Finite Group Databasebeta

← #481686

Subgroups — #481686

Order 512.

1207 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35840C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic, complemented
3#17367C211yesmaximal, characteristic, complemented
4#11760C212yesmaximal, complemented
5#11757C212yesmaximal, complemented
6#11755C212yesmaximal, complemented
7#11754C212yesmaximal, complemented
8C2×#1352C212yesmaximal, complemented
9#11692C212yesmaximal, complemented
10C2×#553C411yescharacteristic, complemented
11C2×#518—21no—
12C2×#518C2211yescharacteristic
13#2973—21no—
14#2635C411yescharacteristic, complemented
15#2633C411yescharacteristic, complemented
16#2617C2211yescharacteristic
17#2611C2211yescharacteristic
18#2598C2211yescharacteristic
19#2543C411yescharacteristic, complemented
20#2539C2211yescharacteristic
21#2487C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2479—21no—
24#2467—21no—
25#2455—21no—
26#2443—21no—
27#2418—21no—
28#2410—21no—
29#2397C411yescharacteristic, complemented
30#2395C411yescharacteristic, complemented
31C2×#399C2211yescharacteristic
32#2361—21no—
33#2358—21no—
34#2353C411yescharacteristic, complemented
35#1740—21no—
36#1740C2212yes—
37C2×#342—21no—
38C2×#342—21no—
39C2×#342C2212yes—
40C2×#342C2212yes—
41#1466—21no—
42#1465—21no—
43#1464C2212yes—
44#1462C2212yes—
45#1459—21no—
46C2×#323C2212yes—
47#1352C2214yes—
48#1352C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C23×D4D411yescharacteristic
52#553—44no—
53C2×#126—21no—
54C2×#125C2×C411yescharacteristic, complemented
55C2×#123—21no—
56C2×#123—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121C2×C411yescharacteristic, complemented
60C2×#121C2×C411yescharacteristic, complemented
61C2×#120—21no—
62C2×#120—21no—
63#518—28no—
64#518—44no—
65#518—44no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—44no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic, complemented
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×#117—22no—
80C2×#117C2×C411yescharacteristic, complemented
81C2×#117D411yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic, complemented
85C22×C22⋊C4—22no—
86C22×C22⋊C4—22no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4D411yescharacteristic
89C22×C22⋊C4D411yescharacteristic
90C22×C22⋊C4C2311yescharacteristic
91C22×C22⋊C4C2311yescharacteristic
92#435—44no—
93#432—22no—
94#407—22no—
95C2×C22⋊C8—22no—
96#400C2×C411yescharacteristic, complemented
97#400C2×C411yescharacteristic, complemented
98#399—21no—
99#399—21no—
100#399—21no—
101#399C2×C411yescharacteristic, complemented
102#399C2×C411yescharacteristic, complemented
103#399C2×C411yescharacteristic, complemented
104#399C2×C411yescharacteristic, complemented
105#399C2311yescharacteristic
106#399C2311yescharacteristic
107#398—21no—
108#398C2×C411yescharacteristic, complemented
109#397—21no—
110#397C2×C411yescharacteristic, complemented
111#397C2×C411yescharacteristic, complemented
112#397C2311yescharacteristic
113#394—21no—
114#394—21no—
115#394—21no—
116#394—21no—
117#394—21no—
118#394C2×C411yescharacteristic, complemented
119#392—21no—
120#392—21no—
121#392C2×C411yescharacteristic, complemented
122#392C2×C411yescharacteristic, complemented
123#392C2311yescharacteristic
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—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#385—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#383C2×C411yescharacteristic, complemented
152#381C2×C411yescharacteristic, complemented
153#380—21no—
154#380C2×C411yescharacteristic, complemented
155#379—21no—
156C4×C4⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4C2×C411yescharacteristic, complemented
160#376C2×C411yescharacteristic, complemented
161C2×#95C2×C411yescharacteristic, complemented
162#342—22no—
163#342—22no—
164#342—22no—
165#342C2314yes—
166#324—22no—
167#323—22no—
168#323—22no—
169#323C2314yes—
170C25C2×D411yescharacteristic
171C25C2×D411yescharacteristic
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C22⋊C411yescharacteristic
184C22×D4C22⋊C411yescharacteristic
185C22×D4C22×C411yescharacteristic
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
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×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C2×M16—42no—
201C22×C8—42no—
202#126—44no—
203#125—42no—
204#123—44no—
205#123—82no—
206#121—42no—
207#121—42no—
208#121—44no—
209#121—44no—
210#120—42no—
211#120—42no—
212#120—82no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—44no—
223C4×D4—432no—
224C4×D4—82no—
225C4×D4—82no—
226#117—28no—
227#117—42no—
228#117—42no—
229#117—44no—
230#117—42no—
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⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—82no—
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×C4⋊C4C2×D411yescharacteristic
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
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×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—82no—
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C22⋊C4C22⋊C411yescharacteristic
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22×C411yescharacteristic
290C2×C22⋊C4C22×C411yescharacteristic
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C22⋊C4C2×D411yescharacteristic
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2411yescharacteristic
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—42no—
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22⋊C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C4⋊C8—42no—
306C22⋊C8—24no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309#97—42no—
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#95C22×C411yescharacteristic
324#95C22×C411yescharacteristic
325#95C22×C411yescharacteristic
326#95C22×C411yescharacteristic
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—21no—
334C24—81no—
335C24—81no—
336C24—81no—
337C24—81no—
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24C2×C22⋊C411yescharacteristic
341C24C2×C22⋊C411yescharacteristic
342C24AGL(1,ℤ/8)12yes—
343C24#13712yes—
344C24C22×D411yescharacteristic
345C2×D4—216no—
346C2×D4—216no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—81no—
368C2×D4—81no—
369C2×D4—81no—
370C2×D4—81no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4#9914yes—
376C2×D4#10414yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—162no—
417C22×C4—162no—
418C22×C4—162no—
419C22×C4—162no—
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×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430M16—162no—
431C2×C8—44no—
432C2×C8—162no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—48no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—28no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—42no—
495C42—42no—
496C42—48no—
497C42—81no—
498C23—24no—
499C23—24no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—24no—
509C23—24no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—82no—
531C23—82no—
532C23—82no—
533C23—162no—
534C23—162no—
535C23—161no—
536C23—161no—
537C23C2×#9912yes—
538C23C2×#9912yes—
539C23#41712yes—
540C23#41712yes—
541C23C2×#10412yes—
542C23C2×#10412yes—
543C23C22×C22⋊C411yescharacteristic
544C23C2×AGL(1,ℤ/8)11yescharacteristic
545C23C2×#13711yescharacteristic
546D4—28no—
547D4—28no—
548D4—28no—
549D4—28no—
550D4—84no—
551D4—82no—
552D4—84no—
553D4—84no—
554D4—162no—
555D4—161no—
556D4—161no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
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—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—48no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
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—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—161no—
609C2×C4—161no—
610C2×C4—161no—
611C2×C4—161no—
612C2×C4—162no—
613C2×C4—162no—
614C2×C4—168no—
615C2×C4—168no—
616C2×C4C2×#9912yes—
617C2×C4C2×#10412yes—
618C8—84no—
619C22—21no—
620C22—22no—
621C22—21no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—21no—
626C22—21no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—84no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—84no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—84no—
654C22—84no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22#150414yes—
659C22C22×#9911yescharacteristic
660C22C2×#41711yescharacteristic
661C22C22×#10411yescharacteristic
662C4—22no—
663C4—22no—
664C4—42no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—81no—
676C4—88no—
677C4—81no—
678C4—81no—
679C4—81no—
680C2—21no—
681C2—21no—
682C2—24no—
683C2—41no—
684C2—42no—
685C2—42no—
686C2—42no—
687C2—41no—
688C2—161no—
689C2C2×#150411yescharacteristic
690C2#3335011yescharacteristic
691C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35840C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20354C2semidirect
3. #17367C2semidirect
4. #11760C2semidirect
5. #11757C2semidirect
6. #11755C2semidirect
7. #11754C2semidirect
8. C2×#1352C2semidirect
9. #11692C2semidirect
10. C2×#553C4semidirect
14. #2635C4semidirect
15. #2633C4semidirect
19. #2543C4semidirect
21. #2487C4semidirect
29. #2397C4semidirect
30. #2395C4semidirect
34. #2353C4semidirect
54. C2×#125C2×C4semidirect
59. C2×#121C2×C4semidirect
60. C2×#121C2×C4semidirect
74. C2×C4×D4C2×C4semidirect
80. C2×#117C2×C4semidirect
84. C22×C4⋊C4C2×C4semidirect
96. #400C2×C4semidirect
97. #400C2×C4semidirect
101. #399C2×C4semidirect
102. #399C2×C4semidirect
103. #399C2×C4semidirect
104. #399C2×C4semidirect
108. #398C2×C4semidirect
110. #397C2×C4semidirect
111. #397C2×C4semidirect
118. #394C2×C4semidirect
121. #392C2×C4semidirect
122. #392C2×C4semidirect
151. #383C2×C4semidirect
152. #381C2×C4semidirect
154. #380C2×C4semidirect
159. C4×C22⋊C4C2×C4semidirect
160. #376C2×C4semidirect
161. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#358402
#203542
#173672
#117602
#117572
#117552
#117542
C2×#13522
#116922