Finite Group Databasebeta

← #205065

Subgroups — #205065

Order 512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic, complemented
3#21252C211yesmaximal, characteristic
4#20816C211yesmaximal, characteristic, complemented
5#17575C211yesmaximal, characteristic, complemented
6#17574C211yesmaximal, characteristic
7C2×C8×M16C211yesmaximal, characteristic, complemented
8#9717C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3167—21no—
12#3167—21no—
13#3167—21no—
14#3167—21no—
15#3167C414yescomplemented
16#3167C2214yes—
17C2×#498C411yescharacteristic, complemented
18C2×#492C411yescharacteristic, complemented
19C2×#487C411yescharacteristic, complemented
20C2×#486C411yescharacteristic, complemented
21C2×#465—22no—
22C2×C4×SD16—22no—
23C2×#431C2211yescharacteristic
24C2×#421—22no—
25C2×C4×M16C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2285C2211yescharacteristic
28#2285C2211yescharacteristic
29#1986—22no—
30#1986C2214yescomplemented
31#1980—22no—
32#1980C2214yes—
33#1844—21no—
34#1834—21no—
35#1807—22no—
36C4×#106C2214yescomplemented
37#1733—22no—
38#1520—22no—
39#1519—22no—
40#1518—22no—
41#1517—22no—
42#1503C2214yescomplemented
43#1502C2214yes—
44C8×M16—22no—
45C8×M16C2214yescomplemented
46C2×#322—21no—
47C2×C82—21no—
48C22×#41—21no—
49C22×SD16—22no—
50C22×M16D412yes—
51C23×C8D412yes—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533C2×C412yes—
59#533C2312yes—
60#532—21no—
61#532—21no—
62#532—21no—
63#532—21no—
64#532—21no—
65#532—21no—
66#532C2×C412yes—
67#532C2312yes—
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4⋊Q8C2×C411yescharacteristic
71C2×#127C2×C411yescharacteristic
72C2×#125—21no—
73C2×#124—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117—21no—
85C22×C42C2311yescharacteristic
86#498—21no—
87#498—21no—
88#498C2×C414yes—
89#492—21no—
90#492—21no—
91#492C2×C414yes—
92#487—21no—
93#487—21no—
94#487C2×C414yes—
95#486—21no—
96#486—21no—
97#486C2×C414yes—
98#465—24no—
99#465—24no—
100#465—24no—
101#465—24no—
102C4×SD16—24no—
103C4×SD16—24no—
104C4×SD16—24no—
105C4×SD16—24no—
106#431—21no—
107#431—21no—
108#431D412yes—
109#431C2312yes—
110C2×#106—22no—
111C2×#106—24no—
112C2×C4⋊C8—21no—
113#421—48no—
114C2×#104—28no—
115C2×#103—22no—
116C2×#102—22no—
117C2×C22⋊C8D412yes—
118C4×M16—21no—
119C4×M16—21no—
120C4×M16D412yes—
121C4×M16C2312yes—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2×C411yescharacteristic
126C2×C4×C8C2×C411yescharacteristic
127C2×C4×C8C2×C412yes—
128C2×C4×C8D411yescharacteristic
129C2×C4×C8D411yescharacteristic
130C2×C4×C8D411yescharacteristic
131C2×C4×C8D411yescharacteristic
132C2×C4×C8D412yes—
133C2×C4×C8C2312yes—
134#384—21no—
135#384—21no—
136#382—21no—
137#382—21no—
138#380—21no—
139#380—21no—
140#380—21no—
141#380—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4C2312yes—
145C4×C4⋊C4C2312yes—
146#340—24no—
147#339—28no—
148#328—28no—
149#327—28no—
150#322—24no—
151C82—24no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—41no—
155C2×#41—41no—
156C2×#41—44no—
157C2×#41—44no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—41no—
165C23×C4—41no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×SD16—24no—
170C2×SD16—24no—
171C2×SD16—48no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16SD1618yescomplemented
175C2×M16C2×D414yes—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—44no—
179C22×C8C22⋊C412yes—
180C22×C8C22⋊C412yes—
181C22×C8C22⋊C412yes—
182C22×C8C22⋊C412yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C412yes—
196C4⋊Q8C22×C412yes—
197#127—21no—
198#127—21no—
199#127—21no—
200#127C22×C412yes—
201#125—22no—
202#125—22no—
203#125—41no—
204#125—41no—
205#124—22no—
206#124—22no—
207#124—41no—
208#124—41no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—22no—
221#121—41no—
222#121—41no—
223#121—41no—
224#121—41no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240C4×D4—41no—
241#117—22no—
242#117—22no—
243#117—41no—
244#117—41no—
245#117—44no—
246#117—44no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—41no—
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42#4111yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#106—24no—
282#106—24no—
283#106—48no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286#104—416no—
287#103—24no—
288#103—24no—
289#102—24no—
290#102—24no—
291C22⋊C8—22no—
292C22⋊C8C2×D414yes—
293#97—22no—
294#97—22no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—22no—
299C4×C8C22⋊C412yes—
300C4×C8C22⋊C412yes—
301C4×C8C22⋊C411yescharacteristic
302C4×C8C22⋊C411yescharacteristic
303C4×C8C22⋊C411yescharacteristic
304C4×C8C22⋊C411yescharacteristic
305C4×C8C22×C412yes—
306C4×C8C22×C412yes—
307C4×C8C2×D412yes—
308C4×C8C2×D412yes—
309#95—22no—
310#95—22no—
311#95—41no—
312#95—41no—
313C24—41no—
314C24#12711yescharacteristic
315#41—44no—
316#41—44no—
317#41—48no—
318#41—84no—
319#41—82no—
320#41—82no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—44no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—82no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4—82no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—84no—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C4×D411yescharacteristic
356C22×C4C4×D411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4#12111yescharacteristic
360C22×C4#12111yescharacteristic
361C22×C4#12111yescharacteristic
362C22×C4#12111yescharacteristic
363C22×C4#12411yescharacteristic
364C22×C4#12411yescharacteristic
365C22×C4#12711yescharacteristic
366C22×C4#12711yescharacteristic
367C22×C4#12711yescharacteristic
368C22×C4C2×SD1614yes—
369C22×C4#13514yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373SD16—48no—
374SD16—48no—
375M16—24no—
376M16—28no—
377M16C2×SD1618yescomplemented
378C2×C8—24no—
379C2×C8—28no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—44no—
388C2×C8C2×C22⋊C412yes—
389C2×C8C2×C22⋊C412yes—
390C2×C8C2×C22⋊C412yes—
391C2×C8C2×C22⋊C412yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—42no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C42—42no—
423C42—42no—
424C42—44no—
425C42—44no—
426C42C2×C22⋊C411yescharacteristic
427C42C2×C22⋊C411yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42C4×D411yescharacteristic
431C42C4×D411yescharacteristic
432C42#12111yescharacteristic
433C42#12111yescharacteristic
434C42#12111yescharacteristic
435C42#12111yescharacteristic
436C42#12411yescharacteristic
437C42#12411yescharacteristic
438C42C23×C411yescharacteristic
439C42C22×D411yescharacteristic
440C42C2×#4111yescharacteristic
441C42C2×#4111yescharacteristic
442C23—41no—
443C23—41no—
444C23—42no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23#39011yescharacteristic
449C23#39011yescharacteristic
450C23#49212yes—
451C23#49512yes—
452C23C2×#12711yescharacteristic
453Q8—44no—
454Q8—44no—
455Q8—82no—
456Q8—82no—
457Q8—82no—
458Q8—82no—
459D4—44no—
460D4—44no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—84no—
466D4—82no—
467D4—82no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
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—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—84no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#39011yescharacteristic
495C2×C4#39011yescharacteristic
496C2×C4#39011yescharacteristic
497C2×C4#39011yescharacteristic
498C2×C4#39011yescharacteristic
499C2×C4#39011yescharacteristic
500C2×C4C4×SD1614yes—
501C2×C4C4×SD1614yes—
502C2×C4#46514yes—
503C2×C4#46514yes—
504C2×C4#49212yes—
505C2×C4#49512yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×#12111yescharacteristic
511C2×C4C2×#12411yescharacteristic
512C2×C4C22×SD1612yes—
513C2×C4C2×#13512yes—
514C8—22no—
515C8—22no—
516C8—22no—
517C8—28no—
518C8—44no—
519C8#42114yescomplemented
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—164no—
532C22#196812yes—
533C22#196812yes—
534C22C2×#39011yescharacteristic
535C22C2×#49211yescharacteristic
536C22C2×#49511yescharacteristic
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4—162no—
547C4—162no—
548C4#196812yes—
549C4#196812yes—
550C4#197612yes—
551C4C2×#42112yes—
552C4C2×C4×SD1612yes—
553C4C2×#46512yes—
554C2—21no—
555C2—21no—
556C2—22no—
557C2—22no—
558C2—162no—
559C2C2×#196811yescharacteristic
560C2C2×#197611yescharacteristic
561C2#2125211yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21306C2semidirect
4. #20816C2semidirect
5. #17575C2semidirect
7. C2×C8×M16C2semidirect
8. #9717C2semidirect
15. #3167C4semidirect
17. C2×#498C4semidirect
18. C2×#492C4semidirect
19. C2×#487C4semidirect
20. C2×#486C4semidirect
30. #1986C22semidirect
36. C4×#106C22semidirect
42. #1503C22semidirect
45. C8×M16C22semidirect
174. C2×M16SD16semidirect
377. M16C2×SD16semidirect
519. C8#421semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#213062
#212522
#208162
#175752
#175742
C2×C8×M162
#97172