Finite Group Databasebeta

← #196545

Subgroups — #196545

Order 512.

1377 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1535C212yesmaximal, complemented
5#9125C212yesmaximal, complemented
6#9123C212yesmaximal, complemented
7#9121C212yesmaximal, complemented
8#9119C212yesmaximal, complemented
9#8032C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C2×#432—21no—
14C2×#432—28no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C2×#432C2212yes—
18C22×C4⋊C8—21no—
19C22×C22⋊C8—21no—
20C22×#97C2212yes—
21#2855C411yescharacteristic
22#2854C411yescharacteristic
23#2853—21no—
24#2852—21no—
25#2852—21no—
26#2851C2211yescharacteristic
27#2851C2211yescharacteristic
28#2850C2211yescharacteristic
29#2850C2211yescharacteristic
30#2849—21no—
31#2848C411yescharacteristic
32#2847C411yescharacteristic
33C4×C22⋊Q8C411yescharacteristic
34C4×C22⋊Q8C411yescharacteristic
35C4×#123—21no—
36C4×#121C411yescharacteristic
37C4×#121C411yescharacteristic
38C4×#120—21no—
39#2290—21no—
40#2290—21no—
41#2283C2211yescharacteristic
42#1817—21no—
43#1797—21no—
44#1797—21no—
45#1779—21no—
46#1749C2212yes—
47#1749C2212yes—
48#1535—28no—
49C2×#330C2214yescomplemented
50C2×#329C2214yescomplemented
51#1295C2214yescomplemented
52#1295C2214yescomplemented
53C24×C4C2×C411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C23×C8—22no—
57#529—24no—
58#529—24no—
59C2×#126—21no—
60C2×#125C2311yescharacteristic
61C2×#124C2311yescharacteristic
62C2×#123—21no—
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×#120—21no—
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#432—28no—
92#432—28no—
93#432—24no—
94#432—24no—
95#432—24no—
96#432—28no—
97#432D418yescomplemented
98C2×C4⋊C8—28no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C22⋊C8—28no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×#97—22no—
109C2×#97—28no—
110C2×#97C2314yes—
111C2×#97C2314yes—
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—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#385—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#383—21no—
157#382C2311yescharacteristic
158#382C2311yescharacteristic
159#381—21no—
160#381—21no—
161#380C2×C411yescharacteristic
162#380C2×C411yescharacteristic
163#380C2×C411yescharacteristic
164#380C2×C411yescharacteristic
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#379—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177#376—21no—
178#336—22no—
179#336—22no—
180#330—28no—
181#329—28no—
182C25C22⋊C411yescharacteristic
183C22×Q8C22×C411yescharacteristic
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C22×C411yescharacteristic
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—21no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
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×C4C22×C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C22×C8—28no—
218C22×C8—24no—
219C22×C8—24no—
220C22×C8—44no—
221C22×C8—44no—
222#126—44no—
223#125—22no—
224#125—22no—
225#124—22no—
226#124—22no—
227#123—44no—
228#123—42no—
229#123—42no—
230C22⋊Q8—24no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233#121—24no—
234#121—42no—
235#121—42no—
236#120—44no—
237C4×Q8—24no—
238C4×D4—24no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—44no—
242#117—42no—
243#117—42no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
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—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
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×C42C22×C411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D414yes—
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42#4114yes—
315C2×C42C2411yescharacteristic
316C4⋊C8—28no—
317C4⋊C8—28no—
318C4⋊C8—44no—
319C22⋊C8—28no—
320C22⋊C8—28no—
321C22⋊C8—44no—
322C22⋊C8—44no—
323C22⋊C8—44no—
324#97—28no—
325#97—24no—
326#97—24no—
327#97—24no—
328#97C2×D418yes—
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—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—41no—
356C24—41no—
357C24C2×C22⋊C411yescharacteristic
358C24C2×C22⋊C411yescharacteristic
359C24C2×C22⋊C411yescharacteristic
360C2×Q8—22no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×D4—22no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
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—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
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—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
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×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—41no—
435C22×C4—48no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—416no—
441C22×C4—42no—
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C4×D414yes—
452C22×C4#12012yes—
453C22×C4#12114yes—
454C22×C4#12312yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C2×C8—28no—
459C2×C8—416no—
460C2×C8—44no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8—44no—
464C2×C8—44no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊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—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—41no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C42—22no—
503C42—22no—
504C42—21no—
505C42—21no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42C4×D414yes—
520C42#12012yes—
521C42#12114yes—
522C42#12312yes—
523C42C22×D412yes—
524C42C2×#4112yes—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—28no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#38612yes—
556C23#41312yes—
557C23#42112yes—
558C23#42112yes—
559C23#42112yes—
560C23C22×C22⋊C411yescharacteristic
561Q8—82no—
562D4—82no—
563D4—84no—
564D4—84no—
565D4—84no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
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—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
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—41no—
625C2×C4—41no—
626C2×C4—41no—
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—84no—
637C2×C4#38612yes—
638C2×C4#38612yes—
639C2×C4#38612yes—
640C2×C4#41312yes—
641C2×C4#42112yes—
642C2×C4C2×C4×D412yes—
643C2×C4C2×#12011yescharacteristic
644C2×C4C2×#12112yes—
645C2×C4C2×#12311yescharacteristic
646C8—48no—
647C8—84no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—416no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22#153514yescomplemented
677C22C2×#38611yescharacteristic
678C22C2×#41311yescharacteristic
679C22C2×#41311yescharacteristic
680C22C2×#42111yescharacteristic
681C22C2×#42111yescharacteristic
682C22C2×#42111yescharacteristic
683C22C2×#42111yescharacteristic
684C4—22no—
685C4—22no—
686C4—21no—
687C4—21no—
688C4—22no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—41no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—81no—
704C2—24no—
705C2—42no—
706C2—42no—
707C2—161no—
708C2C2×#153511yescharacteristic
709C2#1766211yescharacteristic
710C2#1766811yescharacteristic
711C2#1767711yescharacteristic
712C2#2090211yescharacteristic
713C2#2097911yescharacteristic
714C2#2098911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2847C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#432C2semidirect
3. #20715C2semidirect
4. C2×#1535C2semidirect
5. #9125C2semidirect
6. #9123C2semidirect
7. #9121C2semidirect
8. #9119C2semidirect
9. #8032C2semidirect
49. C2×#330C22semidirect
50. C2×#329C22semidirect
51. #1295C22semidirect
52. #1295C22semidirect
97. #432D4semidirect
676. C22#1535semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366212
#207152
C2×#15352
#91252
#91232
#91212
#91192
#80322