Finite Group Databasebeta

← #377530

Subgroups — #377530

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2#21021C211yesmaximal, characteristic, complemented
3C2×#1866C211yesmaximal, characteristic
4C2×#1771C211yesmaximal, characteristic, complemented
5#20384C211yesmaximal, characteristic
6#11233C211yesmaximal, characteristic, complemented
7#11226C211yesmaximal, characteristic, complemented
8#11075C211yesmaximal, characteristic
9#10930C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic, complemented
12#10887C211yesmaximal, characteristic, complemented
13#10859C211yesmaximal, characteristic, complemented
14#10819C211yesmaximal, characteristic, complemented
15#10791C211yesmaximal, characteristic
16C22×#131C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2297C2211yescharacteristic
31#2296C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2068C412yescomplemented
34#2067C412yescomplemented
35#2066C412yescomplemented
36#2065C412yescomplemented
37#1917—22no—
38#1917—22no—
39#1866—24no—
40#1771—24no—
41#1753—22no—
42#1752—22no—
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46#1735C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52C2×#339C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56#1729C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
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×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—41no—
88#427—41no—
89#427—41no—
90C2×#107C2×C412yes—
91C2×#106C2×C412yes—
92#424—22no—
93#424—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2×C412yes—
99C2×C4⋊C8C2×C412yes—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—41no—
105C2×#104—41no—
106#416—22no—
107#416—22no—
108#416—41no—
109#416—41no—
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C412yes—
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C412yes—
116#408—22no—
117#408Q814yes—
118C2×C22⋊C8—22no—
119#405—41no—
120#405—41no—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97—21no—
124C2×#97—22no—
125C2×#97—22no—
126C2×#97—41no—
127C2×#97D412yes—
128C2×#97D412yes—
129C2×C4×C8—22no—
130C2×C4×C8—21no—
131C2×C4×C8—22no—
132C2×C4×C8—41no—
133#391C2×C412yes—
134#390C2×C412yes—
135#389C2×C412yes—
136#389C2×C412yes—
137#386—21no—
138#386—21no—
139#385—21no—
140#385—21no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C22⋊C4—22no—
145C2×#95—21no—
146#341—22no—
147#341—22no—
148#341—41no—
149#340C2×C412yes—
150#340C2×C412yes—
151#339—22no—
152#339—22no—
153#339—41no—
154#339—41no—
155#339—41no—
156#339—41no—
157#336—21no—
158#336—21no—
159#336—22no—
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336D414yes—
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×#41—21no—
175C2×#41—21no—
176C2×#41C2×Q812yes—
177C2×#41#4112yes—
178C22×Q8C22×C411yescharacteristic
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186C23×C4C2411yescharacteristic
187#131—44no—
188#131—44no—
189#131—44no—
190#131—44no—
191C2×M16—24no—
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—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—42no—
215C2×M16—42no—
216C2×M16C2×Q812yes—
217C2×M16#4112yes—
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—24no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—42no—
242C22×C8—42no—
243C22×C8C22×C412yes—
244C22×C8C22×C411yescharacteristic
245C22×C8C22×C411yescharacteristic
246C22×C8C2×D412yes—
247C22×C8C2×D412yes—
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D412yes—
251C22×C8C2×Q812yes—
252C22×C8#4112yes—
253C22×C8#4112yes—
254C22×C8#4112yes—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C412yes—
274C2×C4⋊C4C22×C412yes—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
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×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42C22×C412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42#4112yes—
301#108—44no—
302#107—42no—
303#106—42no—
304C4⋊C8—24no—
305C4⋊C8—24no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308#104—44no—
309#104—44no—
310#103—22no—
311#103—22no—
312#103—42no—
313#102—22no—
314#102—22no—
315#102—42no—
316C22⋊C8—28no—
317#97—24no—
318#97—24no—
319#97—44no—
320#97—42no—
321C4×C8—44no—
322C4×C8—42no—
323#95—24no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328C24—21no—
329C24C2×#4111yescharacteristic
330#41—22no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—22no—
336#41—22no—
337#41—22no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—41no—
345#41—41no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8C4×Q812yes—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4C4×Q812yes—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4C4×D412yes—
392C22×C4C4×D412yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4H(5,𝔽2)11yescharacteristic
404C22×C4H(5,𝔽2)11yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C42−1+411yescharacteristic
407M16—22no—
408M16—22no—
409M16—22no—
410M16—22no—
411M16—48no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416M16—41no—
417M16—41no—
418M16—41no—
419M16—41no—
420M16—41no—
421M16—41no—
422M16—41no—
423M16—41no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—48no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8C4×D412yes—
467C2×C8C4×D412yes—
468C2×C8#12112yes—
469C2×C8#12112yes—
470C2×C8C22⋊Q814yes—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—48no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—82no—
489C22⋊C4—82no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23C2×C4×D411yescharacteristic
516C23#52511yescharacteristic
517C23#52911yescharacteristic
518C23#52911yescharacteristic
519C23#53311yescharacteristic
520C23#54011yescharacteristic
521C23#54211yescharacteristic
522Q8—22no—
523Q8—22no—
524Q8—41no—
525Q8—41no—
526D4—22no—
527D4—22no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—41no—
533D4—41no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×C4×Q811yescharacteristic
573C2×C4C4×#4111yescharacteristic
574C2×C4C4×#4111yescharacteristic
575C2×C4#51811yescharacteristic
576C2×C4#52011yescharacteristic
577C2×C4C2×#12111yescharacteristic
578C2×C4C2×C22⋊Q811yescharacteristic
579C2×C4#52511yescharacteristic
580C2×C4#53211yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#53711yescharacteristic
583C2×C4#54611yescharacteristic
584C2×C4#54611yescharacteristic
585C2×C4#54611yescharacteristic
586C2×C4D4×Q811yescharacteristic
587C2×C4#55111yescharacteristic
588C2×C4#55111yescharacteristic
589C2×C4#55111yescharacteristic
590C2×C4#55211yescharacteristic
591C2×C4#55211yescharacteristic
592C2×C4#55311yescharacteristic
593C2×C4#55311yescharacteristic
594C2×C4#55311yescharacteristic
595C2×C4#55411yescharacteristic
596C2×C4#55411yescharacteristic
597C2×C4#55411yescharacteristic
598C2×C4#55611yescharacteristic
599C8—22no—
600C8—22no—
601C8—22no—
602C8—22no—
603C8—24no—
604C8—42no—
605C8—42no—
606C8—42no—
607C8—81no—
608C8—81no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—44no—
619C22—81no—
620C22—81no—
621C22—82no—
622C22C4×#12111yescharacteristic
623C22C4×C22⋊Q811yescharacteristic
624C22#232411yescharacteristic
625C22#245711yescharacteristic
626C22#246511yescharacteristic
627C22#251511yescharacteristic
628C22#252211yescharacteristic
629C22#295612yes—
630C22#295712yes—
631C4—44no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—82no—
636C4—81no—
637C4—81no—
638C4—82no—
639C4—82no—
640C4#236011yescharacteristic
641C4#236411yescharacteristic
642C4#236811yescharacteristic
643C4#237311yescharacteristic
644C4#241111yescharacteristic
645C4#241711yescharacteristic
646C4#256611yescharacteristic
647C4#260111yescharacteristic
648C2—21no—
649C2—21no—
650C2—21no—
651C2—21no—
652C2—82no—
653C2#1419811yescharacteristic
654C2#2154511yescharacteristic
655C2#2155111yescharacteristic
656C2#2162912yes—
657C2#2216712yes—
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 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21021C2semidirect
4. C2×#1771C2semidirect
6. #11233C2semidirect
7. #11226C2semidirect
11. #10909C2semidirect
12. #10887C2semidirect
13. #10859C2semidirect
14. #10819C2semidirect
33. #2068C4semidirect
34. #2067C4semidirect
35. #2066C4semidirect
36. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
#210212
C2×#18662
C2×#17712
#203842
#112332
#112262
#110752
#109302
#109292
#109092
#108872
#108592
#108192
#107912