Finite Group Databasebeta

← #162252

Subgroups — #162252

Order 512.

1359 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3C2×#1728C211yesmaximal, characteristic, complemented
4#7695C212yesmaximal, complemented
5#7694C212yesmaximal, complemented
6#7681C212yesmaximal, complemented
7#7678C212yesmaximal, complemented
8#7579C212yesmaximal, complemented
9#7578C212yesmaximal, complemented
10C23×M16—21no—
11C2×#525C411yescharacteristic
12C23×C42C2211yescharacteristic
13C22×C22⋊C8—21no—
14#2855C411yescharacteristic
15#2855C411yescharacteristic
16#2855C2211yescharacteristic
17#2855C2211yescharacteristic
18#2854C411yescharacteristic
19#2854C411yescharacteristic
20#2854C2211yescharacteristic
21#2854C2211yescharacteristic
22#2853—21no—
23#2853—21no—
24#2849—21no—
25#2849—21no—
26C4×#123—21no—
27C4×#123—21no—
28C4×#123—21no—
29C4×#123—21no—
30#2297C411yescharacteristic
31#2297C2211yescharacteristic
32#2296C411yescharacteristic
33#2296C2211yescharacteristic
34#2280C411yescharacteristic
35#1817—21no—
36#1817—21no—
37#1817C2212yes—
38#1817C2212yes—
39#1728—21no—
40#1728—21no—
41#1728C2212yes—
42#1728C2212yes—
43C2×#336—21no—
44C2×#336—21no—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C22×M16—28no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—44no—
59C22×M16—44no—
60C23×C8—22no—
61C23×C8—22no—
62#525—44no—
63C2×#123—21no—
64C2×#123—21no—
65C2×#123—21no—
66C2×#123—21no—
67C2×#123—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×#120—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C22×C42—44no—
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
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—44no—
94#391C2×C411yescharacteristic
95#391C2×C411yescharacteristic
96#391C2×C411yescharacteristic
97#391C2311yescharacteristic
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#390C2×C411yescharacteristic
101#390C2311yescharacteristic
102#389C2×C411yescharacteristic
103#389C2×C411yescharacteristic
104#389C2×C411yescharacteristic
105#389C2×C411yescharacteristic
106#389C2×C411yescharacteristic
107#389C2×C411yescharacteristic
108#389C2311yescharacteristic
109#389C2311yescharacteristic
110#388—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—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#381—21no—
143#381—21no—
144#381—21no—
145#381—21no—
146#381—21no—
147#380C2×C411yescharacteristic
148#380C2×C411yescharacteristic
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151#379—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336—22no—
166#336—22no—
167#336—22no—
168#336C2314yes—
169#336C2314yes—
170C25C22⋊C411yescharacteristic
171C2×#41—24no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
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×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C2×M16—28no—
209C2×M16—28no—
210C2×M16—432no—
211C2×M16—44no—
212C2×M16—44no—
213C2×M16—44no—
214C2×M16—44no—
215C22×C8—28no—
216C22×C8—24no—
217C22×C8—24no—
218C22×C8—44no—
219C22×C8—44no—
220C22×C8—44no—
221C22×C8—44no—
222C22×C8—44no—
223#123—44no—
224#123—44no—
225#123—44no—
226#123—44no—
227#123—42no—
228#123—42no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231#121—42no—
232#121—42no—
233#120—44no—
234C4×D4—44no—
235C4×D4—44no—
236#117—24no—
237#117—42no—
238#117—42no—
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⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42—42no—
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×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C22⋊C8—44no—
306C22⋊C8—44no—
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—21no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—41no—
333C24—41no—
334C24#9914yes—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338#41—88no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C22×C4—216no—
355C22×C4—216no—
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—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
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—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
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—416no—
417C22×C4—416no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4#10414yes—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4#13114yes—
432C22×C4AGL(1,ℤ/8)12yes—
433C22×C4#13712yes—
434C22×C4C23×C411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437M16—416no—
438M16—416no—
439C2×C8—28no—
440C2×C8—416no—
441C2×C8—44no—
442C2×C8—44no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—88no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
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—41no—
478C22⋊C4—41no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—42no—
501C42—42no—
502C42—42no—
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—28no—
518C23—28no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23#40812yes—
532C23C2×#9912yes—
533C23C2×#9912yes—
534C23C2×#9912yes—
535C23#41312yes—
536C23#41712yes—
537C23C2×#10412yes—
538C23C22×C22⋊C411yescharacteristic
539Q8—82no—
540D4—82no—
541D4—84no—
542D4—84no—
543D4—84no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
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—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—24no—
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—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
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—41no—
613C2×C4—41no—
614C2×C4—84no—
615C2×C4#40812yes—
616C2×C4#41712yes—
617C2×C4C2×#10412yes—
618C2×C4C2×#10412yes—
619C2×C4#42112yes—
620C2×C4C2×#13112yes—
621C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
622C2×C4C2×#13711yescharacteristic
623C2×C4#57512yes—
624C8—48no—
625C8—84no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—416no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22#146814yes—
654C22#150414yes—
655C22C2×#40811yescharacteristic
656C22C22×#9911yescharacteristic
657C22C2×#41311yescharacteristic
658C22C2×#41711yescharacteristic
659C22#289611yescharacteristic
660C22C22×#10411yescharacteristic
661C22C2×#42111yescharacteristic
662C4—21no—
663C4—21no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—81no—
682C2—24no—
683C2—24no—
684C2—42no—
685C2—161no—
686C2C2×#146811yescharacteristic
687C2#1746211yescharacteristic
688C2#1748111yescharacteristic
689C2C2×#150411yescharacteristic
690C2#1758711yescharacteristic
691C2#1765111yescharacteristic
692C2#1775311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#525C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1817C2semidirect
3. C2×#1728C2semidirect
4. #7695C2semidirect
5. #7694C2semidirect
6. #7681C2semidirect
7. #7678C2semidirect
8. #7579C2semidirect
9. #7578C2semidirect
47. #1272C22semidirect
48. #1271C22semidirect
49. #1271C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366262
C2×#18172
C2×#17282
#76952
#76942
#76812
#76782
#75792
#75782