Finite Group Databasebeta

← #482478

Subgroups — #482478

Order 512.

866 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35138C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#11846C214yesmaximal, complemented
5#11827C214yesmaximal, complemented
6#11786C212yesmaximal, complemented
7#11773C212yesmaximal, complemented
8C22×#125C411yescharacteristic, complemented
9C2×C4×#41C2211yescharacteristic
10#2920—21no—
11#2851—21no—
12#2851—21no—
13#2634C412yescomplemented
14#2633C2212yes—
15#2611C412yescomplemented
16#2611C2212yes—
17#2584—21no—
18#2562—21no—
19#2487C411yescharacteristic, complemented
20#2487C2211yescharacteristic
21#2486C411yescharacteristic, complemented
22#2486C2211yescharacteristic
23#2484—21no—
24#2406—21no—
25#2322—21no—
26#2322—21no—
27#2283—21no—
28#2283C411yescharacteristic, complemented
29#1820C2212yes—
30#1799—21no—
31#1799C2212yes—
32#1782—21no—
33C4×#99—21no—
34C4×#99C2212yes—
35#1467—21no—
36#1466C2212yes—
37#1465—21no—
38#1465C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1352C2218yes—
42#1344C2218yes—
43C22×#41D411yescharacteristic
44C2×#125—21no—
45C2×#125—21no—
46C2×#125—41no—
47C2×#125—41no—
48C2×#125—41no—
49C2×#125—41no—
50C2×#125C2×C412yes—
51C2×#125C2×C412yes—
52C2×#123—21no—
53C2×#123—21no—
54C4×#41—41no—
55C4×#41—41no—
56C4×#41—41no—
57C4×#41—41no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76#432—22no—
77#432—41no—
78#431—22no—
79#431—41no—
80#423—41no—
81C2×#99—22no—
82C2×#99—22no—
83#407—22no—
84C4×M16—41no—
85#400—22no—
86#400C2×C412yes—
87#400C2×C412yes—
88#399C2×C412yes—
89#399C2312yes—
90#398—22no—
91#398C2×C412yes—
92#398C2×C412yes—
93#397C2×C412yes—
94#397C2312yes—
95#395—22no—
96#394—22no—
97#389—22no—
98#389—22no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#386—21no—
110#386—21no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—21no—
116#385—21no—
117#383—41no—
118#383C2×C412yes—
119#382—21no—
120#382—41no—
121#381—21no—
122#381—21no—
123#381—21no—
124#381—22no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—21no—
129#380—22no—
130#380C2×C411yescharacteristic
131#380C2×C411yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—41no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136#376—41no—
137#376C2×C412yes—
138#342—22no—
139#342C2314yes—
140#324—22no—
141#324—22no—
142#323—22no—
143#323C2314yes—
144C2×#41—41no—
145C2×#41—41no—
146C2×#41—42no—
147C2×#41—42no—
148C22×Q8C22⋊C411yescharacteristic
149C22×D4C22⋊C411yescharacteristic
150C22×D4C2×D411yescharacteristic
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—42no—
164#125—42no—
165#125—42no—
166#125—42no—
167#125—42no—
168#125—42no—
169#125—42no—
170#125—41no—
171#125—41no—
172#125—41no—
173#125—41no—
174#123—44no—
175#123—44no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—41no—
179C4×Q8—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197#117—41no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22×C411yescharacteristic
237C2×C22⋊C4C2×D411yescharacteristic
238C2×C22⋊C4C2×D411yescharacteristic
239C2×C22⋊C4C2411yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—42no—
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C4⋊C8—42no—
265C4⋊C8—42no—
266#99—44no—
267#99—44no—
268C22⋊C8—24no—
269C22⋊C8—42no—
270#97—42no—
271C4×C8—42no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95C22×C412yes—
288#95C22×C412yes—
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24C22×D411yescharacteristic
292#41—48no—
293#41—48no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—82no—
345C22×C4—81no—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358M16—84no—
359C2×C8—44no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—42no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—82no—
418C42—84no—
419C42#9911yescharacteristic
420C42#9911yescharacteristic
421C42#9911yescharacteristic
422C42#9911yescharacteristic
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23#42112yes—
437C23C22×C22⋊C411yescharacteristic
438C23#57512yes—
439Q8—44no—
440D4—44no—
441D4—44no—
442D4—44no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
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—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—84no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4C2×#9911yescharacteristic
504C2×C4C2×#9911yescharacteristic
505C2×C4C2×#9911yescharacteristic
506C2×C4C2×#9911yescharacteristic
507C2×C4C2×#9911yescharacteristic
508C2×C4C2×#9911yescharacteristic
509C8—84no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22C22×#9911yescharacteristic
528C22#289611yescharacteristic
529C22C2×#42111yescharacteristic
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—164no—
549C4#212312yes—
550C2—21no—
551C2—21no—
552C2—42no—
553C2—42no—
554C2—81no—
555C2#1758211yescharacteristic
556C2C2×#212311yescharacteristic
557C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11846C2semidirect
5. #11827C2semidirect
6. #11786C2semidirect
7. #11773C2semidirect
8. C22×#125C4semidirect
13. #2634C4semidirect
15. #2611C4semidirect
19. #2487C4semidirect
21. #2486C4semidirect
28. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#351382
#207232
#173612
#118462
#118272
#117862
#117732