Finite Group Databasebeta

← #214221

Subgroups — #214221

Order 512.

1243 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C4×#407C211yesmaximal, characteristic, complemented
4#9078C212yesmaximal, complemented
5#9077C212yesmaximal, complemented
6#8944C212yesmaximal, complemented
7#8942C212yesmaximal, complemented
8#8711C212yesmaximal, complemented
9#8315C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423C2212yes—
13C2×#407—21no—
14C2×#407—21no—
15C2×C4×M16C2212yes—
16#2855C411yescharacteristic
17#2854C411yescharacteristic
18#2853—21no—
19#2852—21no—
20#2852—21no—
21#2851C411yescharacteristic
22#2851C411yescharacteristic
23#2850C411yescharacteristic
24#2850C411yescharacteristic
25#2849—21no—
26#2848C411yescharacteristic
27#2847C411yescharacteristic
28C4×C22⋊Q8C2211yescharacteristic
29C4×C22⋊Q8C2211yescharacteristic
30C4×#123—21no—
31C4×#121C2211yescharacteristic
32C4×#121C2211yescharacteristic
33C4×#120—21no—
34#2290—21no—
35#2290—21no—
36#2283C2211yescharacteristic
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41#1749—21no—
42#1749—21no—
43C4×C22⋊C8C2212yes—
44C4×C22⋊C8C2212yes—
45#1728—21no—
46#1728—21no—
47C2×Q8⋊C8C2214yescomplemented
48C2×D4⋊C8C2214yescomplemented
49#1279C2214yescomplemented
50#1279C2214yescomplemented
51C24×C4C2×C411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C22×M16—22no—
55C22×M16—22no—
56#529—44no—
57C2×#126—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#123—21no—
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#120—21no—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#423—44no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2314yes—
91#407—28no—
92#407—44no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C4×M16—44no—
98C2×#97—22no—
99C2×C4×C8C2314yes—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#389C2×C411yescharacteristic
105#389C2×C411yescharacteristic
106#389C2×C411yescharacteristic
107#389C2×C411yescharacteristic
108#388—21no—
109#388—21no—
110#388—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#383—21no—
145#382C2×C411yescharacteristic
146#382C2×C411yescharacteristic
147#381—21no—
148#381—21no—
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic
154#380C2×C411yescharacteristic
155#379—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2311yescharacteristic
159C4×C4⋊C4C2311yescharacteristic
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#376—21no—
166#336—22no—
167#336—22no—
168#336—22no—
169#336—22no—
170Q8⋊C8—44no—
171D4⋊C8—44no—
172C25C22⋊C411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—21no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22⋊C411yescharacteristic
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C2×M16—28no—
208C2×M16—44no—
209C2×M16—44no—
210C2×M16—44no—
211C2×M16—44no—
212C2×M16—44no—
213C22×C8—24no—
214C22×C8—24no—
215#126—44no—
216#125—42no—
217#124—42no—
218#123—44no—
219#123—44no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223#121—42no—
224#121—42no—
225#121—42no—
226#120—44no—
227C4×Q8—24no—
228C4×D4—24no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—42no—
232C4×D4—42no—
233#117—44no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
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—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C22⋊C4—21no—
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×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C4⋊C8—24no—
301C4⋊C8—24no—
302C4⋊C8—44no—
303C22⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—44no—
306C22⋊C8—44no—
307#97—44no—
308C4×C8—24no—
309C4×C8—24no—
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#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—41no—
335C24—41no—
336C24—41no—
337C24—41no—
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24C2×C22⋊C411yescharacteristic
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—42no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—416no—
421C22×C4—42no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—42no—
425C22×C4#10414yes—
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4#13712yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440M16—416no—
441M16—88no—
442C2×C8—28no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—42no—
499C42—41no—
500C42—41no—
501C42—42no—
502C42H(5,𝔽2)14yes—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23#41712yes—
536C23C2×#10412yes—
537C23#42112yes—
538C23C22×C22⋊C411yescharacteristic
539Q8—44no—
540D4—44no—
541D4—84no—
542D4—84no—
543D4—84no—
544C2×C4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
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—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4#41712yes—
615C2×C4C2×#10412yes—
616C2×C4C2×#10412yes—
617C2×C4#51812yes—
618C2×C4#53412yes—
619C2×C4#53412yes—
620C2×C4#56812yes—
621C2×C4#56812yes—
622C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
623C2×C4C2×#13711yescharacteristic
624C2×C4#57512yes—
625C8—48no—
626C8—84no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—416no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22#230811yescharacteristic
657C22#287911yescharacteristic
658C22#287911yescharacteristic
659C22C2×#41711yescharacteristic
660C22#289611yescharacteristic
661C22C22×#10411yescharacteristic
662C22C2×#42111yescharacteristic
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4#297814yes—
681C4#319412yes—
682C4#319612yes—
683C2—42no—
684C2—42no—
685C2—42no—
686C2—161no—
687C2#1749211yescharacteristic
688C2#1749711yescharacteristic
689C2#1750311yescharacteristic
690C2#1750411yescharacteristic
691C2#2174311yescharacteristic
692C2#2176911yescharacteristic
693C2#2177311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2847C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20626C2semidirect
3. C4×#407C2semidirect
4. #9078C2semidirect
5. #9077C2semidirect
6. #8944C2semidirect
7. #8942C2semidirect
8. #8711C2semidirect
9. #8315C2semidirect
47. C2×Q8⋊C8C22semidirect
48. C2×D4⋊C8C22semidirect
49. #1279C22semidirect
50. #1279C22semidirect

Maximal subgroups

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

subgroupindex
#366212
#206262
C4×#4072
#90782
#90772
#89442
#89422
#87112
#83152