Finite Group Databasebeta

← #480843

Subgroups — #480843

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35453C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#11578C212yesmaximal, complemented
5#11558C212yesmaximal, complemented
6#11548C212yesmaximal, complemented
7#11544C212yesmaximal, complemented
8#11528C212yesmaximal, complemented
9#11526C212yesmaximal, complemented
10C22×#125C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2983—21no—
13#2973—21no—
14#2579C411yescharacteristic
15#2579C2211yescharacteristic
16#2486C411yescharacteristic
17#2486C2211yescharacteristic
18#2484C2211yescharacteristic
19#2484C2211yescharacteristic
20#2475—21no—
21#2475—21no—
22#2473—21no—
23#2473—21no—
24#2462—21no—
25#2462—21no—
26#2460—21no—
27#2460—21no—
28#2406C411yescharacteristic
29#2406C411yescharacteristic
30#2322C411yescharacteristic
31#2322C2211yescharacteristic
32#2313C411yescharacteristic
33C4×#123C411yescharacteristic
34C4×#123C2211yescharacteristic
35#1467—21no—
36#1466C2212yes—
37#1466C2212yes—
38#1465C2212yes—
39#1464—21no—
40#1464—21no—
41#1464C2212yes—
42#1463—21no—
43#1460—21no—
44#1459C2212yes—
45#1458C2212yes—
46C2×#324—21no—
47#1326C2214yes—
48#1326C2214yes—
49#1317C2214yes—
50#1317C2214yes—
51C22×#41C2×C411yescharacteristic
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—44no—
57C2×#123C2×C411yescharacteristic
58C2×#123C2×C411yescharacteristic
59C2×#123C2×C411yescharacteristic
60C2×#123C2×C411yescharacteristic
61#520—44no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C42C2×C411yescharacteristic
82#446—44no—
83#435—44no—
84#432—22no—
85#432—22no—
86#432—22no—
87#431—22no—
88#407—22no—
89C2×C22⋊C8—22no—
90#400—21no—
91#400—21no—
92#398—21no—
93#398—21no—
94#398—21no—
95#398—21no—
96#398—21no—
97#398—21no—
98#397—21no—
99#397—21no—
100#396—21no—
101#396—21no—
102#395—21no—
103#395—21no—
104#394—21no—
105#394—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#388—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#387—21no—
115#387—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#386C2×C411yescharacteristic
129#386C2×C411yescharacteristic
130#386C2×C411yescharacteristic
131#386C2×C411yescharacteristic
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385C2×C411yescharacteristic
141#385C2×C411yescharacteristic
142#385C2311yescharacteristic
143#385C2311yescharacteristic
144#381C2×C411yescharacteristic
145#381C2×C411yescharacteristic
146#381C2×C411yescharacteristic
147#381C2×C411yescharacteristic
148#381C2311yescharacteristic
149#381C2311yescharacteristic
150#380—21no—
151#380—21no—
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic
160#324—22no—
161#324—22no—
162#324—22no—
163#324—22no—
164#323—22no—
165#323—22no—
166#323C2314yes—
167#323C2314yes—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41—42no—
171C22×Q8C22⋊C411yescharacteristic
172C22×D4C22⋊C411yescharacteristic
173C22×D4C22×C411yescharacteristic
174C22×D4C22×C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C2×M16—42no—
189C22×C8—42no—
190#125—44no—
191#125—44no—
192#125—44no—
193#125—44no—
194#123—42no—
195#123—42no—
196#123—42no—
197#123—42no—
198C4×Q8—24no—
199C4×Q8—42no—
200C4×D4—24no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
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—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22×C411yescharacteristic
258C2×C22⋊C4C22×C411yescharacteristic
259C2×C22⋊C4C22×C411yescharacteristic
260C2×C22⋊C4C22×C411yescharacteristic
261C2×C22⋊C4C22×C411yescharacteristic
262C2×C22⋊C4C2×D411yescharacteristic
263C2×C22⋊C4C2×D411yescharacteristic
264C2×C22⋊C4C2411yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—42no—
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C4⋊C8—42no—
277C4⋊C8—42no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284#97—42no—
285#97—42no—
286#97—42no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
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—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C23×C411yescharacteristic
315#41—416no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379M16—162no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—162no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—81no—
409C4⋊C4—81no—
410C4⋊C4—81no—
411C4⋊C4—81no—
412C4⋊C4#9914yes—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—41no—
436C42—41no—
437C42—42no—
438C42—42no—
439C42—81no—
440C42—81no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23#41312yes—
455C23C22×C22⋊C411yescharacteristic
456C23#56812yes—
457Q8—44no—
458D4—44no—
459D4—44no—
460D4—44no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—161no—
523C2×C4—161no—
524C2×C4C2×#9912yes—
525C2×C4C2×#9912yes—
526C2×C4C2×#9912yes—
527C8—84no—
528C8—84no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22#287911yescharacteristic
547C22C22×#9911yescharacteristic
548C22C2×#41311yescharacteristic
549C4—22no—
550C4—22no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—42no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C2—21no—
570C2—21no—
571C2—42no—
572C2—42no—
573C2—81no—
574C2#1740011yescharacteristic
575C2#3334911yescharacteristic
576C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11578C2semidirect
5. #11558C2semidirect
6. #11548C2semidirect
7. #11544C2semidirect
8. #11528C2semidirect
9. #11526C2semidirect

Maximal subgroups

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

subgroupindex
#354532
#173722
#173682
#115782
#115582
#115482
#115442
#115282
#115262