Finite Group Databasebeta

← #163028

Subgroups — #163028

Order 512.

1155 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3#20626C211yesmaximal, characteristic, complemented
4#7715C212yesmaximal, complemented
5#7714C212yesmaximal, complemented
6#7695C212yesmaximal, complemented
7#7694C212yesmaximal, complemented
8#7613C212yesmaximal, complemented
9#7612C212yesmaximal, complemented
10C2×#525C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13C2×#423—21no—
14C2×#407—21no—
15#2855C411yescharacteristic
16#2855C411yescharacteristic
17#2855C2211yescharacteristic
18#2855C2211yescharacteristic
19#2854C411yescharacteristic
20#2854C411yescharacteristic
21#2854C2211yescharacteristic
22#2854C2211yescharacteristic
23#2853—21no—
24#2853—21no—
25#2849—21no—
26#2849—21no—
27C4×#123—21no—
28C4×#123—21no—
29C4×#123—21no—
30C4×#123—21no—
31#2297C411yescharacteristic
32#2297C2211yescharacteristic
33#2296C411yescharacteristic
34#2296C2211yescharacteristic
35#2280C411yescharacteristic
36#1817C2212yes—
37#1797—21no—
38#1797—21no—
39#1797—21no—
40#1797C2212yes—
41#1797C2212yes—
42#1797C2212yes—
43#1779—21no—
44#1749—21no—
45#1749C2212yes—
46#1728C2212yes—
47#1272C2214yescomplemented
48#1271C2214yescomplemented
49#1271C2214yescomplemented
50#1270C2214yescomplemented
51C22×#41C2×C411yescharacteristic
52C24×C4C2×C411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4D411yescharacteristic
55C22×M16—22no—
56#525—44no—
57C2×#123—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×#123—21no—
61C2×#123—21no—
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#120—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C42—44no—
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#432—44no—
86#423—44no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90#407—44no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×#97—22no—
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2×C411yescharacteristic
100#391C2311yescharacteristic
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2311yescharacteristic
105#389C2×C411yescharacteristic
106#389C2×C411yescharacteristic
107#389C2×C411yescharacteristic
108#389C2×C411yescharacteristic
109#389C2×C411yescharacteristic
110#389C2×C411yescharacteristic
111#389C2311yescharacteristic
112#389C2311yescharacteristic
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#381—21no—
146#381—21no—
147#381—21no—
148#381—21no—
149#381—21no—
150#380C2×C411yescharacteristic
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic
154#379—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164C4×C22⋊C4—21no—
165#336—22no—
166#336—22no—
167#336C2314yes—
168#336C2314yes—
169C25C22⋊C411yescharacteristic
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—21no—
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×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C2×M16—44no—
208C2×M16—44no—
209C2×M16—44no—
210C22×C8—24no—
211C22×C8—24no—
212#123—44no—
213#123—44no—
214#123—44no—
215#123—44no—
216#123—42no—
217#123—42no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220#121—42no—
221#121—42no—
222#120—44no—
223C4×D4—44no—
224C4×D4—44no—
225#117—42no—
226#117—42no—
227#117—42no—
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⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
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×C42—24no—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2411yescharacteristic
294C4⋊C8—44no—
295C4⋊C8—44no—
296C4⋊C8—44no—
297C22⋊C8—44no—
298C22⋊C8—44no—
299C22⋊C8—44no—
300C22⋊C8—44no—
301C22⋊C8—44no—
302#97—44no—
303#95—21no—
304#95—21no—
305#95—21no—
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—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—41no—
328C24—41no—
329C24—41no—
330C24—41no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334#41—88no—
335C2×Q8—22no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
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—21no—
366C22×C4—21no—
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—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
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—48no—
412C22×C4—48no—
413C22×C4—416no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—416no—
417C22×C4—42no—
418C22×C4#9914yes—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
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×C4AGL(1,ℤ/8)12yes—
429C22×C4#13712yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433M16—88no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—44no—
437C2×C8—44no—
438C2×C8—44no—
439C2×C8—44no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—42no—
495C42—42no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23C2×#9912yes—
529C23#41312yes—
530C23#41712yes—
531C23C22×C22⋊C411yescharacteristic
532Q8—82no—
533D4—82no—
534D4—84no—
535D4—84no—
536D4—84no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
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—21no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
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—84no—
608C2×C4C2×#9912yes—
609C2×C4C2×#9912yes—
610C2×C4#41712yes—
611C2×C4#42112yes—
612C2×C4#42112yes—
613C2×C4#56812yes—
614C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
615C2×C4C2×#13711yescharacteristic
616C2×C4#57512yes—
617C8—84no—
618C8—84no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—416no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22#287911yescharacteristic
649C22C22×#9911yescharacteristic
650C22C2×#41311yescharacteristic
651C22C2×#41711yescharacteristic
652C22#289611yescharacteristic
653C22C2×#42111yescharacteristic
654C22C2×#42111yescharacteristic
655C4—22no—
656C4—22no—
657C4—21no—
658C4—21no—
659C4—22no—
660C4—22no—
661C4—22no—
662C4—22no—
663C4—22no—
664C4—41no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—81no—
669C4—81no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—81no—
675C2—42no—
676C2—42no—
677C2—42no—
678C2—161no—
679C2#1739211yescharacteristic
680C2#1749311yescharacteristic
681C2#1749811yescharacteristic
682C2#1755911yescharacteristic
683C2#1759211yescharacteristic
684C2#1767411yescharacteristic
685C2#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. #20715C2semidirect
3. #20626C2semidirect
4. #7715C2semidirect
5. #7714C2semidirect
6. #7695C2semidirect
7. #7694C2semidirect
8. #7613C2semidirect
9. #7612C2semidirect
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
#207152
#206262
#77152
#77142
#76952
#76942
#76132
#76122