Finite Group Databasebeta

← #481685

Subgroups — #481685

Order 512.

1235 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35703C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic, complemented
3#17367C211yesmaximal, characteristic, complemented
4#11759C212yesmaximal, complemented
5#11758C212yesmaximal, complemented
6#11756C212yesmaximal, complemented
7#11753C212yesmaximal, complemented
8C2×#1351C212yesmaximal, complemented
9#11692C212yesmaximal, complemented
10C2×#548C411yescharacteristic, complemented
11C2×#518—21no—
12C2×#518C2211yescharacteristic
13#2973—21no—
14#2638C411yescharacteristic, complemented
15#2631C411yescharacteristic, complemented
16#2617C2211yescharacteristic
17#2610C2211yescharacteristic
18#2599C2211yescharacteristic
19#2561C2211yescharacteristic
20#2543C411yescharacteristic, complemented
21#2477—21no—
22#2472—21no—
23#2463—21no—
24#2459—21no—
25#2444—21no—
26#2410—21no—
27#2409C411yescharacteristic, complemented
28#2404C2211yescharacteristic
29#2398C411yescharacteristic, complemented
30#2388C411yescharacteristic, complemented
31C2×#393C2211yescharacteristic
32#2364—21no—
33#2358—21no—
34#2353C411yescharacteristic, complemented
35#1740—21no—
36#1740C2212yes—
37C2×#342—21no—
38C2×#342—21no—
39C2×#342C2212yes—
40C2×#342C2212yes—
41#1466—21no—
42#1465—21no—
43#1464C2212yes—
44#1462C2212yes—
45#1459—21no—
46C2×#323C2212yes—
47#1351C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41C2×C411yescharacteristic, complemented
52C23×Q8C2×C411yescharacteristic, complemented
53C23×D4D411yescharacteristic
54#548—44no—
55C2×#124—21no—
56C2×#124C2×C411yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic, complemented
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C411yescharacteristic, complemented
63C2×#120—21no—
64C2×#120—21no—
65#518—28no—
66#518—44no—
67#518—44no—
68C2×C4×Q8C2×C411yescharacteristic, complemented
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—44no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—22no—
81C2×#117D411yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D411yescharacteristic
87C22×C22⋊C4D411yescharacteristic
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2311yescharacteristic
90#435—44no—
91#432—22no—
92#407—22no—
93C2×C22⋊C8—22no—
94#400—21no—
95#398C2×C411yescharacteristic, complemented
96#397—21no—
97#397—21no—
98#397C2×C411yescharacteristic, complemented
99#397C2×C411yescharacteristic, complemented
100#397C2311yescharacteristic
101#395C2×C411yescharacteristic, complemented
102#395C2×C411yescharacteristic, complemented
103#394—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#394—21no—
108#394C2×C411yescharacteristic, complemented
109#393—21no—
110#393—21no—
111#393C2×C411yescharacteristic, complemented
112#393C2×C411yescharacteristic, complemented
113#393C2×C411yescharacteristic, complemented
114#393C2×C411yescharacteristic, complemented
115#393C2311yescharacteristic
116#393C2311yescharacteristic
117#392—21no—
118#392—21no—
119#392C2×C411yescharacteristic, complemented
120#392C2×C411yescharacteristic, complemented
121#392C2311yescharacteristic
122#391—21no—
123#390—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#386—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#383C2×C411yescharacteristic, complemented
150#381—21no—
151#381C2×C411yescharacteristic, complemented
152#380C2×C411yescharacteristic, complemented
153#379—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2×C411yescharacteristic, complemented
159#376C2×C411yescharacteristic, complemented
160C2×#95—21no—
161C2×#95C2×C411yescharacteristic, complemented
162#342—22no—
163#342—22no—
164#342—22no—
165#342C2314yes—
166#324—22no—
167#323—22no—
168#323—22no—
169#323C2314yes—
170C25C2×D411yescharacteristic
171C25C2×D411yescharacteristic
172C2×#41—42no—
173C2×#41—82no—
174C22×Q8—21no—
175C22×Q8—42no—
176C22×Q8—82no—
177C22×Q8C22×C411yescharacteristic
178C22×Q8C22×C411yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22⋊C411yescharacteristic
189C22×D4C22⋊C411yescharacteristic
190C22×D4C22⋊C411yescharacteristic
191C22×D4C22⋊C411yescharacteristic
192C22×D4C22×C411yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22×C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C2×M16—42no—
208C22×C8—42no—
209#124—42no—
210#124—44no—
211C22⋊Q8—44no—
212C22⋊Q8—42no—
213C22⋊Q8—82no—
214#121—42no—
215#121—44no—
216#121—44no—
217#120—42no—
218#120—42no—
219#120—82no—
220C4×Q8—42no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—44no—
230C4×D4—432no—
231C4×D4—82no—
232#117—28no—
233#117—42no—
234#117—42no—
235#117—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22×C411yescharacteristic
286C2×C22⋊C4C22×C411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C22⋊C4C2×D411yescharacteristic
289C2×C22⋊C4C2×D411yescharacteristic
290C2×C22⋊C4C2×D411yescharacteristic
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C22⋊C4C2411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—42no—
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C4⋊C8—42no—
302C22⋊C8—24no—
303C22⋊C8—42no—
304C22⋊C8—42no—
305#97—42no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95C22×C411yescharacteristic
324#95C22×C411yescharacteristic
325#95C22×C411yescharacteristic
326#95C22×C411yescharacteristic
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—81no—
335C24—81no—
336C24—81no—
337C24—81no—
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24AGL(1,ℤ/8)12yes—
343C24#13712yes—
344C24C22×D411yescharacteristic
345#41—84no—
346#41—84no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×D4—216no—
357C2×D4—216no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4#9914yes—
387C2×D4#10414yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—162no—
428C22×C4—162no—
429C22×C4—162no—
430C22×C4—162no—
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441M16—162no—
442C2×C8—44no—
443C2×C8—162no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—28no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—42no—
499C42—48no—
500C42—81no—
501C42—81no—
502C23—24no—
503C23—24no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—24no—
513C23—24no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—162no—
538C23—161no—
539C23—161no—
540C23—162no—
541C23C2×#9912yes—
542C23C2×#9912yes—
543C23#41712yes—
544C23#41712yes—
545C23C2×#10412yes—
546C23C2×#10412yes—
547C23C22×C22⋊C411yescharacteristic
548C23C2×AGL(1,ℤ/8)11yescharacteristic
549C23C2×#13711yescharacteristic
550Q8—42no—
551Q8—42no—
552Q8—84no—
553Q8—82no—
554D4—28no—
555D4—28no—
556D4—28no—
557D4—28no—
558D4—84no—
559D4—82no—
560D4—84no—
561D4—84no—
562D4—162no—
563D4—161no—
564D4—161no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—48no—
591C2×C4—41no—
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—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—84no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—161no—
618C2×C4—161no—
619C2×C4—161no—
620C2×C4—161no—
621C2×C4—162no—
622C2×C4—168no—
623C2×C4—168no—
624C2×C4C2×#9912yes—
625C2×C4C2×#10412yes—
626C8—84no—
627C22—21no—
628C22—22no—
629C22—21no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—42no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—84no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—84no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—84no—
662C22—84no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22#150414yes—
667C22C22×#9911yescharacteristic
668C22C2×#41711yescharacteristic
669C22C22×#10411yescharacteristic
670C4—22no—
671C4—22no—
672C4—42no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—88no—
686C4—81no—
687C4—81no—
688C2—21no—
689C2—21no—
690C2—24no—
691C2—41no—
692C2—42no—
693C2—42no—
694C2—42no—
695C2—41no—
696C2—161no—
697C2C2×#150411yescharacteristic
698C2#3334811yescharacteristic
699C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35703C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20354C2semidirect
3. #17367C2semidirect
4. #11759C2semidirect
5. #11758C2semidirect
6. #11756C2semidirect
7. #11753C2semidirect
8. C2×#1351C2semidirect
9. #11692C2semidirect
10. C2×#548C4semidirect
14. #2638C4semidirect
15. #2631C4semidirect
20. #2543C4semidirect
27. #2409C4semidirect
29. #2398C4semidirect
30. #2388C4semidirect
34. #2353C4semidirect
51. C22×#41C2×C4semidirect
52. C23×Q8C2×C4semidirect
56. C2×#124C2×C4semidirect
59. C2×C22⋊Q8C2×C4semidirect
62. C2×#121C2×C4semidirect
68. C2×C4×Q8C2×C4semidirect
95. #398C2×C4semidirect
98. #397C2×C4semidirect
99. #397C2×C4semidirect
101. #395C2×C4semidirect
102. #395C2×C4semidirect
108. #394C2×C4semidirect
111. #393C2×C4semidirect
112. #393C2×C4semidirect
113. #393C2×C4semidirect
114. #393C2×C4semidirect
119. #392C2×C4semidirect
120. #392C2×C4semidirect
149. #383C2×C4semidirect
151. #381C2×C4semidirect
152. #380C2×C4semidirect
158. C4×C22⋊C4C2×C4semidirect
159. #376C2×C4semidirect
161. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357032
#203542
#173672
#117592
#117582
#117562
#117532
C2×#13512
#116922