Finite Group Databasebeta

← #209305

Subgroups — #209305

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2935C211yesmaximal, characteristic
2#23570C211yesmaximal, characteristic
3#21596C211yesmaximal, characteristic, complemented
4C2×#1583C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8949C211yesmaximal, characteristic, complemented
7#8945C211yesmaximal, characteristic, complemented
8#8937C211yesmaximal, characteristic, complemented
9#8849C211yesmaximal, characteristic, complemented
10#8845C211yesmaximal, characteristic, complemented
11#8830C211yesmaximal, characteristic, complemented
12#8313C211yesmaximal, characteristic
13#8206C211yesmaximal, characteristic, complemented
14#8202C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C22×#131—21no—
17C2×#532C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21#2935—24no—
22#2935—24no—
23C2×#436C2211yescharacteristic
24C2×#436C2211yescharacteristic
25C2×#435—21no—
26C2×#435—21no—
27C2×C8×D4C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C2×#408—21no—
31C2×C4×M16C2211yescharacteristic
32C4×C22⋊Q8C2211yescharacteristic
33#2058C411yescharacteristic
34#2041C411yescharacteristic
35#2034C411yescharacteristic
36#2034C411yescharacteristic
37#2024C411yescharacteristic
38#2014C411yescharacteristic
39#2000C411yescharacteristic
40#2000C411yescharacteristic
41#1984C2211yescharacteristic
42#1939C2211yescharacteristic
43#1917—21no—
44#1851—21no—
45#1797C2211yescharacteristic
46#1771—21no—
47#1770C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1753—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1583—24no—
54C8×SD16—24no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41—21no—
72C22×SD16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C2×#131—28no—
75C2×#131—44no—
76C2×#131—44no—
77C22×M16—21no—
78C22×M16—21no—
79C22×M16D411yescharacteristic
80C22×M16D411yescharacteristic
81C23×C8—21no—
82C23×C8—21no—
83C23×C8—21no—
84#532—22no—
85#532—22no—
86#532—22no—
87#532—22no—
88C2×C4⋊Q8C2×C411yescharacteristic
89C2×#127C2×C411yescharacteristic
90C2×#124—21no—
91C2×C22⋊Q8C2×C411yescharacteristic
92C2×#121—21no—
93C2×#121C2×C411yescharacteristic
94C2×#121C2×C411yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×D4—21no—
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C8×Q8—22no—
101C8×Q8—22no—
102C4×SD16—22no—
103C4×SD16—22no—
104C4×SD16—22no—
105C4×SD16—22no—
106#436—22no—
107#436—22no—
108#436—22no—
109#436D414yes—
110#435—28no—
111#435—28no—
112C8×D4—22no—
113C8×D4—22no—
114C8×D4—22no—
115C8×D4D414yes—
116C2×#106C2×C411yescharacteristic
117C2×#106C2×C411yescharacteristic
118C2×C4⋊C8—21no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135#408—28no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8C2×C411yescharacteristic
143C2×C22⋊C8C2×C411yescharacteristic
144C4×M16—22no—
145C4×M16—22no—
146C2×#97—21no—
147C2×#97D411yescharacteristic
148C2×#97D411yescharacteristic
149C2×C4×C8—21no—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156#391C2×C411yescharacteristic
157#389C2×C411yescharacteristic
158#387—21no—
159#385—21no—
160#380C2×C411yescharacteristic
161C4×C4⋊C4—21no—
162C4×C4⋊C4C2311yescharacteristic
163C4×C22⋊C4—21no—
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336C2×C411yescharacteristic
175#336C2×C411yescharacteristic
176#334—22no—
177#334—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183D4⋊C8—22no—
184#322—22no—
185C82—22no—
186C2×#41—24no—
187C2×#41—24no—
188C2×#41—42no—
189C2×#41—42no—
190C2×#41—42no—
191C2×#41—42no—
192C22×Q8—21no—
193C22×Q8C22×C411yescharacteristic
194C22×D4—21no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—21no—
198C22×D4C22×C411yescharacteristic
199C22×D4C22×C411yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C23×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4#4111yescharacteristic
205C2×SD16—22no—
206C2×SD16—22no—
207C2×SD16—22no—
208C2×SD16—22no—
209C2×SD16—22no—
210C2×SD16—41no—
211#131—432no—
212C2×M16—24no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—24no—
216C2×M16—22no—
217C2×M16—22no—
218C2×M16—41no—
219C2×M16—42no—
220C2×M16—42no—
221C2×M16—42no—
222C2×M16—42no—
223C2×M16SD1614yescomplemented
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—24no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—42no—
238C22×C8—42no—
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C22×C411yescharacteristic
242C22×C8C22×C411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249#127—22no—
250#127—22no—
251#124—24no—
252#124—24no—
253C22⋊Q8—22no—
254C22⋊Q8—41no—
255#121—24no—
256#121—24no—
257#121—22no—
258#121—22no—
259#121—22no—
260#121—22no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×D4—24no—
266C4×D4—24no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C2×D412yes—
272C4×D4C2×D412yes—
273C4×D4#4112yes—
274C4×D4#4112yes—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42C2411yescharacteristic
300#106—22no—
301#106—22no—
302C4⋊C8—24no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—42no—
310C4⋊C8—42no—
311C4⋊C8C2×D412yes—
312C4⋊C8C2×D412yes—
313#103—22no—
314#103—22no—
315#102—22no—
316#102—22no—
317#102—22no—
318#102—41no—
319#102—41no—
320#102—41no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C22⋊C8—22no—
326C22⋊C8—41no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329#97—22no—
330#97—22no—
331#97—42no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—42no—
343C4×C8C2×D412yes—
344C4×C8C2×D412yes—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350C24—21no—
351C24—21no—
352C24C4×D411yescharacteristic
353#41—48no—
354#41—48no—
355#41—48no—
356#41—48no—
357C2×Q8—24no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8#13112yescomplemented
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4C4×D412yes—
389C2×D4C4×D412yes—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4C4×D411yescharacteristic
423C22×C4C4×D411yescharacteristic
424C22×C4C4×D411yescharacteristic
425C22×C4C2×SD1612yes—
426C22×C4#13512yes—
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4111yescharacteristic
430SD16—28no—
431SD16—44no—
432M16—28no—
433M16—44no—
434M16—48no—
435M16—48no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—48no—
458C2×C8—41no—
459C2×C8—48no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—48no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8#12012yes—
476C2×C8#12012yes—
477C2×C8#13112yescomplemented
478C2×C8C2×SD1612yes—
479C2×C8C2×SD1612yes—
480C2×C8AGL(1,ℤ/8)12yes—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—81no—
494C4⋊C4C4×D412yes—
495C4⋊C4C4×D412yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—42no—
503C22⋊C4—81no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42C22×D411yescharacteristic
515C42C22×D411yescharacteristic
516C42C2×#4111yescharacteristic
517C42C2×#4111yescharacteristic
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23C4×SD1612yes—
539C23C2×C4×D411yescharacteristic
540Q8—22no—
541Q8—22no—
542Q8—48no—
543Q8—41no—
544Q8—41no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4—22no—
549D4—48no—
550D4—48no—
551D4—48no—
552D4—41no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4C4×SD1612yes—
604C2×C4C2×C4×D411yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×#12011yescharacteristic
607C2×C4#52511yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4#52511yescharacteristic
610C2×C4C2×#13111yescharacteristic
611C2×C4C2×#13111yescharacteristic
612C2×C4#56811yescharacteristic
613C2×C4#56811yescharacteristic
614C2×C4C22×SD1611yescharacteristic
615C2×C4C2×#13511yescharacteristic
616C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
617C2×C4#57511yescharacteristic
618C8—22no—
619C8—22no—
620C8—22no—
621C8—42no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—42no—
626C8—42no—
627C8—41no—
628C8—81no—
629C8—81no—
630C8—81no—
631C8#45014yes—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22C4×#12011yescharacteristic
654C22#293311yescharacteristic
655C22#293511yescharacteristic
656C22C2×C4×SD1611yescharacteristic
657C22C4×AGL(1,ℤ/8)11yescharacteristic
658C22#295611yescharacteristic
659C22#295711yescharacteristic
660C4—22no—
661C4—21no—
662C4—21no—
663C4—41no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—41no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4#292911yescharacteristic
672C4#293211yescharacteristic
673C4#293311yescharacteristic
674C4#293511yescharacteristic
675C4C2×#45011yescharacteristic
676C4#300211yescharacteristic
677C4#310111yescharacteristic
678C4#310511yescharacteristic
679C2—42no—
680C2—82no—
681C2—82no—
682C2#1799711yescharacteristic
683C2#1801111yescharacteristic
684C2#1806711yescharacteristic
685C2#1807811yescharacteristic
686C2#2144311yescharacteristic
687C2C4×#45011yescharacteristic
688C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21596C2semidirect
4. C2×#1583C2semidirect
5. C2×C8×SD16C2semidirect
6. #8949C2semidirect
7. #8945C2semidirect
8. #8937C2semidirect
9. #8849C2semidirect
10. #8845C2semidirect
11. #8830C2semidirect
13. #8206C2semidirect
14. #8202C2semidirect
15. #8146C2semidirect
223. C2×M16SD16semidirect
365. C2×Q8#131semidirect
477. C2×C8#131semidirect

Maximal subgroups

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

subgroupindex
C2×#29352
#235702
#215962
C2×#15832
C2×C8×SD162
#89492
#89452
#89372
#88492
#88452
#88302
#83132
#82062
#82022
#81462