Finite Group Databasebeta

← #482612

Subgroups — #482612

Order 512.

827 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic
3#17371C211yesmaximal, characteristic
4#11846C212yesmaximal
5#11842C212yesmaximal
6#11810C212yesmaximal
7#11801C212yesmaximal
8#11760C212yesmaximal
9#11756C212yesmaximal
10C2×#557C411yescharacteristic, complemented
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2634C411yescharacteristic, complemented
14#2634C411yescharacteristic, complemented
15#2621C2211yescharacteristic
16#2611C2211yescharacteristic
17#2611C2211yescharacteristic
18#2599C2211yescharacteristic
19#2599C2211yescharacteristic
20#2581C411yescharacteristic, complemented
21#2546C411yescharacteristic, complemented
22#2475—21no—
23#2475—21no—
24#2462—21no—
25#2462—21no—
26#2446—21no—
27#2434—21no—
28#2399C411yescharacteristic, complemented
29#2399C411yescharacteristic, complemented
30#2396C2211yescharacteristic
31#2365—21no—
32#2362—21no—
33#2356C411yescharacteristic, complemented
34#2310—21no—
35#1820—21no—
36#1820—21no—
37#1820C2212yes—
38#1820C2212yes—
39#1740—21no—
40#1740C2212yes—
41#1467—21no—
42#1467—21no—
43#1464C2212yes—
44#1464C2212yes—
45#1461—21no—
46#1458C2212yes—
47#1352C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1344C2214yes—
51C23×Q8D411yescharacteristic
52#557—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×#125C2×C411yescharacteristic
60C2×C22⋊Q8—21no—
61#519—44no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×#117—22no—
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81#446—44no—
82#432—22no—
83#432—22no—
84#431—22no—
85#400—21no—
86#400—21no—
87#400C2×C411yescharacteristic
88#400C2×C411yescharacteristic
89#400C2×C411yescharacteristic
90#399C2311yescharacteristic
91#398—21no—
92#398—21no—
93#398—21no—
94#398—21no—
95#398C2×C411yescharacteristic
96#398C2×C411yescharacteristic
97#398C2×C411yescharacteristic
98#398C2×C411yescharacteristic
99#398C2×C411yescharacteristic
100#398C2×C411yescharacteristic
101#397—21no—
102#397C2311yescharacteristic
103#397C2311yescharacteristic
104#396—21no—
105#395—21no—
106#395—21no—
107#395—21no—
108#395—21no—
109#395C2×C411yescharacteristic
110#395C2×C411yescharacteristic
111#395C2×C411yescharacteristic
112#393C2311yescharacteristic
113#391—21no—
114#391—21no—
115#391—21no—
116#391—21no—
117#391—44no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—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#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#383C2×C411yescharacteristic
146#383C2×C411yescharacteristic
147#382C2×C411yescharacteristic
148#381—22no—
149#381—22no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C22⋊C4—21no—
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160#342—22no—
161#342—22no—
162#342—22no—
163#342C2314yes—
164#324—22no—
165#324—22no—
166#324—22no—
167#323C2314yes—
168C22×Q8—24no—
169C22×Q8—21no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22⋊C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C4⋊Q8—42no—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183#125—42no—
184#125—42no—
185#125—44no—
186#125—44no—
187C22⋊Q8—44no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—42no—
191C4×Q8—42no—
192C4×Q8—42no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—44no—
200#117—42no—
201#117—42no—
202#117—42no—
203#117—42no—
204#117—42no—
205#117—42no—
206#117—44no—
207#117—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
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—42no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
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⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4C2×D411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C22⋊C4C2411yescharacteristic
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—42no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C4⋊C8—42no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C22⋊C8—24no—
276#97—42no—
277#97—42no—
278C4×C8—42no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
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—44no—
305#95—44no—
306#95C22×C411yescharacteristic
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310C24C22×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8#9914yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C2×C8—44no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—84no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—84no—
428C23—21no—
429C23—21no—
430C23—41no—
431C23—41no—
432C23C22×C22⋊C411yescharacteristic
433C23#57512yes—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—44no—
438Q8—42no—
439Q8—42no—
440Q8—84no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4C2×#9912yes—
503C2×C4C2×#9912yes—
504C2×C4C2×#9912yes—
505C2×C4#42112yes—
506C8—84no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—42no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22C22×#9911yescharacteristic
515C22#289611yescharacteristic
516C22C2×#42111yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—42no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C4—164no—
540C2—21no—
541C2—21no—
542C2—42no—
543C2#1758711yescharacteristic
544C2#3334911yescharacteristic
545C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#557C4semidirect
13. #2634C4semidirect
14. #2634C4semidirect
20. #2581C4semidirect
21. #2546C4semidirect
28. #2399C4semidirect
29. #2399C4semidirect
33. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357492
#206912
#173712
#118462
#118422
#118102
#118012
#117602
#117562