Finite Group Databasebeta

← #6526710

Subgroups — #6526710

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3234C211yesmaximal, characteristic
2C2×#3103C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4#34236C211yesmaximal, characteristic
5#28118C212yesmaximal, complemented
6#28115C212yesmaximal, complemented
7#27651C212yesmaximal, complemented
8#27645C212yesmaximal, complemented
9#25464C214yesmaximal, complemented
10#25332C214yesmaximal, complemented
11#24822C214yesmaximal, complemented
12#24686C214yesmaximal, complemented
13#21953C211yesmaximal, characteristic, complemented
14#21836C211yesmaximal, characteristic, complemented
15#20741C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#3234—28no—
20C2×#501C2212yes—
21C2×#490C2212yes—
22#3103—28no—
23C2×#481C2212yes—
24C2×#481C2212yes—
25C2×#480C2212yes—
26C2×#475C2212yes—
27C2×#478C2214yes—
28C2×#460C2212yes—
29C2×#448C2212yes—
30C2×C4×SD16C2214yes—
31C2×#432C2211yescharacteristic
32C2×#431C2211yescharacteristic
33#2565—21no—
34#2565—21no—
35#2565—21no—
36#2565—21no—
37#2565C2214yes—
38#2565C2214yes—
39C2×#398—22no—
40#2369C2212yes—
41#2366C2212yes—
42#2361—22no—
43#2357C2212yes—
44#2349C2212yes—
45#2339—22no—
46#2337—21no—
47#2337—21no—
48#2319—21no—
49C4×#121C2212yes—
50C4×#121C2212yes—
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#387—22no—
54C2×#384C2211yescharacteristic
55#2283C2211yescharacteristic
56C2×#382C2211yescharacteristic
57#2275—21no—
58C2×C4×C4⋊C4C2211yescharacteristic
59#2085C2214yes—
60#2084C2214yes—
61#2073C2214yes—
62#2071C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2055C2214yes—
66#2055C2214yes—
67#1991C2212yes—
68#1991C2212yes—
69#1989C2214yes—
70#1949C2214yes—
71#1948C2214yes—
72#1940C2214yes—
73#1939C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1877C2214yes—
77#1877C2214yes—
78#1824C2212yes—
79#1824C2212yes—
80#1823C2212yes—
81#1805C2212yes—
82#1805C2212yes—
83#1804C2214yes—
84#1802C2214yes—
85#1797C2212yes—
86#1797C2212yes—
87#1787C2212yes—
88#1763C2214yes—
89C22×#41—21no—
90C23×Q8C2311yescharacteristic
91C22×SD16C2314yes—
92#532—24no—
93#532D418yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#125—21no—
97C2×#125—21no—
98C2×#125—24no—
99C2×#125—24no—
100C2×#125C2312yes—
101C2×#125C2312yes—
102C2×#124—21no—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8C2312yes—
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—22no—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D412yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#501—28no—
131#490—28no—
132#481—28no—
133#481—28no—
134#480—28no—
135#478—216no—
136#475—28no—
137#460—28no—
138#448—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#398—24no—
160#398—24no—
161#398—24no—
162#398—24no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391C2314yes—
170#391C2314yes—
171#390—22no—
172#390C2312yes—
173#390C2312yes—
174#389—22no—
175#389C2312yes—
176#389C2312yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—24no—
182#387—24no—
183#387—24no—
184#387—24no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#384—21no—
192#384—21no—
193#384C2314yes—
194#383—21no—
195#382—21no—
196#382—21no—
197#382C2314yes—
198#382C2311yescharacteristic
199#382C2311yescharacteristic
200#381—21no—
201#381—22no—
202#380—22no—
203#380C2312yes—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C22⋊C4—21no—
215C4×C22⋊C4—22no—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—22no—
218#376—21no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95D412yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#336C2314yes—
231#336C2314yes—
232C2×#41—28no—
233C2×#41—84no—
234C22×Q8—24no—
235C22×Q8—21no—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—42no—
240C22×Q8C2412yes—
241C22×D4—22no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4C2412yes—
245C23×C4—21no—
246C23×C4—22no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×SD16—28no—
258C2×SD16—28no—
259C2×SD16—44no—
260C22×C8C2414yes—
261C22×C8C2414yes—
262C4⋊Q8—22no—
263C4⋊Q8C2×D414yes—
264#127—22no—
265#127C2×D414yes—
266#125—24no—
267#125—24no—
268#125—24no—
269#125—24no—
270#125—44no—
271#125—44no—
272#124—28no—
273#121—28no—
274#121—24no—
275#121C2×D418yes—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×Q8—44no—
282C4×D4—28no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—84no—
288C4×D4—84no—
289C4×D4C2×D418yes—
290C2×C4⋊C4—21no—
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—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—48no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—44no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—42no—
340C2×C42—42no—
341C2×C42—44no—
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—44no—
364#102—28no—
365#102—28no—
366#102—28no—
367C22⋊C8—24no—
368C22⋊C8—24no—
369#97—24no—
370C4×C8—24no—
371#95—21no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—22no—
392C24C22×D411yescharacteristic
393#41—816no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—82no—
406C2×Q82−1+414yes—
407C2×D4—28no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—82no—
417C2×D4C22×D414yes—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4#12114yes—
454C22×C4#12114yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2511yescharacteristic
467SD16—416no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—48no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4C22×D414yes—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—82no—
510C22⋊C4—82no—
511C42—24no—
512C42—24no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42—82no—
521C42—82no—
522C42#12114yes—
523C42#12114yes—
524C42C22×D412yes—
525C42C22×D412yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23C2×#12112yes—
539C23#57714yes—
540C23C23×D411yescharacteristic
541Q8—24no—
542Q8—24no—
543Q8—48no—
544Q8—42no—
545Q8—42no—
546Q8—84no—
547D4—28no—
548D4—48no—
549D4—416no—
550D4—44no—
551D4—84no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—164no—
599C2×C4—164no—
600C2×C4C2×#12112yes—
601C2×C4C2×#12112yes—
602C2×C4C2×#12112yes—
603C2×C4C2×#12112yes—
604C2×C4C2×#12112yes—
605C2×C4#57514yes—
606C2×C4C23×D411yescharacteristic
607C2×C4C22×#4111yescharacteristic
608C2×C4C2×2−1+412yes—
609C2×C4#58512yes—
610C8—44no—
611C8—44no—
612C22—24no—
613C22—22no—
614C22—24no—
615C22—22no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22#304214yes—
624C22C22×#12111yescharacteristic
625C22#348511yescharacteristic
626C22#348611yescharacteristic
627C22C2×#57512yes—
628C22C2×#57712yes—
629C4—22no—
630C4—22no—
631C4—24no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4#348912yes—
646C4#349512yes—
647C2—24no—
648C2—84no—
649C2#3563311yescharacteristic
650C2#6041511yescharacteristic
651C2C2×#304211yescharacteristic
652C2#6172912yes—
653C2#6183612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
3. C2×#2565C2semidirect
5. #28118C2semidirect
6. #28115C2semidirect
7. #27651C2semidirect
8. #27645C2semidirect
9. #25464C2semidirect
10. #25332C2semidirect
11. #24822C2semidirect
12. #24686C2semidirect
13. #21953C2semidirect
14. #21836C2semidirect
15. #20741C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32342
C2×#31032
C2×#25652
#342362
#281182
#281152
#276512
#276452
#254642
#253322
#248222
#246862
#219532
#218362
#207412