Finite Group Databasebeta

← #216741

Subgroups — #216741

Order 512.

1257 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3#14682C211yesmaximal, characteristic, complemented
4#9886C212yesmaximal, complemented
5#9681C212yesmaximal, complemented
6#9155C212yesmaximal
7#9151C212yesmaximal, complemented
8#8574C212yesmaximal, complemented
9#8573C212yesmaximal, complemented
10C2×#529C411yescharacteristic, complemented
11C23×C42C2211yescharacteristic
12C2×#431—21no—
13#2855C411yescharacteristic, complemented
14#2854C411yescharacteristic, complemented
15#2853—21no—
16#2852—21no—
17#2852—21no—
18#2851C411yescharacteristic, complemented
19#2851C2211yescharacteristic
20#2850C411yescharacteristic, complemented
21#2850C2211yescharacteristic
22#2849—21no—
23#2848C2211yescharacteristic
24#2847C2211yescharacteristic
25C4×C22⋊Q8C411yescharacteristic, complemented
26C4×C22⋊Q8C2211yescharacteristic
27C4×#123—21no—
28C4×#121C411yescharacteristic, complemented
29C4×#121C2211yescharacteristic
30C4×#120—21no—
31#2290—21no—
32#2290—21no—
33#2283C411yescharacteristic, complemented
34C2×#382—21no—
35C2×#380C2212yes—
36#1817—21no—
37#1797—21no—
38#1797C2212yes—
39#1779C2212yes—
40C4×C22⋊C8—21no—
41C4×C22⋊C8C2212yes—
42C2×#328C2214yes—
43C2×#327C2214yes—
44C4⋊#95—21no—
45#1434C2212yes—
46#1434C2212yes—
47C22⋊#95—21no—
48C22⋊#95—21no—
49#1294C2214yes—
50#1294C2214yes—
51C24×C4D411yescharacteristic
52C24×C4D411yescharacteristic
53C24×C4C2311yescharacteristic
54#529—44no—
55C2×#126—21no—
56C2×#125C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#123—21no—
59C2×#123—21no—
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2311yescharacteristic
63C2×#121C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121C2311yescharacteristic
66C2×#120—21no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#431—44no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2314yes—
93C2×C4×C8—22no—
94#391C2×C411yescharacteristic
95#391C2×C411yescharacteristic
96#390C2×C411yescharacteristic
97#390C2×C411yescharacteristic
98#389C2×C411yescharacteristic
99#389C2×C411yescharacteristic
100#389C2×C411yescharacteristic
101#389C2×C411yescharacteristic
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#388—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#386—21no—
119#386—21no—
120#386—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#383—21no—
139#382—44no—
140#382C2×C411yescharacteristic
141#382C2×C411yescharacteristic
142#381—21no—
143#381—21no—
144#380—44no—
145#380C2×C411yescharacteristic
146#380C2×C411yescharacteristic
147#380C2×C411yescharacteristic
148#380C2×C411yescharacteristic
149#380C2311yescharacteristic
150#380C2311yescharacteristic
151#379—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161#376—21no—
162C2×#95—22no—
163C2×#95—22no—
164C2×#95—22no—
165C2×#95—22no—
166C2×#95C2314yes—
167#336—22no—
168#336—22no—
169#328—44no—
170#327—44no—
171C25C2×D411yescharacteristic
172C22×Q8C22×C411yescharacteristic
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C22×C411yescharacteristic
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—24no—
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×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8—24no—
208#126—44no—
209#125—42no—
210#124—42no—
211#123—44no—
212#123—44no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217#121—22no—
218#121—22no—
219#121—42no—
220#121—42no—
221#120—44no—
222C4×Q8—42no—
223C4×D4—42no—
224C4×D4—44no—
225C4×D4—44no—
226C4×D4—44no—
227#117—44no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
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—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
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×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C4⋊C8—44no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—44no—
304C22⋊C8—44no—
305C4×C8—44no—
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#95—21no—
324#95—21no—
325#95—21no—
326#95—24no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—41no—
337C24—41no—
338C24—41no—
339C24—41no—
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24#13514yes—
343C24C22×D411yescharacteristic
344C2×Q8—22no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C22×C4—21no—
361C22×C4—21no—
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—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—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
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—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—416no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C23×C411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C2×C8—44no—
450C2×C8—44no—
451C2×C8—44no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—88no—
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—41no—
478C22⋊C4—41no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—88no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—42no—
507C42—42no—
508C42#9914yes—
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—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—24no—
526C23—24no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23#41612yes—
543C23#41612yes—
544C23C22×C22⋊C411yescharacteristic
545C23#53511yescharacteristic
546C23#53611yescharacteristic
547C23C2×#13512yes—
548C23#57512yes—
549Q8—82no—
550D4—82no—
551D4—84no—
552D4—84no—
553D4—82no—
554D4—82no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
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—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
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—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
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—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—84no—
629C2×C4—84no—
630C2×C4—84no—
631C2×C4—84no—
632C2×C4C2×#9912yes—
633C2×C4C2×#9912yes—
634C2×C4C2×#9912yes—
635C2×C4#41012yes—
636C2×C4#42112yes—
637C2×C4#42112yes—
638C2×C4#52012yes—
639C2×C4#53712yes—
640C8—84no—
641C22—21no—
642C22—21no—
643C22—21no—
644C22—21no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—48no—
656C22—48no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22#231111yescharacteristic
673C22C22×#9911yescharacteristic
674C22C2×#41011yescharacteristic
675C22C2×#41611yescharacteristic
676C22#289611yescharacteristic
677C22C2×#42111yescharacteristic
678C22C2×#42111yescharacteristic
679C22#321112yes—
680C22#323212yes—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—164no—
701C2—22no—
702C2—22no—
703C2—42no—
704C2—42no—
705C2—161no—
706C2#1755811yescharacteristic
707C2#1756711yescharacteristic
708C2#1757211yescharacteristic
709C2#1758211yescharacteristic
710C2#2186111yescharacteristic
711C2#2187411yescharacteristic
712C2#2190011yescharacteristic
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. #20714C2semidirect
3. #14682C2semidirect
4. #9886C2semidirect
5. #9681C2semidirect
7. #9151C2semidirect
8. #8574C2semidirect
9. #8573C2semidirect
10. C2×#529C4semidirect
13. #2855C4semidirect
14. #2854C4semidirect
18. #2851C4semidirect
20. #2850C4semidirect
25. C4×C22⋊Q8C4semidirect
28. C4×#121C4semidirect
33. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#366212
#207142
#146822
#98862
#96812
#91552
#91512
#85742
#85732