Finite Group Databasebeta

← #214284

Subgroups — #214284

Order 512.

1461 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36621C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic, complemented
3C2×#1434C211yesmaximal, characteristic, complemented
4#9147C212yesmaximal
5#9143C212yesmaximal, complemented
6#9114C212yesmaximal, complemented
7#9112C212yesmaximal, complemented
8#8714C212yesmaximal, complemented
9#8330C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#431C2212yes—
13#2855C2211yescharacteristic
14#2854C2211yescharacteristic
15#2853—21no—
16#2852—21no—
17#2852—21no—
18#2851C411yescharacteristic, complemented
19#2851C411yescharacteristic, complemented
20#2850C411yescharacteristic, complemented
21#2850C411yescharacteristic, complemented
22#2849—21no—
23#2848C2211yescharacteristic
24#2847C2211yescharacteristic
25C4×C22⋊Q8C411yescharacteristic, complemented
26C4×C22⋊Q8C411yescharacteristic, complemented
27C4×#123—21no—
28C4×#121C411yescharacteristic, complemented
29C4×#121C411yescharacteristic, complemented
30C4×#120—21no—
31#2290—21no—
32#2290—21no—
33#2283C2211yescharacteristic
34C22×#95—21no—
35C22×#95—21no—
36C22×#95C2212yes—
37#1817C2212yes—
38#1797—21no—
39#1797—21no—
40#1779C2212yes—
41C4×C22⋊C8—21no—
42C4×C22⋊C8—21no—
43C2×#328C2214yes—
44C2×#327C2214yes—
45#1434—21no—
46#1434—21no—
47#1434C2212yes—
48#1434C2212yes—
49#1294C2214yes—
50#1294C2214yes—
51C24×C4—22no—
52C24×C4D411yescharacteristic
53C24×C4D411yescharacteristic
54C24×C4C2311yescharacteristic
55#529—44no—
56C2×#126—21no—
57C2×#125C2×C411yescharacteristic, complemented
58C2×#124C2×C411yescharacteristic, complemented
59C2×#123—21no—
60C2×#123—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×C22⋊Q8C2311yescharacteristic
64C2×#121C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#121C2311yescharacteristic
67C2×#120—21no—
68C2×C4×Q8C2×C411yescharacteristic, complemented
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic, complemented
73C2×#117—21no—
74C22×C42—44no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#431—44no—
88C2×C4⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8C2314yes—
92C2×C4×C8—22no—
93#391C2×C411yescharacteristic, complemented
94#391C2×C411yescharacteristic, complemented
95#390C2×C411yescharacteristic, complemented
96#390C2×C411yescharacteristic, complemented
97#389C2×C411yescharacteristic, complemented
98#389C2×C411yescharacteristic, complemented
99#389C2×C411yescharacteristic, complemented
100#389C2×C411yescharacteristic, complemented
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#388—21no—
108#388—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#386—21no—
118#386—21no—
119#386—21no—
120#386—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#385—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#383—21no—
138#382C2×C411yescharacteristic, complemented
139#382C2×C411yescharacteristic, complemented
140#381—21no—
141#381—21no—
142#380C2×C411yescharacteristic
143#380C2×C411yescharacteristic
144#380C2×C411yescharacteristic
145#380C2×C411yescharacteristic
146#380C2311yescharacteristic
147#380C2311yescharacteristic
148#379—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4C2×C411yescharacteristic, complemented
152C4×C4⋊C4C2×C411yescharacteristic, complemented
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158#376—21no—
159C2×#95—28no—
160C2×#95—22no—
161C2×#95—22no—
162C2×#95—22no—
163C2×#95—22no—
164C2×#95—22no—
165C2×#95—44no—
166C2×#95—44no—
167C2×#95C2314yes—
168#336—22no—
169#336—22no—
170#328—44no—
171#327—44no—
172C25C2×D411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—21no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2411yescharacteristic
215C22×C8—24no—
216#126—44no—
217#125—42no—
218#124—42no—
219#123—44no—
220#123—42no—
221#123—42no—
222C22⋊Q8—24no—
223C22⋊Q8—42no—
224C22⋊Q8—42no—
225#121—24no—
226#121—42no—
227#121—42no—
228#120—44no—
229C4×Q8—42no—
230C4×D4—42no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—44no—
234#117—44no—
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⋊C4—21no—
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
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×C22⋊C4—21no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
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×C42—42no—
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×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C4⋊C8—44no—
302C22⋊C8—24no—
303C22⋊C8—24no—
304C22⋊C8—44no—
305C22⋊C8—44no—
306C4×C8—44no—
307#95—28no—
308#95—28no—
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—
328#95—21no—
329#95—24no—
330#95—24no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24—22no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24C2×C22⋊C411yescharacteristic
343C24C2×C22⋊C411yescharacteristic
344C24#13514yes—
345C24C22×D411yescharacteristic
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×D4—22no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
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—28no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
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—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—416no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4#10414yes—
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×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4AGL(1,ℤ/8)12yes—
454C22×C4#13712yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4H(5,𝔽2)14yes—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—44no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—41no—
486C22⋊C4—41no—
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—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—42no—
515C42—42no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23#41012yes—
546C23#41012yes—
547C23#41612yes—
548C23#41612yes—
549C23C2×#10412yes—
550C23#42112yes—
551C23C22×C22⋊C411yescharacteristic
552C23#53412yes—
553C23C2×#13512yes—
554C23C2×AGL(1,ℤ/8)11yescharacteristic
555C23C2×#13711yescharacteristic
556Q8—82no—
557D4—82no—
558D4—82no—
559D4—82no—
560D4—84no—
561D4—84no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—24no—
575C2×C4—24no—
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—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
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—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
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—832no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4#41712yes—
643C2×C4#41712yes—
644C2×C4C2×#10412yes—
645C2×C4C2×#10412yes—
646C2×C4#51812yes—
647C2×C4#53412yes—
648C8—84no—
649C22—216no—
650C22—216no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—21no—
659C22—21no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—21no—
665C22—21no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22#150014yes—
679C22#152114yes—
680C22#230811yescharacteristic
681C22C2×#41011yescharacteristic
682C22C2×#41011yescharacteristic
683C22C2×#41611yescharacteristic
684C22C2×#41711yescharacteristic
685C22C22×#10411yescharacteristic
686C22C2×#42111yescharacteristic
687C22#318812yes—
688C22#319012yes—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—41no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—88no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—81no—
708C4—81no—
709C2—22no—
710C2—22no—
711C2—24no—
712C2—24no—
713C2—161no—
714C2C2×#150011yescharacteristic
715C2#1754611yescharacteristic
716C2C2×#152111yescharacteristic
717C2#1758811yescharacteristic
718C2#2174811yescharacteristic
719C2#2176311yescharacteristic
720C2#2177311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2847C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20714C2semidirect
3. C2×#1434C2semidirect
5. #9143C2semidirect
6. #9114C2semidirect
7. #9112C2semidirect
8. #8714C2semidirect
9. #8330C2semidirect
18. #2851C4semidirect
19. #2851C4semidirect
20. #2850C4semidirect
21. #2850C4semidirect
25. C4×C22⋊Q8C4semidirect
26. C4×C22⋊Q8C4semidirect
28. C4×#121C4semidirect
29. C4×#121C4semidirect
57. C2×#125C2×C4semidirect
58. C2×#124C2×C4semidirect
68. C2×C4×Q8C2×C4semidirect
72. C2×C4×D4C2×C4semidirect
93. #391C2×C4semidirect
94. #391C2×C4semidirect
95. #390C2×C4semidirect
96. #390C2×C4semidirect
97. #389C2×C4semidirect
98. #389C2×C4semidirect
99. #389C2×C4semidirect
100. #389C2×C4semidirect
138. #382C2×C4semidirect
139. #382C2×C4semidirect
151. C4×C4⋊C4C2×C4semidirect
152. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366212
#207142
C2×#14342
#91472
#91432
#91142
#91122
#87142
#83302