Finite Group Databasebeta

← #6850724

Subgroups — #6850724

Order 512.

1845 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36195C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic, complemented
3#33293C211yesmaximal, characteristic, complemented
4#32050C214yesmaximal, complemented
5#32046C214yesmaximal, complemented
6#29814C212yesmaximal, complemented
7#29145C211yesmaximal, characteristic, complemented
8#28519C211yesmaximal, characteristic, complemented
9#27978C212yesmaximal, complemented
10#27897C212yesmaximal, complemented
11#24694C212yesmaximal, complemented
12#24692C212yesmaximal, complemented
13#24132C212yesmaximal, complemented
14#24128C212yesmaximal, complemented
15#21987C212yesmaximal, complemented
16#21701C211yesmaximal, characteristic, complemented
17#21271C211yesmaximal, characteristic, complemented
18C2×#547—22no—
19C2×#547C2212yes—
20C2×#496C2211yescharacteristic
21C2×#477C2212yes—
22C2×#476C2212yes—
23C2×#442C2212yes—
24C2×#440C2212yes—
25C2×#432C2212yes—
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2856C2212yes—
29#2850C2212yes—
30#2572—22no—
31#2542—22no—
32#2542C2212yes—
33#2536C2211yescharacteristic
34#2535C2211yescharacteristic
35#2532—22no—
36#2501—22no—
37#2501—22no—
38#2501C2214yes—
39#2492C2212yes—
40#2491C2212yes—
41#2459—22no—
42#2372—22no—
43#2357C2211yescharacteristic
44#2356C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#124C2211yescharacteristic
47C2×#384C2212yes—
48#2283C2212yes—
49C42×D4C2211yescharacteristic
50#2096C2212yes—
51#2093C2212yes—
52#2072C2214yes—
53#2071C2214yes—
54#2055C2214yes—
55#2054C2214yes—
56#2034C2214yes—
57#2033C2214yes—
58#2023C2214yes—
59#2000C2214yes—
60#1999C2214yes—
61#1993C2212yes—
62#1993C2212yes—
63#1990C2214yes—
64#1970C2212yes—
65#1968C2212yes—
66#1961C2212yes—
67#1961C2211yescharacteristic
68#1959C2211yescharacteristic
69#1959C2212yes—
70#1955C2212yes—
71#1948C2214yes—
72#1947C2214yes—
73#1940C2212yes—
74#1926C2214yes—
75#1925C2214yes—
76#1923C2214yes—
77#1874C2214yes—
78#1872C2214yes—
79#1853C2214yes—
80#1852C2214yes—
81#1847C2212yes—
82#1847C2211yescharacteristic
83#1836C2212yes—
84#1836C2212yes—
85#1836C2212yes—
86#1804C2212yes—
87#1804C2212yes—
88#1767C2211yescharacteristic
89#1749C2212yes—
90C4×#97C2211yescharacteristic
91C22×#41—22no—
92C22×SD16C2312yes—
93C22×D8C2312yes—
94#547—44no—
95#547—44no—
96#547—416no—
97C2×C4⋊Q8—22no—
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#124—22no—
102C2×#124C2311yescharacteristic
103C2×#123—24no—
104C2×#123—24no—
105C2×#123—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8C2312yes—
109C2×#121—22no—
110C2×#121—22no—
111C2×#121C2312yes—
112C2×#121C2314yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—22no—
115C2×C4×D4D412yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C22×C4⋊C4—22no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4C2314yes—
122C22×C42C2312yes—
123#496—24no—
124#477—44no—
125#476—44no—
126#442—44no—
127#440—44no—
128#432—28no—
129C2×#107C2314yes—
130C2×#107C2312yes—
131C2×#106C2314yes—
132C2×#106C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2312yes—
141C2×#102C2312yes—
142C2×#102C2312yes—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97C2311yescharacteristic
146C2×#97C2312yes—
147C2×#97C2312yes—
148C2×#97C2311yescharacteristic
149C2×C4×C8C2312yes—
150#400—22no—
151#399—22no—
152#399—22no—
153#399—24no—
154#399—24no—
155#398—21no—
156#397—21no—
157#394—21no—
158#394—22no—
159#393—22no—
160#393—22no—
161#392—21no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391C2312yes—
166#391C2311yescharacteristic
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2312yes—
171#390C2312yes—
172#390C2314yes—
173#390C2311yescharacteristic
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#384—22no—
186#384—22no—
187#384—22no—
188#384C2311yescharacteristic
189#384C2314yes—
190#384C2312yes—
191#384C2312yes—
192#383—22no—
193#383—21no—
194#382C2311yescharacteristic
195#382C2311yescharacteristic
196#382C2312yes—
197#382C2311yescharacteristic
198#381—22no—
199#381—21no—
200#380C2312yes—
201#380C2312yes—
202C4×C4⋊C4D412yes—
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4C2312yes—
212#376—22no—
213#376—21no—
214C43C2311yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2312yes—
223C2×#41—416no—
224C2×#41—48no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8C2411yescharacteristic
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4C2×D411yescharacteristic
232C22×D4C2411yescharacteristic
233C22×D4C2412yes—
234C23×C4—24no—
235C23×C4—22no—
236C23×C4—21no—
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2412yes—
240C2×SD16—24no—
241C2×SD16—42no—
242C2×SD16—42no—
243C2×D8—24no—
244C2×D8—42no—
245C2×D8—42no—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C22×C8C2412yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—44no—
252C4⋊Q8—48no—
253#127—22no—
254#127—22no—
255#124—22no—
256#124—22no—
257#124—48no—
258#123—416no—
259#123—416no—
260#123—416no—
261C22⋊Q8—48no—
262C22⋊Q8—416no—
263C22⋊Q8—44no—
264#121—44no—
265#121—44no—
266#121—48no—
267#121—48no—
268#121—44no—
269C4×Q8—24no—
270C4×D4—28no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—48no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2414yes—
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C22⋊C4C2×D412yes—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2411yescharacteristic
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—42no—
330#107—44no—
331#106—42no—
332#106—44no—
333C4⋊C8—28no—
334C4⋊C8—24no—
335#103—42no—
336#103—44no—
337#102—44no—
338#102—44no—
339#102—42no—
340#102—42no—
341#102—42no—
342#102—44no—
343C22⋊C8—28no—
344C22⋊C8—44no—
345#97—24no—
346#97—24no—
347#97—22no—
348#97#4114yescomplemented
349C4×C8—42no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—22no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95C2×D411yescharacteristic
365#95C2×D411yescharacteristic
366#95C2×D412yes—
367#95C2×D412yes—
368C24—21no—
369C24—22no—
370C24C22×D412yes—
371#41—832no—
372C2×Q8—21no—
373C2×Q8—22no—
374C2×Q8—21no—
375C2×Q8—22no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×D4—24no—
382C2×D4—22no—
383C2×D4—21no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—21no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—82no—
401C2×D4—82no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C2511yescharacteristic
434SD16—48no—
435D8—48no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8C2×#4114yes—
445C4⋊C4—22no—
446C4⋊C4—24no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C22⋊C4—24no—
467C22⋊C4—24no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—42no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—82no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4AGL(1,ℤ/8)18yes—
483C42—24no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—42no—
490C42—42no—
491C42—44no—
492C42—82no—
493C42C2×#4112yes—
494C42H(5,𝔽2)14yes—
495C422−1+412yes—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23C2×AGL(1,ℤ/8)14yes—
513C23C23×D411yescharacteristic
514Q8—42no—
515Q8—42no—
516Q8—88no—
517Q8—164no—
518D4—28no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—88no—
526D4—816no—
527D4—164no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—88no—
565C2×C4—816no—
566C2×C4—162no—
567C2×C4#53512yes—
568C2×C4#53512yes—
569C2×C4#54714yes—
570C2×C4C2×AGL(1,ℤ/8)14yes—
571C2×C4C2×AGL(1,ℤ/8)14yes—
572C2×C4C22×#4111yescharacteristic
573C2×C4C2×H(5,𝔽2)12yes—
574C2×C4C2×2−1+411yescharacteristic
575C2×C4#58512yes—
576C8—24no—
577C8—44no—
578C8—42no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—81no—
587C22—82no—
588C22—84no—
589C22—82no—
590C22—82no—
591C22—84no—
592C22—82no—
593C22—81no—
594C22—81no—
595C22—82no—
596C22C2×#53511yescharacteristic
597C22C2×#54711yescharacteristic
598C22#350011yescharacteristic
599C22C22×AGL(1,ℤ/8)12yes—
600C22#358612yes—
601C4—24no—
602C4—24no—
603C4—24no—
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—82no—
610C4—82no—
611C4—81no—
612C4—82no—
613C4—84no—
614C4—81no—
615C4#350612yes—
616C4#354712yes—
617C2—44no—
618C2—162no—
619C2—161no—
620C2#3949111yescharacteristic
621C2#6079811yescharacteristic
622C2#6082811yescharacteristic
623C2#6125212yes—
624C2#6258312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36195C2semidirect
2. #36179C2semidirect
3. #33293C2semidirect
4. #32050C2semidirect
5. #32046C2semidirect
6. #29814C2semidirect
7. #29145C2semidirect
8. #28519C2semidirect
9. #27978C2semidirect
10. #27897C2semidirect
11. #24694C2semidirect
12. #24692C2semidirect
13. #24132C2semidirect
14. #24128C2semidirect
15. #21987C2semidirect
16. #21701C2semidirect
17. #21271C2semidirect
348. #97#41semidirect

Maximal subgroups

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

subgroupindex
#361952
#361792
#332932
#320502
#320462
#298142
#291452
#285192
#279782
#278972
#246942
#246922
#241322
#241282
#219872
#217012
#212712