Finite Group Databasebeta

← #162028

Subgroups — #162028

Order 512.

1221 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#7715C212yesmaximal, complemented
5#7714C212yesmaximal, complemented
6#7681C212yesmaximal, complemented
7#7678C212yesmaximal, complemented
8#7598C212yesmaximal, complemented
9#7597C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C22×C4⋊C8—21no—
14C2×#407—21no—
15C22×C22⋊C8—21no—
16#2855C411yescharacteristic
17#2855C411yescharacteristic
18#2855C2211yescharacteristic
19#2855C2211yescharacteristic
20#2854C411yescharacteristic
21#2854C411yescharacteristic
22#2854C2211yescharacteristic
23#2854C2211yescharacteristic
24#2853—21no—
25#2853—21no—
26#2849—21no—
27#2849—21no—
28C4×#123—21no—
29C4×#123—21no—
30C4×#123—21no—
31C4×#123—21no—
32#2297C411yescharacteristic
33#2297C2211yescharacteristic
34#2296C411yescharacteristic
35#2296C2211yescharacteristic
36#2280C411yescharacteristic
37#1797—21no—
38#1797—21no—
39#1797—21no—
40#1797—21no—
41#1797C2212yes—
42#1797C2212yes—
43#1797C2212yes—
44#1797C2212yes—
45#1728C2212yes—
46C2×#336C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57#525—44no—
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#123—21no—
62C2×#123—21no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#120—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#423—44no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—44no—
92#407—44no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—44no—
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2311yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2311yescharacteristic
106#389C2×C411yescharacteristic
107#389C2×C411yescharacteristic
108#389C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#389C2×C411yescharacteristic
112#389C2311yescharacteristic
113#389C2311yescharacteristic
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#381—21no—
147#381—21no—
148#381—21no—
149#381—21no—
150#381—21no—
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic
154#380C2×C411yescharacteristic
155#379—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
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—22no—
184C23×C4—22no—
185C23×C4—21no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
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×C4C22×C411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C2×M16—44no—
210C2×M16—44no—
211C2×M16—44no—
212C22×C8—24no—
213C22×C8—24no—
214C22×C8—44no—
215C22×C8—44no—
216C22×C8—44no—
217#123—44no—
218#123—44no—
219#123—44no—
220#123—42no—
221#123—42no—
222#123—42no—
223#123—42no—
224C22⋊Q8—42no—
225C22⋊Q8—42no—
226#121—42no—
227#121—42no—
228#120—44no—
229C4×D4—44no—
230C4×D4—44no—
231#117—24no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
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⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
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—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—42no—
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×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C4⋊C8—44no—
301C4⋊C8—44no—
302C4⋊C8—44no—
303C4⋊C8—44no—
304C22⋊C8—44no—
305C22⋊C8—44no—
306C22⋊C8—44no—
307C22⋊C8—44no—
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—
327#95—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—41no—
335C24—41no—
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339#41—88no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×D4—22no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
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—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
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—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—41no—
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—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4#9914yes—
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×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4#13114yes—
435C22×C4AGL(1,ℤ/8)12yes—
436C22×C4AGL(1,ℤ/8)12yes—
437C22×C4#13712yes—
438C22×C4#13712yes—
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442M16—88no—
443C2×C8—44no—
444C2×C8—44no—
445C2×C8—44no—
446C2×C8—44no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—88no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊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—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—42no—
506C42—42no—
507C42—42no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—28no—
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—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#40812yes—
539C23C2×#9912yes—
540C23#41712yes—
541C23#41712yes—
542C23#42112yes—
543C23C22×C22⋊C411yescharacteristic
544Q8—82no—
545D4—82no—
546D4—84no—
547D4—84no—
548D4—84no—
549C2×C4—24no—
550C2×C4—24no—
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—24no—
566C2×C4—24no—
567C2×C4—28no—
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—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
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—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—84no—
621C2×C4#40812yes—
622C2×C4C2×#9912yes—
623C2×C4C2×#9912yes—
624C2×C4#41312yes—
625C2×C4#41712yes—
626C2×C4#41712yes—
627C2×C4#42112yes—
628C2×C4C2×#13112yes—
629C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
630C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
631C2×C4C2×#13711yescharacteristic
632C2×C4C2×#13711yescharacteristic
633C8—84no—
634C8—84no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—416no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#151314yes—
664C22C2×#40811yescharacteristic
665C22C22×#9911yescharacteristic
666C22C2×#41311yescharacteristic
667C22C2×#41711yescharacteristic
668C22C2×#41711yescharacteristic
669C22C2×#42111yescharacteristic
670C22C2×#42111yescharacteristic
671C4—21no—
672C4—21no—
673C4—22no—
674C4—21no—
675C4—21no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4#146814yes—
692C2—24no—
693C2—42no—
694C2—42no—
695C2—161no—
696C2C2×#146811yescharacteristic
697C2#1746611yescharacteristic
698C2#1748111yescharacteristic
699C2C2×#151311yescharacteristic
700C2#1759211yescharacteristic
701C2#1766211yescharacteristic
702C2#1775211yescharacteristic
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. #20626C2semidirect
3. C2×#1797C2semidirect
4. #7715C2semidirect
5. #7714C2semidirect
6. #7681C2semidirect
7. #7678C2semidirect
8. #7598C2semidirect
9. #7597C2semidirect
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
#206262
C2×#17972
#77152
#77142
#76812
#76782
#75982
#75972