Finite Group Databasebeta

← #206420

Subgroups — #206420

Order 512.

1002 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60947C211yesmaximal, characteristic
2#21415C211yesmaximal, characteristic
3C2×#1994C211yesmaximal, characteristic, complemented
4#20463C211yesmaximal, characteristic, complemented
5#17817C211yesmaximal, characteristic, complemented
6#17481C211yesmaximal, characteristic
7#17479C211yesmaximal, characteristic, complemented
8#9851C212yesmaximal, complemented
9#9850C212yesmaximal, complemented
10#9848C212yesmaximal, complemented
11#9846C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3248—22no—
15#3231C412yescomplemented
16#3229C411yescharacteristic, complemented
17#3226C411yescharacteristic, complemented
18#3210C412yescomplemented
19#3209C411yescharacteristic, complemented
20#3206C411yescharacteristic, complemented
21#3190—22no—
22#3170C2212yes—
23#3167C2212yes—
24#3108—22no—
25#3107—22no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1994—22no—
32#1994C2212yes—
33#1994C2212yes—
34#1994C2211yescharacteristic
35#1994C2211yescharacteristic
36#1962C2211yescharacteristic
37#1962C2211yescharacteristic
38#1896—22no—
39#1767—22no—
40C4×#104—22no—
41#1749—22no—
42#1744C2212yes—
43C2×#339—22no—
44C4×#97C2212yes—
45#1574—22no—
46#1573—22no—
47#1567C2212yes—
48#1566C2212yes—
49#1497—22no—
50#1495—22no—
51#1493C2212yes—
52#1493C2212yes—
53#1493C2212yes—
54#1492C2212yes—
55#1479—22no—
56#1477—22no—
57C22×M16—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533C2×C412yes—
65#533C2×C411yescharacteristic
66#533C2×C411yescharacteristic
67#533C2312yes—
68#533C2311yescharacteristic
69#532C2×C411yescharacteristic
70#532C2×C411yescharacteristic
71#532C2311yescharacteristic
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×C4⋊Q8C2×C411yescharacteristic
74#529—21no—
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80C2×#125—21no—
81#525—21no—
82#525—21no—
83#525—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—21no—
86C4×#41—21no—
87C4×#41—21no—
88C4×#41—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8—21no—
91C2×#117—21no—
92C22×C42—21no—
93C22×C42—21no—
94C22×C42D412yes—
95C22×C42C2311yescharacteristic
96#500C2×C412yescomplemented
97#498C2×C412yescomplemented
98#494C2×C412yescomplemented
99#492C2×C412yescomplemented
100#487C2×C412yescomplemented
101#487C2×C412yescomplemented
102#487C2×C412yescomplemented
103#486C2×C412yescomplemented
104#485—24no—
105#483—22no—
106#480—22no—
107#479—24no—
108#475—22no—
109#474—22no—
110#467—24no—
111#465—24no—
112#464—24no—
113#462—22no—
114#461—22no—
115#452—24no—
116#451—22no—
117#448—22no—
118C4×Q16—24no—
119C4×SD16—24no—
120#432—21no—
121#432—21no—
122#431—21no—
123#431—21no—
124#431—21no—
125#431—21no—
126#431C2×C411yescharacteristic
127#431C2×C411yescharacteristic
128#431C2×C412yes—
129#431D412yes—
130#431D412yes—
131#431D412yes—
132#431D412yes—
133#423—21no—
134#423—21no—
135#423—21no—
136#423C2311yescharacteristic
137#423C2312yes—
138#423C2311yescharacteristic
139C2×C4⋊C8—21no—
140C2×C4⋊C8—21no—
141C2×C4⋊C8—21no—
142C2×C4⋊C8D412yes—
143C2×#104—22no—
144C2×#104—22no—
145#407—21no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8D412yes—
148C4×M16—21no—
149C2×#97—24no—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152C43—21no—
153C43—21no—
154C43D412yes—
155C43C2311yescharacteristic
156C43C2311yescharacteristic
157#339—22no—
158#339—22no—
159#339—24no—
160#339—24no—
161#336—24no—
162#335—24no—
163#334—24no—
164Q8⋊C8—24no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167D4⋊C8—24no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C22×Q8—21no—
172C22×Q8—21no—
173C23×C4—22no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×Q16—24no—
178C2×SD16—24no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—41no—
183C2×M16—44no—
184C22×C8—22no—
185C22×C8—41no—
186C22×C8C2×D414yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8C22×C412yes—
194C4⋊Q8C22×C412yes—
195C4⋊Q8C22×C411yescharacteristic
196C4⋊Q8C22×C411yescharacteristic
197C4⋊Q8C22×C411yescharacteristic
198#127C22×C411yescharacteristic
199#126—41no—
200#126—41no—
201#125—21no—
202#125—22no—
203#125—21no—
204#125—22no—
205#125—21no—
206#125—41no—
207#125—41no—
208#124—21no—
209#124—21no—
210#124—21no—
211#123—41no—
212#123—41no—
213#123—41no—
214C22⋊Q8—22no—
215C22⋊Q8—21no—
216C22⋊Q8—22no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—22no—
221C22⋊Q8—21no—
222C22⋊Q8—22no—
223C22⋊Q8—21no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228#121—21no—
229#121—21no—
230#121—21no—
231#121—21no—
232#121—21no—
233#121—21no—
234#120—41no—
235C4×Q8—22no—
236C4×Q8—21no—
237C4×Q8—22no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—41no—
246C4×Q8—41no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261#117—21no—
262#117—22no—
263#117—21no—
264#117—22no—
265#117—21no—
266#117—41no—
267#117—41no—
268#117—41no—
269#117—41no—
270#117—41no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C42—21no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—42no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—41no—
290C2×C42—41no—
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42#4111yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305#107—24no—
306#106—24no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—22no—
314C4⋊C8—21no—
315C4⋊C8—41no—
316C4⋊C8—41no—
317C4⋊C8C22⋊C412yes—
318C4⋊C8C22⋊C412yes—
319C4⋊C8C22⋊C414yes—
320#104—44no—
321#104—44no—
322#103—24no—
323#103—24no—
324#103—24no—
325#102—24no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328C22⋊C8—41no—
329C22⋊C8—41no—
330C22⋊C8C22⋊C412yes—
331C22⋊C8C22⋊C412yes—
332C22⋊C8C22⋊C414yes—
333#97—48no—
334#97—41no—
335C4×C8—22no—
336C4×C8—41no—
337C4×C8C22×C412yes—
338C4×C8C22×C412yes—
339C4×C8C2×D414yes—
340C4×C8C2×D414yes—
341C24#12111yescharacteristic
342#41—42no—
343#41—42no—
344#41—44no—
345#41—82no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—44no—
354C2×Q8—82no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12411yescharacteristic
387C22×C4#12411yescharacteristic
388C22×C4#12711yescharacteristic
389C22×C4#12711yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393Q16—84no—
394SD16—84no—
395M16—42no—
396M16—42no—
397M16—44no—
398M16—82no—
399C2×C8—24no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—42no—
405C2×C8—82no—
406C2×C8C2×C22⋊C414yes—
407C2×C8C2×C22⋊C414yes—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C22⋊C4—42no—
433C22⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C42—24no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—44no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—44no—
452C42—44no—
453C42—41no—
454C42—42no—
455C42—41no—
456C42—41no—
457C42—42no—
458C42C2×C22⋊C411yescharacteristic
459C42C2×C22⋊C411yescharacteristic
460C42C4×D411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42C4×D411yescharacteristic
464C42#12111yescharacteristic
465C42#12111yescharacteristic
466C42#12411yescharacteristic
467C42#12411yescharacteristic
468C42#12711yescharacteristic
469C42#12711yescharacteristic
470C42#13714yes—
471C42C23×C411yescharacteristic
472C42C22×D411yescharacteristic
473C42C22×D411yescharacteristic
474C42C2×#4111yescharacteristic
475C23—22no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23#39011yescharacteristic
480C23#39011yescharacteristic
481C23C2×#12111yescharacteristic
482Q8—44no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—42no—
487Q8—82no—
488Q8—82no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—82no—
494D4—82no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4#39011yescharacteristic
532C2×C4#39011yescharacteristic
533C2×C4#39011yescharacteristic
534C2×C4#39011yescharacteristic
535C2×C4#39011yescharacteristic
536C2×C4#39011yescharacteristic
537C2×C4#44414yes—
538C2×C4C22×C22⋊C411yescharacteristic
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12711yescharacteristic
544C2×C4C2×#13712yes—
545C2×C4#57512yes—
546C8—24no—
547C8—24no—
548C8—44no—
549C8—81no—
550C8—82no—
551C8—81no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—44no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22#197512yes—
564C22C2×#39011yescharacteristic
565C22#305612yes—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—44no—
574C4—44no—
575C4—81no—
576C4—82no—
577C4—81no—
578C4—81no—
579C4—82no—
580C4#197512yes—
581C4#290112yes—
582C4C4×#13712yes—
583C4C2×#44412yes—
584C4#315211yescharacteristic
585C4#315311yescharacteristic
586C2—22no—
587C2—22no—
588C2—41no—
589C2—161no—
590C2#2121011yescharacteristic
591C2#2123711yescharacteristic
592C2C2×#197511yescharacteristic
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. #20463C2semidirect
5. #17817C2semidirect
7. #17479C2semidirect
8. #9851C2semidirect
9. #9850C2semidirect
10. #9848C2semidirect
11. #9846C2semidirect
15. #3231C4semidirect
16. #3229C4semidirect
17. #3226C4semidirect
18. #3210C4semidirect
19. #3209C4semidirect
20. #3206C4semidirect
96. #500C2×C4semidirect
97. #498C2×C4semidirect
98. #494C2×C4semidirect
99. #492C2×C4semidirect
100. #487C2×C4semidirect
101. #487C2×C4semidirect
102. #487C2×C4semidirect
103. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#609472
#214152
C2×#19942
#204632
#178172
#174812
#174792
#98512
#98502
#98482
#98462