Finite Group Databasebeta

← #373344

Subgroups — #373344

Order 512.

1004 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21148C211yesmaximal, characteristic, complemented
2C2×#1938C211yesmaximal, characteristic
3#21083C211yesmaximal, characteristic
4#20839C211yesmaximal, characteristic
5C2×#1765C211yesmaximal, characteristic
6#11291C211yesmaximal, characteristic, complemented
7#11209C211yesmaximal, characteristic, complemented
8#11068C211yesmaximal, characteristic
9#11053C211yesmaximal, characteristic
10#10887C211yesmaximal, characteristic, complemented
11#10883C211yesmaximal, characteristic, complemented
12#10883C211yesmaximal, characteristic, complemented
13#10844C211yesmaximal, characteristic, complemented
14#10791C211yesmaximal, characteristic
15#10696C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C22×#104C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#416C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2297C2211yescharacteristic
32#2296C2211yescharacteristic
33#1940—22no—
34#1939—22no—
35#1938—24no—
36#1934C412yescomplemented
37#1934C412yescomplemented
38#1933C412yescomplemented
39#1933C412yescomplemented
40#1851—22no—
41#1850—22no—
42#1765—24no—
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1735C2211yescharacteristic
48#1735C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C4×#95C2211yescharacteristic
61C22×#41C2311yescharacteristic
62C2×#131—22no—
63C2×#131—22no—
64C2×#131—22no—
65C2×#131—22no—
66C2×#131—41no—
67C2×#131—41no—
68C22×M16—21no—
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8—21no—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#424—22no—
85#424—22no—
86#424—22no—
87#424—22no—
88#424—41no—
89#424—41no—
90#423—22no—
91#423—22no—
92#423—41no—
93#423—41no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—41no—
106#416—41no—
107#416—41no—
108#416D414yes—
109#416Q814yes—
110C2×#103C2×C412yes—
111C2×#103C2×C412yes—
112C2×#102C2×C412yes—
113C2×#102C2×C412yes—
114#408—22no—
115#408—41no—
116C2×C22⋊C8—22no—
117#405—22no—
118#405—41no—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97—21no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—41no—
125C2×#97—41no—
126C2×C4×C8—21no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129C2×C4×C8C2×C412yes—
130C2×C4×C8C2×C412yes—
131#391C2×C411yescharacteristic
132#391C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#389C2×C412yes—
136#389C2×C412yes—
137#386—22no—
138#385—22no—
139C4×C4⋊C4—22no—
140C4×C22⋊C4—22no—
141C2×#95—22no—
142#341—22no—
143#341—22no—
144#341—41no—
145#341—41no—
146#340C2×C412yes—
147#340C2×C412yes—
148#340C2×C412yes—
149#340C2×C412yes—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—22no—
154#339—22no—
155#339—41no—
156#336—22no—
157#336—21no—
158#336—21no—
159#336—22no—
160#336—22no—
161#336D412yes—
162#336D412yes—
163C2×#41—21no—
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×#41C2×D412yes—
174C2×#41C2×Q812yes—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4#4112yes—
182C23×C4C2411yescharacteristic
183#131—28no—
184#131—44no—
185#131—44no—
186#131—44no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
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—42no—
210C2×M16—42no—
211C2×M16—42no—
212C2×M16#4112yes—
213C2×M16#4112yes—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—42no—
239C22×C8—42no—
240C22×C8—42no—
241C22×C8C22×C412yes—
242C22×C8C22×C412yes—
243C22×C8C2×D412yes—
244C22×C8C2×D412yes—
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×Q812yes—
248C22×C8#4112yes—
249C22×C8#4112yes—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117C2×D412yes—
261#117C2×Q812yes—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C412yes—
267C2×C4⋊C4C22×C412yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D412yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295C4⋊C8—44no—
296C4⋊C8—44no—
297C4⋊C8—44no—
298C4⋊C8—44no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301#104—28no—
302#104—44no—
303#103—42no—
304#103C4⋊C418yes—
305#102—42no—
306#102C4⋊C418yes—
307C22⋊C8—44no—
308#97—44no—
309#97—42no—
310#97—44no—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8—42no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—44no—
319C24—21no—
320C24C2×#4111yescharacteristic
321#41—22no—
322#41—22no—
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—22no—
328#41—22no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—41no—
334#41—41no—
335#41—41no—
336#41—41no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C2×C4⋊C412yes—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4C2×C4⋊C412yes—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4C4×D412yes—
381C22×C4C4×D412yes—
382C22×C4C23×C411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×Q811yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4H(5,𝔽2)11yescharacteristic
393C22×C4H(5,𝔽2)11yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395C22×C42−1+411yescharacteristic
396M16—22no—
397M16—22no—
398M16—22no—
399M16—22no—
400M16—48no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16—41no—
410M16—41no—
411M16—41no—
412M16—41no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—48no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8C2×C4⋊C414yes—
455C2×C8#12412yes—
456C2×C8#12412yes—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—44no—
465C4⋊C4—82no—
466C4⋊C4C2×C4⋊C414yes—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23C2×C4×D411yescharacteristic
503C23#52911yescharacteristic
504C23#52911yescharacteristic
505C23#54711yescharacteristic
506C23#55311yescharacteristic
507C23#55311yescharacteristic
508C23#55411yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—41no—
512Q8—41no—
513D4—22no—
514D4—22no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—42no—
519D4—41no—
520D4—41no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4#44414yes—
556C2×C4C22×C4⋊C411yescharacteristic
557C2×C4C4×#4111yescharacteristic
558C2×C4C4×#4111yescharacteristic
559C2×C4C4×#4111yescharacteristic
560C2×C4#51811yescharacteristic
561C2×C4#52011yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C2×C4#53211yescharacteristic
566C2×C4#53311yescharacteristic
567C2×C4#53311yescharacteristic
568C2×C4#53811yescharacteristic
569C2×C4#53911yescharacteristic
570C2×C4#54011yescharacteristic
571C2×C4#54211yescharacteristic
572C2×C4D4211yescharacteristic
573C2×C4#54611yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4#54611yescharacteristic
576C2×C4#54611yescharacteristic
577C2×C4D4×Q811yescharacteristic
578C2×C4#55111yescharacteristic
579C2×C4#55211yescharacteristic
580C2×C4#55311yescharacteristic
581C2×C4#55411yescharacteristic
582C2×C4#55411yescharacteristic
583C2×C4#55611yescharacteristic
584C8—24no—
585C8—22no—
586C8—22no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—42no—
591C8—81no—
592C8—81no—
593C22—22no—
594C22—22no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—44no—
603C22—82no—
604C22—81no—
605C22—81no—
606C22C4×#12411yescharacteristic
607C22D4×C4⋊C411yescharacteristic
608C22#236411yescharacteristic
609C22#246611yescharacteristic
610C22#247011yescharacteristic
611C22#252211yescharacteristic
612C22#258711yescharacteristic
613C22C2×#44412yes—
614C22#295712yes—
615C4—24no—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—81no—
623C4—81no—
624C4#229611yescharacteristic
625C4#233511yescharacteristic
626C4#234911yescharacteristic
627C4#237311yescharacteristic
628C4#246411yescharacteristic
629C4#247211yescharacteristic
630C4#249211yescharacteristic
631C4#255611yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—82no—
637C2#1411611yescharacteristic
638C2#2165911yescharacteristic
639C2#2166011yescharacteristic
640C2#2268612yes—
641C2#2314712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21148C2semidirect
6. #11291C2semidirect
7. #11209C2semidirect
10. #10887C2semidirect
11. #10883C2semidirect
12. #10883C2semidirect
13. #10844C2semidirect
15. #10696C2semidirect
36. #1934C4semidirect
37. #1934C4semidirect
38. #1933C4semidirect
39. #1933C4semidirect

Maximal subgroups

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

subgroupindex
#211482
C2×#19382
#210832
#208392
C2×#17652
#112912
#112092
#110682
#110532
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