Finite Group Databasebeta

← #206262

Subgroups — #206262

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic, complemented
4#21859C211yesmaximal, characteristic, complemented
5#21410C211yesmaximal, characteristic, complemented
6#21200C211yesmaximal, characteristic, complemented
7C2×#1453C211yesmaximal, characteristic, complemented
8#9840C211yesmaximal, characteristic, complemented
9#9837C211yesmaximal, characteristic, complemented
10#9832C211yesmaximal, characteristic, complemented
11#9829C211yesmaximal, characteristic, complemented
12#9827C211yesmaximal, characteristic, complemented
13#9826C211yesmaximal, characteristic, complemented
14#9822C211yesmaximal, characteristic, complemented
15#9820C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3207—24no—
19#3207C418yescomplemented
20C2×#493C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#463—22no—
25C2×#449—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#417—22no—
29C2×C4×M16C2211yescharacteristic
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1959—22no—
39#1957C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1957C2211yescharacteristic
44#1957C2211yescharacteristic
45#1957C2211yescharacteristic
46#1957C2211yescharacteristic
47#1956C2211yescharacteristic
48#1956C2211yescharacteristic
49#1948—22no—
50#1895—22no—
51#1875—22no—
52#1874—22no—
53#1853—22no—
54#1844C2211yescharacteristic
55#1844C2211yescharacteristic
56#1834C2211yescharacteristic
57#1834C2211yescharacteristic
58#1804—22no—
59#1802—22no—
60#1762—22no—
61#1453—24no—
62C2×#322C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×#322C2211yescharacteristic
65C2×C82C2211yescharacteristic
66C22×#41—21no—
67C22×#41—21no—
68C22×SD16—22no—
69C22×D8—22no—
70C22×M16D412yes—
71#532—22no—
72#532—22no—
73#532—22no—
74#532—22no—
75#532—22no—
76#532—22no—
77#532C2×C414yes—
78#532C2×C414yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#127C2311yescharacteristic
82C2×#127C2311yescharacteristic
83C2×#124—21no—
84C2×#124—21no—
85C2×#121—21no—
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—21no—
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4—21no—
93C2×#117—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97C22×C42C2311yescharacteristic
98#493—22no—
99#493C2×C414yescomplemented
100#492—22no—
101#492C2×C414yescomplemented
102#487—22no—
103#487C2×C414yescomplemented
104#486—22no—
105#486C2×C414yescomplemented
106#463—216no—
107#449—216no—
108#431—22no—
109#431—22no—
110#431C2×C414yes—
111#431D414yes—
112C2×#106—22no—
113C2×#106—22no—
114C2×C4⋊C8D412yes—
115C2×C4⋊C8D412yes—
116#417—48no—
117C2×#103—22no—
118C2×#103—22no—
119C2×#103—22no—
120C2×#102—22no—
121C2×#102—22no—
122C2×#102—22no—
123C2×#102—22no—
124C2×#102—22no—
125C2×#102—22no—
126C2×#102—22no—
127C2×#102—22no—
128C2×#102—22no—
129C2×C22⋊C8D412yes—
130C2×C22⋊C8D412yes—
131C4×M16—22no—
132C4×M16—22no—
133C2×#97D412yes—
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2311yescharacteristic
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138C2×C4×C8C2311yescharacteristic
139C2×C4×C8C2311yescharacteristic
140#384—21no—
141#384—21no—
142#384—21no—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#382—21no—
148#380—21no—
149#380—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#340—22no—
160#322—22no—
161#322—22no—
162#322—22no—
163C82—22no—
164C2×#41—24no—
165C2×#41—24no—
166C2×#41—44no—
167C2×#41—44no—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41—82no—
171C2×#41—82no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—41no—
181C22×D4—41no—
182C23×C4—41no—
183C23×C4—41no—
184C23×C4—41no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C2×SD16—28no—
189C2×SD16—82no—
190C2×SD16—82no—
191C2×D8—28no—
192C2×D8—82no—
193C2×D8—82no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—42no—
197C22×C8C2×D412yes—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C22×C412yes—
210C4⋊Q8C22×C412yes—
211#127—21no—
212#127—21no—
213#127—21no—
214#127—21no—
215#127—21no—
216#127—21no—
217#127C22×C412yes—
218#127C22×C412yes—
219#124—24no—
220#124—24no—
221#124—42no—
222#124—42no—
223#121—24no—
224#121—24no—
225#121—24no—
226#121—24no—
227#121—42no—
228#121—42no—
229#121—42no—
230#121—42no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—42no—
236C4×D4—42no—
237C4×D4—42no—
238C4×D4—42no—
239#117—82no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
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—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—82no—
261C2×C4⋊C4—82no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C42—41no—
268C2×C42C22×C412yes—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274C2×C42#4112yes—
275C2×C42C2411yescharacteristic
276#106—82no—
277#106—82no—
278C4⋊C8—24no—
279C4⋊C8C22⋊C418yes—
280#103—28no—
281#103—44no—
282#103—44no—
283#102—28no—
284#102—44no—
285#102—44no—
286#102—44no—
287#102—44no—
288#102—82no—
289#102—82no—
290#102—82no—
291#102—82no—
292C22⋊C8—24no—
293C22⋊C8C22⋊C418yes—
294#97—24no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—21no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8C22×C412yes—
304C4×C8C22×C412yes—
305C4×C8C2×D412yes—
306C4×C8C2×D412yes—
307#95—41no—
308#95—41no—
309C24—41no—
310C24—41no—
311C24#12711yescharacteristic
312#41—416no—
313#41—88no—
314#41—88no—
315#41—88no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—82no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C4×D412yes—
377C22×C4C4×D412yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12412yes—
381C22×C4#12711yescharacteristic
382C22×C4#12711yescharacteristic
383C22×C4#12711yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387SD16—88no—
388D8—88no—
389M16—48no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8C2×C22⋊C414yes—
406C2×C8C2×C22⋊C414yes—
407C2×C8AGL(1,ℤ/8)14yes—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—82no—
436C42—82no—
437C42C2×C22⋊C412yes—
438C42C4×D412yes—
439C42C4×D412yes—
440C42#12112yes—
441C42#12112yes—
442C42#12412yes—
443C42C23×C411yescharacteristic
444C42C22×D411yescharacteristic
445C42C2×#4111yescharacteristic
446C42C2×#4111yescharacteristic
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—42no—
454C23—42no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23#39012yes—
468C23C2×#12711yescharacteristic
469Q8—44no—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473D4—44no—
474D4—44no—
475D4—44no—
476D4—44no—
477D4—44no—
478D4—44no—
479D4—88no—
480D4—88no—
481D4—82no—
482D4—82no—
483D4—82no—
484D4—82no—
485D4—82no—
486D4—82no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—88no—
502C2×C4—88no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—164no—
520C2×C4#39012yes—
521C2×C4#39012yes—
522C2×C4#39012yes—
523C2×C4C22×C22⋊C411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×#12411yescharacteristic
529C2×C4C2×AGL(1,ℤ/8)12yes—
530C2×C4#57512yes—
531C2×C4#57612yes—
532C2×C4#57712yes—
533C2×C4#57712yes—
534C2×C4#57712yes—
535C8—24no—
536C8—24no—
537C8—24no—
538C8—42no—
539C8—42no—
540C8—42no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—161no—
555C22—161no—
556C22—161no—
557C22—161no—
558C22—161no—
559C22—161no—
560C22—161no—
561C22—161no—
562C22C2×#39011yescharacteristic
563C22#315111yescharacteristic
564C22#315311yescharacteristic
565C22#315411yescharacteristic
566C22#315411yescharacteristic
567C22#315411yescharacteristic
568C22#315511yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—82no—
573C4—82no—
574C4—161no—
575C4—161no—
576C4—161no—
577C4—161no—
578C4#289712yes—
579C4C4×AGL(1,ℤ/8)12yes—
580C4#295212yes—
581C4#305512yes—
582C4#306012yes—
583C4#313612yes—
584C2—42no—
585C2—162no—
586C2—162no—
587C2#1971811yescharacteristic
588C2#1971911yescharacteristic
589C2#1972111yescharacteristic
590C2#1972211yescharacteristic
591C2#2120611yescharacteristic
592C2#2122911yescharacteristic
593C2#2123611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3207C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21981C2semidirect
3. #21863C2semidirect
4. #21859C2semidirect
5. #21410C2semidirect
6. #21200C2semidirect
7. C2×#1453C2semidirect
8. #9840C2semidirect
9. #9837C2semidirect
10. #9832C2semidirect
11. #9829C2semidirect
12. #9827C2semidirect
13. #9826C2semidirect
14. #9822C2semidirect
15. #9820C2semidirect
19. #3207C4semidirect
99. #493C2×C4semidirect
101. #492C2×C4semidirect
103. #487C2×C4semidirect
105. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
#219812
#218632
#218592
#214102
#212002
C2×#14532
#98402
#98372
#98322
#98292
#98272
#98262
#98222
#98202