Finite Group Databasebeta

← #480690

Subgroups — #480690

Order 512.

914 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35530C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic, complemented
3#17368C211yesmaximal, characteristic, complemented
4#11555C212yesmaximal, complemented
5#11554C212yesmaximal, complemented
6#11552C212yesmaximal, complemented
7#11549C212yesmaximal, complemented
8#11512C212yesmaximal, complemented
9#11508C212yesmaximal, complemented
10C2×#537C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2975—21no—
13#2973—21no—
14#2502C411yescharacteristic
15#2502C2211yescharacteristic
16#2501C411yescharacteristic
17#2501C2211yescharacteristic
18#2500C411yescharacteristic
19#2500C2211yescharacteristic
20#2499C411yescharacteristic
21#2499C2211yescharacteristic
22#2443—21no—
23#2443—21no—
24#2443—21no—
25#2443—21no—
26#2423—21no—
27#2423—21no—
28#2422—21no—
29#2422—21no—
30#2317C411yescharacteristic
31#2311C2211yescharacteristic
32#2297C411yescharacteristic
33#2297C411yescharacteristic
34C2×#384C2211yescharacteristic
35#1467—21no—
36#1465—21no—
37#1465—21no—
38#1465—21no—
39#1464C2212yes—
40#1463C2212yes—
41#1463C2212yes—
42#1463C2212yes—
43#1461—21no—
44#1459—21no—
45#1458C2212yes—
46C2×#324C2212yes—
47#1325C2214yes—
48#1325C2214yes—
49#1319C2214yes—
50#1318C2214yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53#537—44no—
54C2×#123—21no—
55C2×#123—21no—
56C2×#123—21no—
57C2×#123—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66#520—44no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C42C2×C411yescharacteristic
86#436—44no—
87#435—44no—
88#432—22no—
89#431—22no—
90#407—22no—
91#407—22no—
92#407—22no—
93C2×C22⋊C8—22no—
94#399—21no—
95#399—21no—
96#399—21no—
97#399—21no—
98#398—21no—
99#398—21no—
100#398—21no—
101#398—21no—
102#397—21no—
103#397—21no—
104#397—21no—
105#397—21no—
106#394—21no—
107#394—21no—
108#394—21no—
109#394—21no—
110#391—21no—
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#391C2311yescharacteristic
115#390—21no—
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2311yescharacteristic
120#389—21no—
121#389—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—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#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#384—21no—
143#384—21no—
144#384C2×C411yescharacteristic
145#384C2×C411yescharacteristic
146#384C2×C411yescharacteristic
147#384C2×C411yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382C2×C411yescharacteristic
151#382C2×C411yescharacteristic
152#380—21no—
153#380—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4C2×C411yescharacteristic
157C4×C22⋊C4C2×C411yescharacteristic
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic
160#324—22no—
161#324—22no—
162#324—22no—
163#324—22no—
164#324C2314yes—
165#324C2314yes—
166#323—22no—
167#323—22no—
168C2×#41—24no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C22×C8—42no—
199#123—82no—
200#123—82no—
201#123—82no—
202#123—82no—
203C22⋊Q8—44no—
204C22⋊Q8—42no—
205C22⋊Q8—42no—
206#121—42no—
207#121—44no—
208#121—44no—
209#121—44no—
210#121—42no—
211C4×Q8—42no—
212C4×Q8—42no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219#117—42no—
220#117—42no—
221#117—42no—
222#117—42no—
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—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—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—82no—
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⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C42—21no—
281C2×C42—42no—
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C4⋊C8—42no—
293C4⋊C8—42no—
294C22⋊C8—24no—
295C22⋊C8—24no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300#97—42no—
301C4×C8—42no—
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#95—21no—
318C24—21no—
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321C24C2×C22⋊C411yescharacteristic
322#41—416no—
323#41—88no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—81no—
350C2×D4#9914yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
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—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400M16—162no—
401M16—162no—
402M16—162no—
403C2×C8—44no—
404C2×C8—44no—
405C2×C8—162no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—42no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23C2×#9912yes—
489C23C2×#9912yes—
490C23C22×C22⋊C411yescharacteristic
491Q8—44no—
492Q8—82no—
493D4—44no—
494D4—44no—
495D4—44no—
496D4—82no—
497D4—84no—
498D4—162no—
499D4—162no—
500C2×C4—28no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—48no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
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—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—84no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—161no—
556C2×C4—161no—
557C2×C4—162no—
558C2×C4C2×#9912yes—
559C2×C4#41312yes—
560C2×C4#56812yes—
561C8—84no—
562C8—84no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#287911yescharacteristic
588C22C22×#9911yescharacteristic
589C22C2×#41311yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—42no—
594C4—42no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—42no—
613C2—81no—
614C2—161no—
615C2#1739211yescharacteristic
616C2#3335911yescharacteristic
617C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17370C2semidirect
3. #17368C2semidirect
4. #11555C2semidirect
5. #11554C2semidirect
6. #11552C2semidirect
7. #11549C2semidirect
8. #11512C2semidirect
9. #11508C2semidirect

Maximal subgroups

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

subgroupindex
#355302
#173702
#173682
#115552
#115542
#115522
#115492
#115122
#115082