Finite Group Databasebeta

← #481716

Subgroups — #481716

Order 512.

1137 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36010C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic
3#17367C211yesmaximal, characteristic
4#11760C212yesmaximal, complemented
5#11759C212yesmaximal, complemented
6#11756C212yesmaximal, complemented
7#11755C212yesmaximal, complemented
8#11716C212yesmaximal, complemented
9C2×#1344C212yesmaximal, complemented
10C2×#552C411yescharacteristic, complemented
11C2×#518—21no—
12C2×#518C2211yescharacteristic
13#2973—21no—
14#2638C411yescharacteristic, complemented
15#2633C411yescharacteristic, complemented
16#2626C2211yescharacteristic
17#2611C2211yescharacteristic
18#2599C2211yescharacteristic
19#2564C411yescharacteristic, complemented
20#2561C2211yescharacteristic
21#2486C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2480—21no—
24#2475—21no—
25#2466—21no—
26#2460—21no—
27#2419—21no—
28#2405—21no—
29#2398C411yescharacteristic, complemented
30#2395C411yescharacteristic, complemented
31C2×#397C2211yescharacteristic
32#2359—21no—
33#2353C411yescharacteristic, complemented
34#2342—21no—
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#1351C2214yes—
49#1344C2214yes—
50#1344C2214yes—
51C23×D4D411yescharacteristic
52#552—44no—
53C2×#126—21no—
54C2×#125—21no—
55C2×#125C2×C411yescharacteristic, complemented
56C2×#123—21no—
57C2×#123—21no—
58C2×C22⋊Q8C2×C411yescharacteristic, complemented
59C2×C22⋊Q8C2×C411yescharacteristic, complemented
60#518—28no—
61#518—44no—
62#518—44no—
63C2×C4×Q8C2×C411yescharacteristic, complemented
64C2×C4×D4—44no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×#117—21no—
74C2×#117—22no—
75C2×#117C2×C411yescharacteristic, complemented
76C2×#117D411yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic, complemented
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85C22×C22⋊C4C2311yescharacteristic
86#435—44no—
87#432—22no—
88#407—22no—
89C2×C22⋊C8—22no—
90#400—21no—
91#400C2×C411yescharacteristic, complemented
92#399C2×C411yescharacteristic, complemented
93#399C2×C411yescharacteristic, complemented
94#399C2311yescharacteristic
95#398—21no—
96#398C2×C411yescharacteristic, complemented
97#398C2×C411yescharacteristic, complemented
98#397—21no—
99#397—21no—
100#397—21no—
101#397—21no—
102#397C2×C411yescharacteristic, complemented
103#397C2×C411yescharacteristic, complemented
104#397C2×C411yescharacteristic, complemented
105#397C2×C411yescharacteristic, complemented
106#397C2311yescharacteristic
107#397C2311yescharacteristic
108#396—21no—
109#396—21no—
110#395C2×C411yescharacteristic, complemented
111#394—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#393C2×C411yescharacteristic, complemented
116#393C2×C411yescharacteristic, complemented
117#393C2311yescharacteristic
118#389—21no—
119#389—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#386—21no—
134#386—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#383C2×C411yescharacteristic, complemented
144#381—21no—
145#381—21no—
146#381C2×C411yescharacteristic, complemented
147#380—21no—
148#380—21no—
149#380C2×C411yescharacteristic, complemented
150#379—21no—
151#379—21no—
152#379—21no—
153C4×C4⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2×C411yescharacteristic, complemented
159#376C2×C411yescharacteristic, complemented
160C2×#95—21no—
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×Q8C22×C411yescharacteristic
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C2×M16—42no—
195C22×C8—42no—
196#126—44no—
197#125—42no—
198#125—44no—
199#123—44no—
200#123—44no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C4×Q8—42no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—432no—
213#117—28no—
214#117—42no—
215#117—44no—
216#117—42no—
217#117—44no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
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—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22×C411yescharacteristic
277C2×C22⋊C4C22×C411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C22⋊C4C2×D411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2411yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—42no—
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C4⋊C8—42no—
294C22⋊C8—24no—
295C22⋊C8—42no—
296C22⋊C8—42no—
297#97—42no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
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#95C22×C411yescharacteristic
318#95C22×C411yescharacteristic
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321C24—22no—
322C24—22no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—81no—
328C24—81no—
329C24—81no—
330C24—81no—
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24AGL(1,ℤ/8)12yes—
336C24#13712yes—
337C24C22×D411yescharacteristic
338C2×Q8—41no—
339C2×Q8—81no—
340C2×D4—216no—
341C2×D4—216no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4#9914yes—
353C2×D4#10414yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—162no—
392C22×C4—162no—
393C22×C4—162no—
394C22×C4—162no—
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405M16—162no—
406C2×C8—44no—
407C2×C8—162no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—48no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—28no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—44no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—81no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C42—22no—
466C42—22no—
467C42—41no—
468C42—41no—
469C42—42no—
470C42—42no—
471C42—48no—
472C42—81no—
473C42—81no—
474C23—24no—
475C23—24no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—24no—
485C23—24no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23—162no—
503C23—161no—
504C23—161no—
505C23—162no—
506C23C2×#9912yes—
507C23C2×#9912yes—
508C23#41712yes—
509C23#41712yes—
510C23C2×#10412yes—
511C23C2×#10412yes—
512C23C22×C22⋊C411yescharacteristic
513C23C2×AGL(1,ℤ/8)11yescharacteristic
514C23C2×#13711yescharacteristic
515Q8—82no—
516D4—28no—
517D4—28no—
518D4—28no—
519D4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—161no—
575C2×C4—161no—
576C2×C4—161no—
577C2×C4—161no—
578C2×C4—162no—
579C2×C4—168no—
580C2×C4—168no—
581C2×C4C2×#9912yes—
582C2×C4C2×#10412yes—
583C8—84no—
584C22—21no—
585C22—22no—
586C22—21no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—84no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22#150414yes—
617C22C22×#9911yescharacteristic
618C22C2×#41711yescharacteristic
619C22C22×#10411yescharacteristic
620C4—22no—
621C4—22no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—88no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—81no—
639C2—21no—
640C2—21no—
641C2—24no—
642C2—41no—
643C2—42no—
644C2—42no—
645C2—42no—
646C2—41no—
647C2C2×#150411yescharacteristic
648C2#3335011yescharacteristic
649C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
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
4. #11760C2semidirect
5. #11759C2semidirect
6. #11756C2semidirect
7. #11755C2semidirect
8. #11716C2semidirect
9. C2×#1344C2semidirect
10. C2×#552C4semidirect
14. #2638C4semidirect
15. #2633C4semidirect
19. #2564C4semidirect
21. #2486C4semidirect
29. #2398C4semidirect
30. #2395C4semidirect
33. #2353C4semidirect
55. C2×#125C2×C4semidirect
58. C2×C22⋊Q8C2×C4semidirect
59. C2×C22⋊Q8C2×C4semidirect
63. C2×C4×Q8C2×C4semidirect
75. C2×#117C2×C4semidirect
78. C22×C4⋊C4C2×C4semidirect
91. #400C2×C4semidirect
92. #399C2×C4semidirect
93. #399C2×C4semidirect
96. #398C2×C4semidirect
97. #398C2×C4semidirect
102. #397C2×C4semidirect
103. #397C2×C4semidirect
104. #397C2×C4semidirect
105. #397C2×C4semidirect
110. #395C2×C4semidirect
115. #393C2×C4semidirect
116. #393C2×C4semidirect
143. #383C2×C4semidirect
146. #381C2×C4semidirect
149. #380C2×C4semidirect
158. C4×C22⋊C4C2×C4semidirect
159. #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
#360102
#203542
#173672
#117602
#117592
#117562
#117552
#117162
C2×#13442