Finite Group Databasebeta

← #6526711

Subgroups — #6526711

Order 512.

1954 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3236C211yesmaximal, characteristic, complemented
2C2×#3101C211yesmaximal, characteristic
3C2×#2566C211yesmaximal, characteristic
4#34235C211yesmaximal, characteristic, complemented
5#28123C212yesmaximal
6#28120C212yesmaximal
7#27653C212yesmaximal, complemented
8#27647C212yesmaximal, complemented
9#25461C214yesmaximal, complemented
10#25331C214yesmaximal, complemented
11#24819C214yesmaximal, complemented
12#24685C214yesmaximal, complemented
13#21953C211yesmaximal, characteristic, complemented
14#21836C211yesmaximal, characteristic, complemented
15#20739C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#3236—28no—
20C2×#501C2212yescomplemented
21C2×#491C2212yescomplemented
22#3101—28no—
23C2×#484C2212yes—
24C2×#483C2212yes—
25C2×#483C2212yes—
26C2×#479C2214yescomplemented
27C2×#476C2212yes—
28C2×#461C2212yes—
29C2×#450C2212yes—
30C2×C4×SD16C2214yes—
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2566—21no—
34#2566—21no—
35#2566—21no—
36#2566—21no—
37#2566C2214yes—
38#2566C2214yes—
39C2×#396—22no—
40#2365C2212yescomplemented
41#2365C2212yescomplemented
42#2360—22no—
43#2356C2212yescomplemented
44#2351C2212yescomplemented
45#2343—22no—
46#2336—21no—
47#2336—21no—
48#2320—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51C2×#390C2211yescharacteristic
52C2×#390C2211yescharacteristic
53C2×#385—22no—
54C2×#384C2211yescharacteristic
55#2283C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2276—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yescomplemented
60#2084C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2067C2214yes—
64#2065C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#1991C2212yescomplemented
68#1991C2212yescomplemented
69#1989C2214yescomplemented
70#1949C2214yescomplemented
71#1948C2214yescomplemented
72#1940C2214yescomplemented
73#1939C2214yescomplemented
74#1926C2214yes—
75#1925C2214yes—
76#1880C2214yes—
77#1880C2214yes—
78#1824C2212yes—
79#1823C2212yes—
80#1823C2212yes—
81#1805C2214yescomplemented
82#1804C2212yes—
83#1804C2212yes—
84#1803C2214yescomplemented
85#1797C2212yes—
86#1797C2212yes—
87#1786C2212yes—
88#1762C2214yes—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×SD16C2314yes—
92#533—24no—
93#533D418yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#125—21no—
97C2×#125—21no—
98C2×#125—21no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125C2312yes—
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2312yes—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—22no—
120C22×C4⋊C4D412yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—22no—
126C22×C22⋊C4—22no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#501—28no—
131#491—28no—
132#484—28no—
133#483—28no—
134#483—28no—
135#479—216no—
136#476—28no—
137#461—28no—
138#450—28no—
139C4×SD16—216no—
140#432—24no—
141#431—24no—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×C4⋊C8C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2312yes—
158C2×C4×C8C2312yes—
159#396—24no—
160#396—24no—
161#396—24no—
162#396—24no—
163#391—22no—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390—21no—
169#390—21no—
170#390C2314yes—
171#390C2314yes—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#385—24no—
190#385—24no—
191#384—21no—
192#384—21no—
193#384C2314yes—
194#383—21no—
195#383—22no—
196#382—21no—
197#382—21no—
198#382—22no—
199#382C2314yes—
200#382C2311yescharacteristic
201#382C2311yescharacteristic
202#381—21no—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220#376—21no—
221#376—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C25—21no—
233C22×Q8—21no—
234C22×Q8C2412yes—
235C22×D4—24no—
236C22×D4—21no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—42no—
240C22×D4C2412yes—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—44no—
247C23×C4—84no—
248C23×C4C2×D412yes—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×SD16—28no—
254C2×SD16—28no—
255C2×SD16—44no—
256C22×C8C2414yes—
257C22×C8C2414yes—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8C2×D414yes—
261C4⋊Q8C2×D414yes—
262#125—28no—
263#125—24no—
264#125—24no—
265#125—24no—
266#125—24no—
267#125—44no—
268#125—44no—
269C22⋊Q8—28no—
270C22⋊Q8—24no—
271C22⋊Q8C2×D418yes—
272C4×Q8—28no—
273C4×Q8—24no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×Q8—48no—
277C4×Q8C2×D418yes—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—44no—
283C4×D4—44no—
284#117—28no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—84no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—42no—
341C2×C42—42no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D414yes—
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#106—28no—
356#106—28no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—44no—
360C4⋊C8—44no—
361#103—28no—
362#103—28no—
363#103—28no—
364#102—28no—
365#102—28no—
366#102—44no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—24no—
370C4×C8—24no—
371#95—24no—
372#95—24no—
373#95—21no—
374#95—22no—
375#95—21no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—24no—
392C24—22no—
393C24—44no—
394C24—42no—
395C24—42no—
396C24—44no—
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—44no—
403C2×Q8C22×D414yes—
404C2×D4—24no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—42no—
413C2×D4—42no—
414C2×D42−1+414yes—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—816no—
443C22×C4#12114yes—
444C22×C4#12114yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4112yes—
456C22×C4C2511yescharacteristic
457SD16—416no—
458C2×C8—24no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C4⋊C4—24no—
465C4⋊C4—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4C22×D414yes—
496C22⋊C4—28no—
497C22⋊C4—48no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C42—28no—
505C42—24no—
506C42—24no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—44no—
512C42—44no—
513C42—44no—
514C42—82no—
515C42—82no—
516C42#12114yes—
517C42#12114yes—
518C42C22×D412yes—
519C42C22×D412yes—
520C42C2×#4112yes—
521C42C2×#4112yes—
522C23—21no—
523C23—21no—
524C23—22no—
525C23—24no—
526C23—48no—
527C23—416no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—88no—
533C23—84no—
534C23—84no—
535C23C2×#12112yes—
536C23#57714yes—
537C23C23×D411yescharacteristic
538Q8—28no—
539Q8—48no—
540Q8—44no—
541D4—24no—
542D4—24no—
543D4—48no—
544D4—42no—
545D4—42no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—88no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4C2×#12112yes—
593C2×C4C2×#12112yes—
594C2×C4C2×#12112yes—
595C2×C4C2×#12112yes—
596C2×C4C2×#12112yes—
597C2×C4#57514yes—
598C2×C4C23×D411yescharacteristic
599C2×C4C22×#4111yescharacteristic
600C2×C4C2×2−1+412yes—
601C2×C4#58512yes—
602C8—44no—
603C8—44no—
604C22—24no—
605C22—22no—
606C22—24no—
607C22—22no—
608C22—44no—
609C22—48no—
610C22—816no—
611C22—82no—
612C22—84no—
613C22—82no—
614C22#304314yes—
615C22C22×#12111yescharacteristic
616C22#348511yescharacteristic
617C22#348611yescharacteristic
618C22C2×#57512yes—
619C22C2×#57712yes—
620C4—22no—
621C4—22no—
622C4—24no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—44no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4#348912yes—
637C4#349512yes—
638C2—24no—
639C2—84no—
640C2#3563311yescharacteristic
641C2#6041511yescharacteristic
642C2C2×#304311yescharacteristic
643C2#6173612yes—
644C2#6184112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3236C2semidirect
4. #34235C2semidirect
7. #27653C2semidirect
8. #27647C2semidirect
9. #25461C2semidirect
10. #25331C2semidirect
11. #24819C2semidirect
12. #24685C2semidirect
13. #21953C2semidirect
14. #21836C2semidirect
20. C2×#501C22semidirect
21. C2×#491C22semidirect
26. C2×#479C22semidirect
40. #2365C22semidirect
41. #2365C22semidirect
43. #2356C22semidirect
44. #2351C22semidirect
59. #2085C22semidirect
60. #2084C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
67. #1991C22semidirect
68. #1991C22semidirect
69. #1989C22semidirect
70. #1949C22semidirect
71. #1948C22semidirect
72. #1940C22semidirect
73. #1939C22semidirect
81. #1805C22semidirect
84. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32362
C2×#31012
C2×#25662
#342352
#281232
#281202
#276532
#276472
#254612
#253312
#248192
#246852
#219532
#218362
#207392