Finite Group Databasebeta

← #208384

Subgroups — #208384

Order 512.

1757 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C2×C4×#95C211yesmaximal, characteristic, complemented
4#9910C212yesmaximal, complemented
5#9713C212yesmaximal, complemented
6#9147C212yesmaximal
7#9143C212yesmaximal, complemented
8#8219C212yesmaximal, complemented
9#8218C212yesmaximal, complemented
10C2×#529C411yescharacteristic, complemented
11C23×C42—21no—
12C23×C42C2211yescharacteristic
13C2×#432—21no—
14#2855C411yescharacteristic, complemented
15#2854C411yescharacteristic, complemented
16#2853—21no—
17#2852—21no—
18#2852—21no—
19#2851C411yescharacteristic, complemented
20#2851C2211yescharacteristic
21#2850C411yescharacteristic, complemented
22#2850C2211yescharacteristic
23#2849—21no—
24#2848C2211yescharacteristic
25#2847C2211yescharacteristic
26C4×C22⋊Q8C411yescharacteristic, complemented
27C4×C22⋊Q8C2211yescharacteristic
28C4×#123—21no—
29C4×#121C411yescharacteristic, complemented
30C4×#121C2211yescharacteristic
31C4×#120—21no—
32#2290—21no—
33#2290—21no—
34#2283C411yescharacteristic, complemented
35C22×#95—21no—
36C22×#95C2212yes—
37#1817—21no—
38#1797—21no—
39#1797C2212yes—
40#1779C2212yes—
41#1749—21no—
42#1749C2212yes—
43C2×#328C2214yes—
44C2×#327C2214yes—
45C4×#95—21no—
46C4×#95—21no—
47C4×#95C2212yes—
48C4×#95C2212yes—
49#1294C2214yes—
50#1294C2214yes—
51C24×C4—22no—
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4C2311yescharacteristic
55#529—44no—
56C2×#126—21no—
57C2×#125C2×C411yescharacteristic, complemented
58C2×#124C2×C411yescharacteristic, complemented
59C2×#123—21no—
60C2×#123—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#121C2311yescharacteristic
67C2×#120—21no—
68C2×C4×Q8C2×C411yescharacteristic, complemented
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic, complemented
73C2×#117—21no—
74C22×C42—28no—
75C22×C42—22no—
76C22×C42—22no—
77C22×C42—44no—
78C22×C42—44no—
79C22×C42—44no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—44no—
93C2×C4⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2314yes—
97C2×#97—22no—
98#391C2×C411yescharacteristic, complemented
99#391C2×C411yescharacteristic, complemented
100#390C2×C411yescharacteristic, complemented
101#390C2×C411yescharacteristic, complemented
102#389C2×C411yescharacteristic, complemented
103#389C2×C411yescharacteristic, complemented
104#389C2×C411yescharacteristic, complemented
105#389C2×C411yescharacteristic, complemented
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#388—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#386—21no—
123#386—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#385—21no—
133#385—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#383—21no—
143#382C2×C411yescharacteristic, complemented
144#382C2×C411yescharacteristic, complemented
145#381—21no—
146#381—21no—
147#380C2×C411yescharacteristic
148#380C2×C411yescharacteristic
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#380C2311yescharacteristic
152#380C2311yescharacteristic
153#379—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2×C411yescharacteristic, complemented
157C4×C4⋊C4C2×C411yescharacteristic, complemented
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163#376—21no—
164C2×#95—28no—
165C2×#95—22no—
166C2×#95—22no—
167C2×#95—22no—
168C2×#95—28no—
169C2×#95C2314yes—
170#336—22no—
171#336—22no—
172#328—28no—
173#327—28no—
174C25C2×D411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C22×C411yescharacteristic
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—28no—
186C23×C4—28no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—21no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C411yescharacteristic
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2411yescharacteristic
217C22×C8—24no—
218#126—44no—
219#125—42no—
220#124—42no—
221#123—44no—
222#123—44no—
223C22⋊Q8—24no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226#121—24no—
227#121—42no—
228#121—42no—
229#120—42no—
230#120—42no—
231C4×Q8—42no—
232C4×D4—42no—
233C4×D4—44no—
234C4×D4—44no—
235C4×D4—44no—
236#117—44no—
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⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
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×C42—24no—
273C2×C42—24no—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—432no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C22⋊C411yescharacteristic
297C2×C42C22⋊C411yescharacteristic
298C2×C42C22⋊C411yescharacteristic
299C2×C42C22⋊C411yescharacteristic
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C4⋊C8—44no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—44no—
314C22⋊C8—44no—
315#97—44no—
316#95—28no—
317#95—28no—
318#95—28no—
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—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—24no—
340#95#4118yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—21no—
348C24C2×C22⋊C411yescharacteristic
349C24C2×C22⋊C411yescharacteristic
350C24AGL(1,ℤ/8)12yes—
351C24#13712yes—
352C24C22×D411yescharacteristic
353C2×Q8—22no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—232no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—44no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4#9914yes—
450C22×C4#10414yes—
451C22×C4#10414yes—
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C23×C411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4114yes—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8—44no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—416no—
521C42—416no—
522C42—42no—
523C42—42no—
524C42—42no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—21no—
546C23—21no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C2×#9912yes—
555C23#41012yes—
556C23#41712yes—
557C23#41712yes—
558C23C2×#10412yes—
559C23C2×#10412yes—
560C23C22×C22⋊C411yescharacteristic
561C23#52512yes—
562C23C2×AGL(1,ℤ/8)11yescharacteristic
563C23C2×#13711yescharacteristic
564C23#57512yes—
565Q8—82no—
566D4—82no—
567D4—82no—
568D4—82no—
569D4—84no—
570D4—84no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—216no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—464no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—48no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—84no—
649C2×C4C2×#9912yes—
650C2×C4C2×#9912yes—
651C2×C4C2×#10412yes—
652C2×C4C2×#10412yes—
653C2×C4C2×#10412yes—
654C2×C4C2×#10412yes—
655C2×C4C4×#4112yes—
656C2×C4#53212yes—
657C8—84no—
658C22—216no—
659C22—216no—
660C22—21no—
661C22—21no—
662C22—22no—
663C22—21no—
664C22—21no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22#150414yes—
688C22#150914yes—
689C22#229711yescharacteristic
690C22C22×#9911yescharacteristic
691C22C2×#41011yescharacteristic
692C22C2×#41711yescharacteristic
693C22#289611yescharacteristic
694C22C22×#10411yescharacteristic
695C22C22×#10411yescharacteristic
696C22#311412yes—
697C22#311712yes—
698C4—22no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—416no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4—81no—
718C2—22no—
719C2—22no—
720C2—24no—
721C2—24no—
722C2—161no—
723C2C2×#150911yescharacteristic
724C2C2×#150411yescharacteristic
725C2#1754111yescharacteristic
726C2#1754211yescharacteristic
727C2#2139111yescharacteristic
728C2#2141511yescharacteristic
729C2#2142011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20715C2semidirect
3. C2×C4×#95C2semidirect
4. #9910C2semidirect
5. #9713C2semidirect
7. #9143C2semidirect
8. #8219C2semidirect
9. #8218C2semidirect
10. C2×#529C4semidirect
14. #2855C4semidirect
15. #2854C4semidirect
19. #2851C4semidirect
21. #2850C4semidirect
26. C4×C22⋊Q8C4semidirect
29. C4×#121C4semidirect
34. #2283C4semidirect
57. C2×#125C2×C4semidirect
58. C2×#124C2×C4semidirect
68. C2×C4×Q8C2×C4semidirect
72. C2×C4×D4C2×C4semidirect
98. #391C2×C4semidirect
99. #391C2×C4semidirect
100. #390C2×C4semidirect
101. #390C2×C4semidirect
102. #389C2×C4semidirect
103. #389C2×C4semidirect
104. #389C2×C4semidirect
105. #389C2×C4semidirect
143. #382C2×C4semidirect
144. #382C2×C4semidirect
156. C4×C4⋊C4C2×C4semidirect
157. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366212
#207152
C2×C4×#952
#99102
#97132
#91472
#91432
#82192
#82182