Finite Group Databasebeta

← #6506478

Subgroups — #6506478

Order 512.

2280 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37426C211yesmaximal, characteristic, complemented
2#35676C211yesmaximal, characteristic
3#29118C212yesmaximal, complemented
4#29108C212yesmaximal, complemented
5#27285C214yesmaximal, complemented
6#27276C214yesmaximal, complemented
7#27260C214yesmaximal, complemented
8#27257C214yesmaximal, complemented
9#25865C212yesmaximal, complemented
10#25859C212yesmaximal, complemented
11#21770C212yesmaximal, complemented
12#20603C212yesmaximal, complemented
13#20530C211yesmaximal, characteristic, complemented
14C22×2−1+4C2211yescharacteristic
15C2×#535C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C2×#489C2212yes—
19C2×#488C2214yes—
20C2×#486C2212yes—
21C2×#417C2212yescomplemented
22C2×#419C2212yescomplemented
23C2×#416C2212yescomplemented
24C22×#103C2212yescomplemented
25C2×#408C2212yes—
26#2851C2212yes—
27#2804—22no—
28#2799—22no—
29#2784—22no—
30#2748—22no—
31#2628C2212yes—
32#2442C2214yescomplemented
33#2439C2214yescomplemented
34#2436—22no—
35#2436C2214yes—
36#2431—22no—
37#2431C2214yes—
38#2404—22no—
39#2404—22no—
40#2311C2212yes—
41#2310C2211yescharacteristic
42#2297C2211yescharacteristic
43#2297C2211yescharacteristic
44#2074C2214yescomplemented
45#2074C2212yescomplemented
46#2073C2212yescomplemented
47#2042C2214yes—
48#2042C2214yes—
49#2042C2214yes—
50#2041C2214yes—
51#2026C2212yescomplemented
52#2025C2212yescomplemented
53#2024C2214yescomplemented
54#1961C2212yes—
55#1958C2214yes—
56#1958C2212yes—
57#1935C2214yescomplemented
58#1934C2214yescomplemented
59#1933C2214yescomplemented
60#1933C2214yescomplemented
61#1895C2214yes—
62#1893C2214yes—
63#1875C2214yes—
64#1874C2214yes—
65#1825C2214yes—
66#1788C2214yes—
67#1788C2214yescomplemented
68#1787C2214yes—
69#1787C2214yescomplemented
70#1786C2214yes—
71#1753C2214yescomplemented
72C8×C22⋊C4C2214yescomplemented
73#1750C2212yes—
74C4×C22⋊C8C2212yes—
75C2×2−1+4—28no—
76C2×2−1+4—44no—
77C2×2−1+4—42no—
78C2×2−1+4—42no—
79C22×#41—22no—
80C22×#41—21no—
81C22×#41C2312yes—
82C22×#41C2311yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8—21no—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C22×M16C2312yes—
89C23×C8C2312yes—
90#535—28no—
91#535—42no—
92#535—42no—
93C2×C4⋊Q8C2312yes—
94C2×#127—21no—
95C2×#127C2312yes—
96C2×#125C2312yes—
97C2×#124—21no—
98C2×#124—22no—
99C2×#124—44no—
100C2×#124C2314yes—
101C2×#124C2312yes—
102C2×#121—22no—
103C2×#121C2314yes—
104C2×#120—22no—
105C2×#120—21no—
106C2×#120—21no—
107C2×#117C2311yescharacteristic
108C2×#117C2312yes—
109C2×#117C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C42C2311yescharacteristic
113#489—44no—
114#488—48no—
115#486—44no—
116#419—44no—
117#417—44no—
118#416—44no—
119C2×#103—44no—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132#408—28no—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137C2×C4×C8C2314yes—
138#400—22no—
139#400—24no—
140#399—22no—
141#398—24no—
142#396—24no—
143#395—22no—
144#393—22no—
145#393—24no—
146#393—24no—
147#393—24no—
148#392—24no—
149#392—24no—
150#391—22no—
151#391—22no—
152#391—24no—
153#391D412yes—
154#391C2314yes—
155#391C2312yes—
156#390—22no—
157#390D412yes—
158#390C2312yes—
159#389C2314yes—
160#389C2314yes—
161#387—24no—
162#387—24no—
163#386—22no—
164#386—22no—
165#386—22no—
166#386—24no—
167#386—24no—
168#386—24no—
169#386—24no—
170#386D412yes—
171#386D412yes—
172#385—24no—
173#385—24no—
174#382C2312yes—
175#382C2312yes—
176#381—21no—
177#381—21no—
178#380—24no—
179#380C2314yes—
180#380C2314yes—
181C4×C22⋊C4—22no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2314yes—
190C25—21no—
1912−1+4—432no—
1922−1+4—432no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—42no—
201C2×#41—41no—
202C2×#41—41no—
203C2×#41—42no—
204C2×#41—42no—
205C2×#41—42no—
206C2×#41—48no—
207C2×#41—44no—
208C2×#41—48no—
209C2×#41—44no—
210C2×#41—48no—
211C2×#41—44no—
212C2×#41#4118yes—
213C22×Q8—24no—
214C22×Q8—21no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—24no—
218C22×Q8—24no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8—24no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8—41no—
225C22×Q8—41no—
226C22×Q8—44no—
227C22×Q8—44no—
228C22×Q8—44no—
229C22×Q8C2×D411yescharacteristic
230C22×Q8C2412yes—
231C22×Q8C2411yescharacteristic
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—24no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—24no—
239C22×D4—42no—
240C22×D4—41no—
241C22×D4—41no—
242C22×D4C2×D412yes—
243C22×D4C2×D411yescharacteristic
244C22×D4C2412yes—
245C22×D4C2411yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4C2×D412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8—24no—
258C22×C8—42no—
259C22×C8—42no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—42no—
263C4⋊Q8—42no—
264#127—24no—
265#127—42no—
266#127—44no—
267#125—44no—
268#124—28no—
269#124—24no—
270#124—44no—
271#124—42no—
272#124—48no—
273#124—44no—
274#121—28no—
275#121—28no—
276#121—48no—
277#120—44no—
278#120—44no—
279#120—48no—
280#117—24no—
281#117—41no—
282#117—41no—
283#117—42no—
284#117—42no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2414yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2414yes—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4C2×D411yescharacteristic
308C2×C22⋊C4C2×D411yescharacteristic
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D414yes—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2412yes—
316C2×C42C2412yes—
317#103—28no—
318#103—28no—
319#103—28no—
320#103—44no—
321#103—44no—
322#103—44no—
323#102—28no—
324#102—44no—
325#102—44no—
326#102—44no—
327#102—44no—
328#102—44no—
329C22⋊C8—28no—
330C22⋊C8—28no—
331#97—42no—
332C4×C8—42no—
333C4×C8—44no—
334#95—24no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95C2×D414yes—
345#95C2×D414yes—
346C24—21no—
347C24—22no—
348C24—42no—
349C24—42no—
350C24—42no—
351C24—84no—
352C24C22×D411yescharacteristic
353C24C22×D412yes—
354#41—216no—
355#41—48no—
356#41—48no—
357#41—48no—
358#41—48no—
359#41—48no—
360#41—48no—
361#41—48no—
362#41—432no—
363#41—464no—
364C2×Q8—24no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—216no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—48no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—48no—
384C2×Q8—432no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8—82no—
388C2×Q8—84no—
389C2×Q8#12414yes—
390C2×Q8H(5,𝔽2)12yes—
391C2×Q8H(5,𝔽2)14yes—
392C2×D4—24no—
393C2×D4—22no—
394C2×D4—24no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—24no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—41no—
401C2×D4—42no—
402C2×D4—41no—
403C2×D4—42no—
404C2×D4—44no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—48no—
414C2×D4—42no—
415C2×D4—84no—
416C2×D4—84no—
417C2×D4—84no—
418C2×D4—84no—
419C2×D4—84no—
420C2×D4—84no—
421C2×D4#12418yes—
422C2×D4#12414yes—
423C2×D4H(5,𝔽2)12yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—44no—
448C22×C4—42no—
449C22×C4—44no—
450C22×C4—82no—
451C22×C4—84no—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2511yescharacteristic
461M16—84no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—84no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—42no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—82no—
478C4⋊C4—84no—
479C4⋊C4—82no—
480C22⋊C4—44no—
481C22⋊C4—44no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—82no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—42no—
498C42—42no—
499C42—82no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—21no—
504C23—22no—
505C23—21no—
506C23—22no—
507C23—22no—
508C23—44no—
509C23—41no—
510C23—41no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—84no—
517C23—88no—
518C23—164no—
519C23—164no—
520C23C2×#12412yes—
521C23C2×#12414yes—
522C23C23×D411yescharacteristic
523Q8—24no—
524Q8—24no—
525Q8—24no—
526Q8—44no—
527Q8—48no—
528Q8—416no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—164no—
533D4—24no—
534D4—44no—
535D4—48no—
536D4—48no—
537D4—416no—
538D4—42no—
539D4—44no—
540D4—44no—
541D4—84no—
542D4—164no—
543D4—164no—
544D4—168no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—48no—
565C2×C4—416no—
566C2×C4—48no—
567C2×C4—416no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—84no—
581C2×C4—164no—
582C2×C4C2×#12412yes—
583C2×C4C2×#12414yes—
584C2×C4C22×#4112yes—
585C2×C4C2×H(5,𝔽2)12yes—
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C2×C4C2×H(5,𝔽2)11yescharacteristic
588C8—44no—
589C8—44no—
590C22—22no—
591C22—24no—
592C22—28no—
593C22—24no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—168no—
609C22C22×#12411yescharacteristic
610C22#348411yescharacteristic
611C22#348411yescharacteristic
612C22#358414yes—
613C4—22no—
614C4—24no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—84no—
621C4—82no—
622C4—84no—
623C4#349012yes—
624C4#349012yes—
625C2—44no—
626C2—42no—
627C2—82no—
628C2—162no—
629C2#3536711yescharacteristic
630C2#6062412yes—
631C2#6163812yes—
632C2#6166312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35676C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37426C2semidirect
3. #29118C2semidirect
4. #29108C2semidirect
5. #27285C2semidirect
6. #27276C2semidirect
7. #27260C2semidirect
8. #27257C2semidirect
9. #25865C2semidirect
10. #25859C2semidirect
11. #21770C2semidirect
12. #20603C2semidirect
13. #20530C2semidirect
21. C2×#417C22semidirect
22. C2×#419C22semidirect
23. C2×#416C22semidirect
24. C22×#103C22semidirect
32. #2442C22semidirect
33. #2439C22semidirect
44. #2074C22semidirect
45. #2074C22semidirect
46. #2073C22semidirect
51. #2026C22semidirect
52. #2025C22semidirect
53. #2024C22semidirect
57. #1935C22semidirect
58. #1934C22semidirect
59. #1933C22semidirect
60. #1933C22semidirect
67. #1788C22semidirect
69. #1787C22semidirect
71. #1753C22semidirect
72. C8×C22⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
#374262
#356762
#291182
#291082
#272852
#272762
#272602
#272572
#258652
#258592
#217702
#206032
#205302