Finite Group Databasebeta

← #206412

Subgroups — #206412

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60701C211yesmaximal, characteristic
2#21415C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic, complemented
4C42×M16C211yesmaximal, characteristic, complemented
5#17796C211yesmaximal, characteristic, complemented
6#17504C211yesmaximal, characteristic
7#17501C211yesmaximal, characteristic, complemented
8#9851C212yesmaximal, complemented
9#9850C212yesmaximal, complemented
10#9845C212yesmaximal, complemented
11#9843C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3185C412yescomplemented
15#3182C412yescomplemented
16#3181—22no—
17#3172—22no—
18#3170C2212yes—
19#3167C2212yes—
20#3165C411yescharacteristic, complemented
21#3152C411yescharacteristic, complemented
22#3141C411yescharacteristic, complemented
23#3139C411yescharacteristic, complemented
24#3056—22no—
25C4×#137—22no—
26#2920C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×C4×M16—22no—
29C2×C4×M16C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1994—22no—
33#1994C2212yes—
34#1994C2212yes—
35#1994C2211yescharacteristic
36#1994C2211yescharacteristic
37#1962C2211yescharacteristic
38#1962C2211yescharacteristic
39#1896—22no—
40C4×#104—22no—
41C2×#339—22no—
42C4×#97—22no—
43C4×#97C2212yes—
44C42×C8C2212yes—
45#1569—22no—
46#1567C2212yes—
47#1566C2212yes—
48#1565—22no—
49#1493C2212yes—
50#1493C2212yes—
51#1493C2212yes—
52#1492C2212yes—
53#1491—22no—
54#1489—22no—
55#1485—22no—
56#1483—22no—
57C2×#137—22no—
58C22×M16—21no—
59C22×M16—21no—
60C22×M16D412yes—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533—21no—
67#533C2×C411yescharacteristic
68#533C2×C412yes—
69#533C2×C411yescharacteristic
70#533C2312yes—
71#533C2311yescharacteristic
72#532C2×C411yescharacteristic
73#532C2×C411yescharacteristic
74#532C2311yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83C2×#125—21no—
84#525—21no—
85#525—21no—
86#525—21no—
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C4×#41—21no—
90C4×#41—21no—
91C4×#41—21no—
92C2×C4×Q8—21no—
93C2×C4×Q8—21no—
94C2×#117—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42D412yes—
98C22×C42C2311yescharacteristic
99#501—22no—
100#500C2×C412yescomplemented
101#498C2×C412yescomplemented
102#497—22no—
103#494C2×C412yescomplemented
104#492C2×C412yescomplemented
105#491—22no—
106#489—22no—
107#487C2×C412yescomplemented
108#487C2×C412yescomplemented
109#487C2×C412yescomplemented
110#486C2×C412yescomplemented
111#470—24no—
112#468—24no—
113#467—24no—
114#465—24no—
115#441—24no—
116#440—24no—
117C4×Q16—24no—
118C4×SD16—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—22no—
133C2×#104—22no—
134#419—22no—
135#417—22no—
136#407—21no—
137C4×M16—216no—
138C4×M16—21no—
139C4×M16—21no—
140C4×M16—21no—
141C4×M16C2×C411yescharacteristic
142C4×M16C2×C411yescharacteristic
143C4×M16C2×C412yes—
144C4×M16D412yes—
145C4×M16D412yes—
146C4×M16D412yes—
147C4×M16D412yes—
148C2×#97—24no—
149C2×#97—21no—
150C2×#97D412yes—
151C2×C4×C8—24no—
152C2×C4×C8C2312yes—
153C2×C4×C8C2312yes—
154C43—21no—
155C43—21no—
156C43D412yes—
157C43C2311yescharacteristic
158C43C2311yescharacteristic
159#339—22no—
160#339—22no—
161#339—24no—
162#339—24no—
163#335—24no—
164#334—24no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167Q8⋊C8—24no—
168D4⋊C8—24no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C23×C4—22no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178#137—48no—
179C2×Q16—24no—
180C2×SD16—24no—
181C2×M16—28no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—28no—
185C2×M16—22no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—41no—
189C2×M16—44no—
190C2×M16C22⋊C412yes—
191C2×M16C22⋊C414yes—
192C2×M16C22⋊C412yes—
193C22×C8—22no—
194C22×C8—41no—
195C22×C8C2×D414yes—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8C22×C412yes—
203C4⋊Q8C22×C412yes—
204C4⋊Q8C22×C411yescharacteristic
205C4⋊Q8C22×C411yescharacteristic
206C4⋊Q8C22×C411yescharacteristic
207#127C22×C411yescharacteristic
208#126—41no—
209#126—41no—
210#125—21no—
211#125—22no—
212#125—21no—
213#125—22no—
214#125—21no—
215#125—41no—
216#125—41no—
217#124—21no—
218#124—21no—
219#124—21no—
220#123—41no—
221#123—41no—
222#123—41no—
223C22⋊Q8—22no—
224C22⋊Q8—21no—
225C22⋊Q8—22no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—22no—
230C22⋊Q8—21no—
231C22⋊Q8—22no—
232C22⋊Q8—21no—
233C22⋊Q8—41no—
234C22⋊Q8—41no—
235C22⋊Q8—41no—
236C22⋊Q8—41no—
237#121—21no—
238#121—21no—
239#121—21no—
240#121—21no—
241#121—21no—
242#121—21no—
243#120—41no—
244C4×Q8—22no—
245C4×Q8—21no—
246C4×Q8—22no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—41no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—41no—
265C4×D4—41no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270#117—21no—
271#117—22no—
272#117—21no—
273#117—22no—
274#117—21no—
275#117—41no—
276#117—41no—
277#117—41no—
278#117—41no—
279#117—41no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C42—28no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42#4111yescharacteristic
311C2×C42#4111yescharacteristic
312C2×C42#4111yescharacteristic
313C2×C42C2411yescharacteristic
314#107—24no—
315#106—24no—
316C4⋊C8—22no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—22no—
321C4⋊C8—21no—
322C4⋊C8—41no—
323C4⋊C8—41no—
324#104—44no—
325#104—44no—
326#103—24no—
327#103—24no—
328#103—24no—
329#102—24no—
330C22⋊C8—41no—
331C22⋊C8—41no—
332#97—216no—
333#97—22no—
334#97—22no—
335#97—41no—
336#97C22⋊C414yes—
337#97C22⋊C412yes—
338#97C22⋊C412yes—
339C4×C8—216no—
340C4×C8—22no—
341C4×C8—41no—
342C4×C8C22×C412yes—
343C4×C8C22×C412yes—
344C4×C8C2×D414yes—
345C4×C8C2×D414yes—
346C24#12111yescharacteristic
347#41—42no—
348#41—42no—
349#41—44no—
350#41—82no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—44no—
359C2×Q8—82no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12411yescharacteristic
392C22×C4#12411yescharacteristic
393C22×C4#12711yescharacteristic
394C22×C4#12711yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4111yescharacteristic
398Q16—48no—
399SD16—48no—
400M16—28no—
401M16—28no—
402M16—24no—
403M16—28no—
404M16—42no—
405M16—42no—
406M16—44no—
407M16—82no—
408M16#10418yescomplemented
409C2×C8—28no—
410C2×C8—24no—
411C2×C8—28no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—42no—
415C2×C8—82no—
416C2×C8C2×C22⋊C414yes—
417C2×C8C2×C22⋊C414yes—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C22⋊C4—42no—
443C22⋊C4—41no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—24no—
454C42—24no—
455C42—28no—
456C42—24no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—44no—
461C42—44no—
462C42—42no—
463C42—41no—
464C42—42no—
465C42—41no—
466C42—41no—
467C42—42no—
468C42C2×C22⋊C411yescharacteristic
469C42C2×C22⋊C411yescharacteristic
470C42C4×D411yescharacteristic
471C42C4×D411yescharacteristic
472C42C4×D411yescharacteristic
473C42C4×D411yescharacteristic
474C42#12111yescharacteristic
475C42#12111yescharacteristic
476C42#12411yescharacteristic
477C42#12411yescharacteristic
478C42#12711yescharacteristic
479C42#12711yescharacteristic
480C42#13714yescomplemented
481C42C23×C411yescharacteristic
482C42C22×D411yescharacteristic
483C42C22×D411yescharacteristic
484C42C2×#4111yescharacteristic
485C23—22no—
486C23—22no—
487C23—41no—
488C23—41no—
489C23#39011yescharacteristic
490C23#39011yescharacteristic
491C23C2×#12111yescharacteristic
492Q8—44no—
493Q8—42no—
494Q8—42no—
495Q8—44no—
496Q8—42no—
497Q8—82no—
498Q8—82no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—82no—
503D4—82no—
504D4—82no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4#39011yescharacteristic
542C2×C4#39011yescharacteristic
543C2×C4#39011yescharacteristic
544C2×C4#39011yescharacteristic
545C2×C4#39011yescharacteristic
546C2×C4#39011yescharacteristic
547C2×C4C2×#10414yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12411yescharacteristic
553C2×C4C2×#12711yescharacteristic
554C2×C4C2×#13712yes—
555C2×C4#57512yes—
556C8—28no—
557C8—24no—
558C8—81no—
559C8—82no—
560C8—81no—
561C8C2×#10418yes—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—44no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#196212yes—
574C22C2×#39011yescharacteristic
575C22#305612yes—
576C4—28no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—21no—
584C4—44no—
585C4—81no—
586C4—82no—
587C4—81no—
588C4—81no—
589C4—82no—
590C4#196212yes—
591C4C22×#10412yes—
592C4C4×#13712yes—
593C4#295612yes—
594C4#315211yescharacteristic
595C4#315311yescharacteristic
596C2—22no—
597C2—22no—
598C2—41no—
599C2—161no—
600C2C2×#196211yescharacteristic
601C2#2123711yescharacteristic
602C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#3444C22
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
3. C2×#1994C2semidirect
4. C42×M16C2semidirect
5. #17796C2semidirect
7. #17501C2semidirect
8. #9851C2semidirect
9. #9850C2semidirect
10. #9845C2semidirect
11. #9843C2semidirect
14. #3185C4semidirect
15. #3182C4semidirect
20. #3165C4semidirect
21. #3152C4semidirect
22. #3141C4semidirect
23. #3139C4semidirect
100. #500C2×C4semidirect
101. #498C2×C4semidirect
103. #494C2×C4semidirect
104. #492C2×C4semidirect
107. #487C2×C4semidirect
108. #487C2×C4semidirect
109. #487C2×C4semidirect
110. #486C2×C4semidirect
408. M16#104semidirect
480. C42#137semidirect

Maximal subgroups

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

subgroupindex
#607012
#214152
C2×#19942
C42×M162
#177962
#175042
#175012
#98512
#98502
#98452
#98432