Finite Group Databasebeta

← #196510

Subgroups — #196510

Order 512.

1445 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#432C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic, complemented
4C2×#1529C212yesmaximal, complemented
5#9126C212yesmaximal, complemented
6#9124C212yesmaximal, complemented
7#9121C212yesmaximal, complemented
8#9119C212yesmaximal, complemented
9#8035C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C2×#432—21no—
14C2×#432—28no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C22×C4⋊C8—21no—
18C2×#407—21no—
19C22×C22⋊C8—21no—
20C2×C4×M16C2212yes—
21C22×#97C2212yes—
22#2855C411yescharacteristic
23#2854C411yescharacteristic
24#2853—21no—
25#2852—21no—
26#2852—21no—
27#2851C2211yescharacteristic
28#2851C2211yescharacteristic
29#2850C2211yescharacteristic
30#2850C2211yescharacteristic
31#2849—21no—
32#2848C411yescharacteristic
33#2847C411yescharacteristic
34C4×C22⋊Q8C411yescharacteristic
35C4×C22⋊Q8C411yescharacteristic
36C4×#123—21no—
37C4×#121C411yescharacteristic
38C4×#121C411yescharacteristic
39C4×#120—21no—
40#2290—21no—
41#2290—21no—
42#2283C2211yescharacteristic
43#1749C2212yes—
44#1749C2212yes—
45C4×C22⋊C8—21no—
46C4×C22⋊C8—21no—
47#1728—21no—
48#1529—28no—
49C2×#330C2214yescomplemented
50C2×#329C2214yescomplemented
51#1295C2214yescomplemented
52#1295C2214yescomplemented
53C24×C4C2×C411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C22×M16—22no—
57C23×C8—22no—
58#529—24no—
59#529—24no—
60C2×#126—21no—
61C2×#125C2311yescharacteristic
62C2×#124C2311yescharacteristic
63C2×#123—21no—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#120—21no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93#432—28no—
94#432—24no—
95#432—28no—
96#432D418yescomplemented
97C2×C4⋊C8—28no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100#407—28no—
101C2×C22⋊C8—28no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C4×M16—24no—
107C4×M16—24no—
108C2×#97—22no—
109C2×#97—28no—
110C2×#97C2314yes—
111C2×#97C2314yes—
112C2×C4×C8—22no—
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—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#386—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#385—21no—
157#383—21no—
158#382C2311yescharacteristic
159#382C2311yescharacteristic
160#381—21no—
161#381—21no—
162#380C2×C411yescharacteristic
163#380C2×C411yescharacteristic
164#380C2×C411yescharacteristic
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#379—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178#376—21no—
179#336—22no—
180#336—22no—
181#330—28no—
182#329—28no—
183C25C22⋊C411yescharacteristic
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—21no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
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×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C2×M16—28no—
219C2×M16—44no—
220C2×M16—44no—
221C22×C8—28no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—44no—
225C22×C8—44no—
226#126—44no—
227#125—22no—
228#125—22no—
229#124—22no—
230#124—22no—
231#123—44no—
232#123—42no—
233#123—42no—
234C22⋊Q8—24no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237#121—24no—
238#121—42no—
239#121—42no—
240#120—44no—
241C4×Q8—24no—
242C4×D4—24no—
243C4×D4—44no—
244C4×D4—44no—
245C4×D4—44no—
246#117—42no—
247#117—42no—
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⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22⋊C411yescharacteristic
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D414yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4114yes—
319C2×C42C2411yescharacteristic
320C4⋊C8—28no—
321C4⋊C8—28no—
322C22⋊C8—28no—
323C22⋊C8—28no—
324C22⋊C8—28no—
325C22⋊C8—28no—
326#97—28no—
327#97—24no—
328#97—24no—
329#97—24no—
330#97C2×D418yes—
331C4×C8—44no—
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—
349#95—21no—
350#95—21no—
351#95—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—41no—
359C24—41no—
360C24C2×C22⋊C411yescharacteristic
361C24C2×C22⋊C411yescharacteristic
362C24C2×C22⋊C411yescharacteristic
363C2×Q8—22no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C22×C4—22no—
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—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
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—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
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—41no—
436C22×C4—41no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C4×D414yes—
455C22×C4#12012yes—
456C22×C4#12114yes—
457C22×C4#12312yes—
458C22×C4C23×C411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461M16—416no—
462C2×C8—28no—
463C2×C8—28no—
464C2×C8—416no—
465C2×C8—44no—
466C2×C8—44no—
467C2×C8—44no—
468C2×C8—44no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊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—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—42no—
501C22⋊C4—41no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C42—22no—
507C42—22no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—22no—
512C42—21no—
513C42—21no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—42no—
521C42—42no—
522C42—42no—
523C42C4×D414yes—
524C42#12012yes—
525C42#12114yes—
526C42#12312yes—
527C42C22×D412yes—
528C42C2×#4112yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—28no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#38612yes—
560C23#41312yes—
561C23#41312yes—
562C23C2×#10412yes—
563C23#42112yes—
564C23C22×C22⋊C411yescharacteristic
565Q8—82no—
566D4—82no—
567D4—84no—
568D4—84no—
569D4—84no—
570C2×C4—24no—
571C2×C4—24no—
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—28no—
587C2×C4—28no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
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—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
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—41no—
637C2×C4—41no—
638C2×C4—84no—
639C2×C4#38612yes—
640C2×C4#38612yes—
641C2×C4#38612yes—
642C2×C4C2×#10412yes—
643C2×C4C2×#10412yes—
644C2×C4#42112yes—
645C2×C4#42112yes—
646C2×C4C2×C4×D412yes—
647C2×C4C2×#12011yescharacteristic
648C2×C4C2×#12112yes—
649C2×C4C2×#12311yescharacteristic
650C8—48no—
651C8—48no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—416no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22#152914yescomplemented
681C22C2×#38611yescharacteristic
682C22C2×#41311yescharacteristic
683C22C2×#41311yescharacteristic
684C22C22×#10411yescharacteristic
685C22C2×#42111yescharacteristic
686C22C2×#42111yescharacteristic
687C22C2×#42111yescharacteristic
688C4—22no—
689C4—22no—
690C4—21no—
691C4—21no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—81no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4#187114yes—
708C2—24no—
709C2—42no—
710C2—42no—
711C2—161no—
712C2C2×#152911yescharacteristic
713C2#1765111yescharacteristic
714C2#1766911yescharacteristic
715C2#1767411yescharacteristic
716C2C2×#187111yescharacteristic
717C2#2097911yescharacteristic
718C2#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. C4×#407C2semidirect
4. C2×#1529C2semidirect
5. #9126C2semidirect
6. #9124C2semidirect
7. #9121C2semidirect
8. #9119C2semidirect
9. #8035C2semidirect
49. C2×#330C22semidirect
50. C2×#329C22semidirect
51. #1295C22semidirect
52. #1295C22semidirect
96. #432D4semidirect
680. C22#1529semidirect

Maximal subgroups

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

subgroupindex
C22×#4322
#366212
C4×#4072
C2×#15292
#91262
#91242
#91212
#91192
#80352