#46914yes—
606C23C2×#12111yescharacteristic
607C23C2×#12111yescharacteristic
608C23C2×#12111yescharacteristic
609Q8—22no—
610Q8—22no—
611Q8—42no—
612Q8—44no—
613Q8—42no—
614Q8—44no—
615Q8—42no—
616Q8—84no—
617Q8—84no—
618Q8—84no—
619Q8—82no—
620D4—22no—
621D4—22no—
622D4—44no—
623D4—44no—
624D4—44no—
625D4—42no—
626D4—42no—
627D4—42no—
628D4—82no—
629D4—82no—
630D4—84no—
631D4—84no—
632D4—82no—
633D4—82no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—82no—
657C2×C4—82no—
658C2×C4—81no—
659C2×C4—81no—
660C2×C4—82no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4—84no—
665C2×C4—84no—
666C2×C4#46914yes—
667C2×C4#46914yes—
668C2×C4#46914yes—
669C2×C4C2×#12111yescharacteristic
670C2×C4C2×#12111yescharacteristic
671C2×C4C2×#12111yescharacteristic
672C2×C4C2×#12111yescharacteristic
673C2×C4C2×#12111yescharacteristic
674C2×C4C2×#12111yescharacteristic
675C2×C4C2×#12111yescharacteristic
676C2×C4C2×#12111yescharacteristic
677C2×C4C2×#12111yescharacteristic
678C2×C4C2×AGL(1,ℤ/8)12yes—
679C2×C4C2×AGL(1,ℤ/8)12yes—
680C2×C4C2×AGL(1,ℤ/8)12yes—
681C2×C4C2×AGL(1,ℤ/8)12yes—
682C2×C4C2×AGL(1,ℤ/8)12yes—
683C2×C4C2×AGL(1,ℤ/8)12yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C8—28no—
688C8—44no—
689C8—44no—
690C8—44no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—44no—
698C22—42no—
699C22—42no—
700C22—42no—
701C22—84no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—164no—
706C22C2×#46912yes—
707C22C2×#46912yes—
708C22C2×#46912yes—
709C22C22×#12111yescharacteristic
710C4—21no—
711C4—21no—
712C4—21no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—82no—
720C4—84no—
721C4C2×#46912yes—
722C4C2×#46912yes—
723C4C2×#46912yes—
724C4#306912yes—
725C4C22×AGL(1,ℤ/8)11yescharacteristic
726C4C22×AGL(1,ℤ/8)11yescharacteristic
727C4#358612yes—
728C2—22no—
729C2—22no—
730C2—22no—
731C2—82no—
732C2—162no—
733C2#6042711yescharacteristic
734C2C22×#46911yescharacteristic
735C2C2×#306911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#60415C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3226C2semidirect
3. C2×#3039C2semidirect
4. #60277C2semidirect
5. #19452C2semidirect
6. #19440C2semidirect
7. #19428C2semidirect
8. #19301C2semidirect
9. #19296C2semidirect
10. #17813C2semidirect
11. #17704C2semidirect
12. C2×#1546C2semidirect
21. #3226C22semidirect
22. C2×#500C22semidirect
23. C2×#486C22semidirect
31. #3039C22semidirect
40. C2×#462C22semidirect
42. C2×#461C22semidirect
44. #2989C22semidirect
45. #2983C22semidirect
46. #2975C22semidirect
57. #1702C22semidirect
58. #1701C22semidirect
59. #1694C22semidirect
60. #1693C22semidirect
61. #1570C22semidirect
62. #1570C22semidirect
63. #1566C22semidirect
64. #1547C22semidirect
65. #1546C22semidirect
66. #1546C22semidirect
67. #1546C22semidirect
68. #1545C22semidirect
70. C2×#332C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#604152
C2×#30392
#602772
#194522
#194402
#194282
#193012
#192962
#178132
#177042
C2×#15462