Finite Group Databasebeta

← #206411

Subgroups — #206411

Order 512.

1264 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60700C211yesmaximal, characteristic
2#21415C211yesmaximal, characteristic, complemented
3C2×#1962C211yesmaximal, characteristic, complemented
4C42×M16C211yesmaximal, characteristic, complemented
5#17796C211yesmaximal, characteristic, complemented
6#17504C211yesmaximal, characteristic, complemented
7#17500C211yesmaximal, characteristic, complemented
8#9851C212yesmaximal, complemented
9#9849C212yesmaximal, complemented
10#9844C212yesmaximal, complemented
11#9843C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3184C412yescomplemented
15#3182C412yescomplemented
16#3180—22no—
17#3171—22no—
18#3169C2212yes—
19#3167C2212yes—
20#3165C411yescharacteristic, complemented
21#3151C411yescharacteristic, complemented
22#3141C411yescharacteristic, complemented
23#3138C411yescharacteristic, complemented
24#3055—22no—
25C4×AGL(1,ℤ/8)—22no—
26#2920C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C22×#104—22no—
29C2×C4×M16—22no—
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C2×C43C2211yescharacteristic
33#1994C2211yescharacteristic, complemented
34#1994C2211yescharacteristic, complemented
35#1962—22no—
36#1962C2212yes—
37#1962C2212yes—
38#1962C2211yescharacteristic, complemented
39#1962C2211yescharacteristic, complemented
40#1896—22no—
41C4×#104—22no—
42C4×#97—22no—
43C4×#97C2212yescomplemented
44C42×C8C2212yescomplemented
45#1569—22no—
46#1567C2212yescomplemented
47#1566C2212yescomplemented
48#1565—22no—
49#1493C2212yescomplemented
50#1492C2212yes—
51#1492C2212yes—
52#1492C2212yescomplemented
53#1490—22no—
54#1489—22no—
55#1484—22no—
56#1483—22no—
57C22×#41—21no—
58C2×AGL(1,ℤ/8)—22no—
59C22×M16—21no—
60C22×M16—21no—
61C22×M16D412yes—
62#533C2×C411yescharacteristic
63#533C2×C411yescharacteristic
64#533C2311yescharacteristic
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532C2×C411yescharacteristic
72#532C2×C412yes—
73#532C2×C411yescharacteristic
74#532C2312yes—
75#532C2311yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83#529—21no—
84C2×#124—21no—
85#525—21no—
86#525—21no—
87#525—21no—
88C2×#121—21no—
89C2×#121—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42D412yes—
98C22×C42C2311yescharacteristic
99#501—22no—
100#500C2×C412yescomplemented
101#498C2×C412yescomplemented
102#496—22no—
103#493C2×C412yescomplemented
104#492C2×C412yescomplemented
105#490—22no—
106#489—22no—
107#487C2×C412yescomplemented
108#486C2×C412yescomplemented
109#486C2×C412yescomplemented
110#486C2×C412yescomplemented
111#469—24no—
112#468—24no—
113#466—24no—
114#465—24no—
115#442—24no—
116#440—24no—
117C4×SD16—24no—
118C4×D8—24no—
119#432—21no—
120#432—21no—
121#431—21no—
122#431—21no—
123#428—22no—
124#423—21no—
125#423—21no—
126#423—21no—
127#423C2311yescharacteristic
128#423C2312yes—
129#423C2311yescharacteristic
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×#104—24no—
133C2×#104—24no—
134C2×#104—22no—
135C2×#104—22no—
136C2×#104—48no—
137#418—22no—
138#417—22no—
139#407—21no—
140C4×M16—216no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16—21no—
144C4×M16C2×C411yescharacteristic
145C4×M16C2×C411yescharacteristic
146C4×M16C2×C412yes—
147C4×M16D412yes—
148C4×M16D412yes—
149C4×M16D412yes—
150C4×M16D412yes—
151C2×#97—24no—
152C2×#97—21no—
153C2×#97D412yes—
154C2×C4×C8—24no—
155C2×C4×C8C2312yes—
156C2×C4×C8C2312yes—
157C43—21no—
158C43—21no—
159C43D412yes—
160C43C2311yescharacteristic
161C43C2311yescharacteristic
162#339—22no—
163#339—22no—
164#335—24no—
165#334—24no—
166Q8⋊C8—24no—
167D4⋊C8—24no—
168D4⋊C8—24no—
169D4⋊C8—24no—
170C2×#41—21no—
171C2×#41—22no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—22no—
175C2×#41—44no—
176C2×#41—44no—
177C2×#41—41no—
178C2×#41—41no—
179C22×Q8—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—41no—
184C23×C4—22no—
185C23×C4—41no—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189AGL(1,ℤ/8)—48no—
190C2×SD16—24no—
191C2×D8—24no—
192C2×M16—28no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—28no—
196C2×M16—22no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—41no—
200C2×M16—44no—
201C2×M16C22⋊C412yes—
202C2×M16C22⋊C414yes—
203C2×M16C22⋊C412yes—
204C22×C8—22no—
205C22×C8—41no—
206C22×C8C2×D414yes—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8C22×C412yes—
211C4⋊Q8C22×C411yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214#127—21no—
215#127—21no—
216#127—21no—
217#127C22×C412yes—
218#127C22×C411yescharacteristic
219#126—41no—
220#126—41no—
221#125—21no—
222#125—21no—
223#125—21no—
224#124—21no—
225#124—22no—
226#124—21no—
227#124—22no—
228#124—21no—
229#124—41no—
230#124—41no—
231#123—41no—
232#123—41no—
233#123—41no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240#121—22no—
241#121—21no—
242#121—22no—
243#121—21no—
244#121—21no—
245#121—21no—
246#121—22no—
247#121—21no—
248#121—22no—
249#121—21no—
250#121—41no—
251#121—41no—
252#121—41no—
253#121—41no—
254#120—41no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×D4—22no—
262C4×D4—21no—
263C4×D4—22no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—41no—
280C4×D4—41no—
281#117—21no—
282#117—21no—
283#117—21no—
284#117—41no—
285#117—41no—
286#117—41no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C42—28no—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—21no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—41no—
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42#4111yescharacteristic
314C2×C42#4111yescharacteristic
315C2×C42#4111yescharacteristic
316C2×C42C2411yescharacteristic
317#107—24no—
318#106—24no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—22no—
322C4⋊C8—21no—
323C4⋊C8—22no—
324C4⋊C8—21no—
325C4⋊C8—41no—
326C4⋊C8—41no—
327#104—44no—
328#104—48no—
329#104—44no—
330#104—48no—
331#103—24no—
332#102—24no—
333#102—24no—
334#102—24no—
335C22⋊C8—41no—
336C22⋊C8—41no—
337#97—216no—
338#97—22no—
339#97—22no—
340#97—41no—
341#97C22⋊C414yes—
342#97C22⋊C412yes—
343#97C22⋊C412yes—
344C4×C8—216no—
345C4×C8—22no—
346C4×C8—41no—
347C4×C8C22×C412yes—
348C4×C8C22×C412yes—
349C4×C8C2×D414yes—
350C4×C8C2×D414yes—
351C24—41no—
352C24#12111yescharacteristic
353#41—48no—
354#41—44no—
355#41—42no—
356#41—42no—
357#41—44no—
358#41—44no—
359#41—82no—
360#41—84no—
361#41—82no—
362#41—82no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—44no—
368C2×Q8—41no—
369C2×D4—21no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—44no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—44no—
385C2×D4—82no—
386C2×D4—84no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12411yescharacteristic
416C22×C4#12411yescharacteristic
417C22×C4#12711yescharacteristic
418C22×C4#12711yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422SD16—48no—
423D8—48no—
424M16—28no—
425M16—28no—
426M16—24no—
427M16—28no—
428M16—44no—
429M16—42no—
430M16—42no—
431M16—82no—
432M16#10418yescomplemented
433C2×C8—28no—
434C2×C8—24no—
435C2×C8—28no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—42no—
439C2×C8—82no—
440C2×C8C2×C22⋊C414yes—
441C2×C8C2×C22⋊C414yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—41no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C42—24no—
472C42—24no—
473C42—28no—
474C42—24no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—44no—
479C42—44no—
480C42—42no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42C2×C22⋊C411yescharacteristic
485C42C2×C22⋊C411yescharacteristic
486C42C4×D411yescharacteristic
487C42C4×D411yescharacteristic
488C42C4×D411yescharacteristic
489C42C4×D411yescharacteristic
490C42#12111yescharacteristic
491C42#12111yescharacteristic
492C42#12411yescharacteristic
493C42#12411yescharacteristic
494C42#12711yescharacteristic
495C42#12711yescharacteristic
496C42AGL(1,ℤ/8)14yescomplemented
497C42C23×C411yescharacteristic
498C42C22×D411yescharacteristic
499C42C22×D411yescharacteristic
500C42C2×#4111yescharacteristic
501C23—22no—
502C23—22no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—82no—
507C23—82no—
508C23—82no—
509C23#39011yescharacteristic
510C23#39011yescharacteristic
511C23C2×#12111yescharacteristic
512Q8—42no—
513Q8—44no—
514Q8—42no—
515Q8—42no—
516Q8—82no—
517D4—42no—
518D4—44no—
519D4—44no—
520D4—42no—
521D4—44no—
522D4—42no—
523D4—82no—
524D4—82no—
525D4—84no—
526D4—82no—
527D4—82no—
528D4—84no—
529D4—82no—
530D4—82no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—84no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4#39011yescharacteristic
569C2×C4#39011yescharacteristic
570C2×C4#39011yescharacteristic
571C2×C4#39011yescharacteristic
572C2×C4#39011yescharacteristic
573C2×C4#39011yescharacteristic
574C2×C4C2×#10414yes—
575C2×C4C22×C22⋊C411yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×#12411yescharacteristic
580C2×C4C2×#12711yescharacteristic
581C2×C4C2×AGL(1,ℤ/8)12yes—
582C2×C4#57512yes—
583C8—28no—
584C8—24no—
585C8—82no—
586C8—81no—
587C8—81no—
588C8C2×#10418yescomplemented
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—44no—
597C22—81no—
598C22—82no—
599C22—82no—
600C22—81no—
601C22—81no—
602C22—82no—
603C22—164no—
604C22#196212yes—
605C22C2×#39011yescharacteristic
606C22#305512yes—
607C4—28no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—44no—
616C4—81no—
617C4—82no—
618C4—81no—
619C4—81no—
620C4#196212yes—
621C4C22×#10412yes—
622C4C4×AGL(1,ℤ/8)12yes—
623C4#295612yes—
624C4#315111yescharacteristic
625C4#315311yescharacteristic
626C2—22no—
627C2—22no—
628C2—41no—
629C2—161no—
630C2—162no—
631C2C2×#196211yescharacteristic
632C2#2123611yescharacteristic
633C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60700C2
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. #21415C2semidirect
3. C2×#1962C2semidirect
4. C42×M16C2semidirect
5. #17796C2semidirect
6. #17504C2semidirect
7. #17500C2semidirect
8. #9851C2semidirect
9. #9849C2semidirect
10. #9844C2semidirect
11. #9843C2semidirect
14. #3184C4semidirect
15. #3182C4semidirect
20. #3165C4semidirect
21. #3151C4semidirect
22. #3141C4semidirect
23. #3138C4semidirect
33. #1994C22semidirect
34. #1994C22semidirect
38. #1962C22semidirect
39. #1962C22semidirect
43. C4×#97C22semidirect
44. C42×C8C22semidirect
46. #1567C22semidirect
47. #1566C22semidirect
49. #1493C22semidirect
52. #1492C22semidirect
100. #500C2×C4semidirect
101. #498C2×C4semidirect
103. #493C2×C4semidirect
104. #492C2×C4semidirect
107. #487C2×C4semidirect
108. #486C2×C4semidirect
109. #486C2×C4semidirect
110. #486C2×C4semidirect
432. M16#104semidirect
496. C42AGL(1,ℤ/8)semidirect
588. C8C2×#104semidirect

Maximal subgroups

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

subgroupindex
#607002
#214152
C2×#19622
C42×M162
#177962
#175042
#175002
#98512
#98492
#98442
#98432