Finite Group Databasebeta

← #360486

Subgroups — #360486

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21068C211yesmaximal, characteristic
2#21067C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic, complemented
4C2×#1815C211yesmaximal, characteristic
5#20663C211yesmaximal, characteristic
6#20454C211yesmaximal, characteristic, complemented
7C2×#1765C211yesmaximal, characteristic
8#11271C211yesmaximal, characteristic, complemented
9#11270C211yesmaximal, characteristic, complemented
10#11079C211yesmaximal, characteristic
11#11078C211yesmaximal, characteristic
12#10892C211yesmaximal, characteristic, complemented
13#10891C211yesmaximal, characteristic, complemented
14#10870C211yesmaximal, characteristic, complemented
15#10860C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#106C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#417C2211yescharacteristic
25C2×#417C2211yescharacteristic
26C2×#416C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C22×#97C2211yescharacteristic
29#2311C2211yescharacteristic
30#2311C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33#2274C2211yescharacteristic
34#2274C2211yescharacteristic
35#1928—22no—
36#1927—22no—
37#1926—22no—
38#1925—22no—
39#1847—24no—
40#1815—24no—
41#1809—22no—
42#1808—22no—
43#1776—22no—
44C4×#106—22no—
45#1765C418yescomplemented
46C2×#343C2211yescharacteristic
47C2×#343C2211yescharacteristic
48C2×#343C2211yescharacteristic
49C2×#343C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1736C2211yescharacteristic
52#1736C2211yescharacteristic
53#1736C2211yescharacteristic
54#1736C2211yescharacteristic
55#1736C2211yescharacteristic
56#1736C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62C22×#41C2311yescharacteristic
63C2×#131—22no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—22no—
67C2×#131—22no—
68C2×#131C2×C414yes—
69C22×M16—21no—
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C23×C8—21no—
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—41no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—22no—
89#427—22no—
90#427—22no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—41no—
98C2×#107D412yes—
99C2×#106—21no—
100C2×#106—21no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106—22no—
104C2×#106—41no—
105C2×#106D412yes—
106C2×#106D414yes—
107C2×#104—22no—
108C2×#104—22no—
109C2×#104C2×C414yes—
110#417—22no—
111#417—22no—
112#417—41no—
113#417—41no—
114#416C2×C414yes—
115#416Q814yes—
116C2×#103—21no—
117C2×#103—21no—
118C2×#103D412yes—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102D412yes—
122#405—22no—
123#405C2×C414yes—
124C2×#97—22no—
125C2×#97—21no—
126C2×#97—22no—
127C2×#97C2×C414yes—
128C2×C4×C8—21no—
129#391—21no—
130#391—21no—
131#390—21no—
132#390—21no—
133#386—41no—
134#386—41no—
135#384—22no—
136#384—22no—
137#382—22no—
138#382—22no—
139#380—22no—
140#380—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C22⋊C4—41no—
146C4×C22⋊C4—41no—
147C2×#95—41no—
148#343—22no—
149#343—22no—
150#343—41no—
151#343—41no—
152#341C2×C414yes—
153#340—22no—
154#340—22no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159#339—22no—
160#339—22no—
161#339C2×C414yes—
162#336—22no—
163#336—22no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C2×#41—21no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41C22×C412yes—
175C2×#41C2×Q812yes—
176C22×Q8C2×D411yescharacteristic
177C22×D4—21no—
178C22×D4C2×D411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C2411yescharacteristic
186#131—28no—
187#131—28no—
188#131—44no—
189#131—44no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—24no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—42no—
214C2×M16—42no—
215C2×M16C22×C412yes—
216C2×M16C22×C412yes—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—24no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8C22×C412yes—
229C22×C8C22×C412yes—
230C22×C8C2×D412yes—
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D412yes—
234C22×C8C2×Q812yes—
235C22×C8#4112yes—
236C22×C8#4112yes—
237C22×C8#4112yes—
238C22×C8#4112yes—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117C22×C412yes—
250#117C2×Q812yes—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—84no—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C22×C412yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296#108—44no—
297#107—24no—
298#107—44no—
299#107—42no—
300#107—42no—
301#106—24no—
302#106—24no—
303#106—24no—
304#106—44no—
305#104—28no—
306#103—42no—
307#103—42no—
308#103C4⋊C418yescomplemented
309#102—42no—
310#102—42no—
311#102C4⋊C418yescomplemented
312#97—28no—
313#97—24no—
314C4×C8—24no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—42no—
320C24—21no—
321C24H(5,𝔽2)11yescharacteristic
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—22no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8C2×C4⋊C412yes—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4C2×C4⋊C412yes—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×Q811yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4H(5,𝔽2)11yescharacteristic
399C22×C42−1+411yescharacteristic
400C22×C42−1+411yescharacteristic
401M16—22no—
402M16—22no—
403M16—22no—
404M16—22no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—48no—
410M16—41no—
411M16—41no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—48no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8C2×C4⋊C414yes—
438C2×C8#11714yes—
439C2×C8C4×D412yes—
440C2×C8C4×D412yes—
441C2×C8C4×D414yes—
442C2×C8#12112yes—
443C2×C8#12112yes—
444C2×C8#12312yes—
445C2×C8#12312yes—
446C2×C8#12414yes—
447C2×C8#12614yes—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—88no—
470C4⋊C4C2×C4⋊C414yes—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—162no—
478C22⋊C4—162no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—162no—
489C42—162no—
490C42#11714yes—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23#52011yescharacteristic
499C23#54011yescharacteristic
500C23#54211yescharacteristic
501C23D4211yescharacteristic
502C23#54611yescharacteristic
503C23#55311yescharacteristic
504C23#55411yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—41no—
508Q8—41no—
509D4—22no—
510D4—22no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—42no—
515D4—41no—
516D4—41no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—161no—
551C2×C4—161no—
552C2×C4—161no—
553C2×C4—161no—
554C2×C4C22×C4⋊C411yescharacteristic
555C2×C4C2×#11711yescharacteristic
556C2×C4C2×#11711yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4#52011yescharacteristic
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C2×#12411yescharacteristic
563C2×C4C2×#12611yescharacteristic
564C2×C4#53211yescharacteristic
565C2×C4#53211yescharacteristic
566C2×C4#53311yescharacteristic
567C2×C4#53311yescharacteristic
568C2×C4#54111yescharacteristic
569C2×C4#54211yescharacteristic
570C2×C4D4211yescharacteristic
571C2×C4#54711yescharacteristic
572C2×C4#54711yescharacteristic
573C2×C4#54711yescharacteristic
574C2×C4#54811yescharacteristic
575C2×C4D4×Q811yescharacteristic
576C2×C4D4×Q811yescharacteristic
577C2×C4#55011yescharacteristic
578C2×C4#55211yescharacteristic
579C2×C4#55411yescharacteristic
580C2×C4#55511yescharacteristic
581C2×C4#55611yescharacteristic
582C8—24no—
583C8—42no—
584C8—42no—
585C8—81no—
586C8—81no—
587C8#38814yes—
588C8#38814yes—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—42no—
598C22—44no—
599C22—81no—
600C22—81no—
601C22—82no—
602C22#231411yescharacteristic
603C22D4×C4⋊C411yescharacteristic
604C22#235211yescharacteristic
605C22#242211yescharacteristic
606C22#242811yescharacteristic
607C22#249211yescharacteristic
608C22#249711yescharacteristic
609C22#328812yes—
610C22#328912yes—
611C4—44no—
612C4—42no—
613C4—44no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—161no—
619C4—161no—
620C4#228511yescharacteristic
621C4C2×#38811yescharacteristic
622C4D4×C4⋊C411yescharacteristic
623C4#235411yescharacteristic
624C4#242311yescharacteristic
625C4#242911yescharacteristic
626C4#249111yescharacteristic
627C4#249611yescharacteristic
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—82no—
633C2#1387611yescharacteristic
634C2#2269312yes—
635C2#2505412yes—
636C2#2543611yescharacteristic
637C2#2543711yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1847C2semidirect
6. #20454C2semidirect
8. #11271C2semidirect
9. #11270C2semidirect
12. #10892C2semidirect
13. #10891C2semidirect
14. #10870C2semidirect
15. #10860C2semidirect
45. #1765C4semidirect
308. #103C4⋊C4semidirect
311. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#210682
#210672
C2×#18472
C2×#18152
#206632
#204542
C2×#17652
#112712
#112702
#110792
#110782
#108922
#108912
#108702
#108602