Finite Group Databasebeta

← #6795851

Subgroups — #6795851

Order 512.

1509 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2#34943C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#32202C214yesmaximal, complemented
5#30665C214yesmaximal, complemented
6#29926C212yesmaximal, complemented
7#29206C211yesmaximal, characteristic, complemented
8#28284C211yesmaximal, characteristic, complemented
9#28223C212yesmaximal
10#27626C212yesmaximal, complemented
11#26089C212yesmaximal, complemented
12#25340C212yesmaximal, complemented
13#24751C212yesmaximal, complemented
14#24679C212yesmaximal, complemented
15D4×C4⋊C8C211yesmaximal, characteristic, complemented
16#21962C212yesmaximal, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#477C2212yes—
19C2×#474C2212yes—
20C2×#465C2214yes—
21C2×#461C2211yescharacteristic
22C2×C4×SD16C2212yes—
23C2×C8×D4C2212yes—
24C22×C4⋊C8C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2572—22no—
28#2571—21no—
29#2571C2211yescharacteristic
30#2563—21no—
31#2563C2211yescharacteristic
32#2563C2211yescharacteristic
33#2538C2212yes—
34#2537C2211yescharacteristic
35#2497—22no—
36#2495—21no—
37#2495C2212yes—
38#2492C2214yes—
39#2439—22no—
40#2420—21no—
41#2420C2212yes—
42#2398—22no—
43#2369—21no—
44#2355C2212yes—
45#2354—21no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#121—21no—
49C4×#121C2212yes—
50#2285C2212yes—
51#2283—21no—
52#2283C2212yes—
53C42×D4C2211yescharacteristic
54C42×D4C2211yescharacteristic
55#2097C2214yes—
56#2076C2214yes—
57#2074C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2058C2214yes—
61#2055C2214yes—
62#2036C2214yes—
63#2035C2214yes—
64#2034C2214yes—
65#1990C2212yes—
66#1988C2212yes—
67#1981C2212yes—
68#1981C2211yescharacteristic
69#1949C2214yes—
70#1949C2214yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1947C2212yes—
75#1947C2212yes—
76#1947C2212yes—
77#1939C2212yes—
78#1927C2212yes—
79#1925C2214yes—
80#1920C2214yes—
81#1892C2214yes—
82#1877C2212yes—
83#1850C2214yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1808C2214yes—
87#1807C2214yes—
88#1803C2212yes—
89#1802C2212yes—
90#1802C2212yes—
91#1802C2211yescharacteristic
92#1797C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96C23×C8C2314yes—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C22⋊Q8C2314yes—
103C2×#121—21no—
104C2×#121C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D411yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×C4×D4C2311yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C22×C42—21no—
116C22×C42C2312yes—
117#477—44no—
118#474—44no—
119#465—48no—
120#461—24no—
121C4×SD16—44no—
122C8×D4—44no—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×C4⋊C8—28no—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2311yescharacteristic
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2311yescharacteristic
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#102C2314yes—
141C2×#102C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C4×C8C2312yes—
144#400—24no—
145#400—22no—
146#400—22no—
147#397—22no—
148#397—22no—
149#396—24no—
150#394—22no—
151#394—22no—
152#393—24no—
153#393—22no—
154#393—22no—
155#391—21no—
156#391—24no—
157#391C2312yes—
158#391C2314yes—
159#390—22no—
160#390—21no—
161#390C2312yes—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—24no—
173#388—22no—
174#387—24no—
175#386—22no—
176#386—22no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#383—22no—
184#383—22no—
185#382—22no—
186#382—21no—
187#382—21no—
188#382—22no—
189#382C2312yes—
190#382C2312yes—
191#381—22no—
192#381—22no—
193#380—22no—
194#380—22no—
195#380—22no—
196#380—21no—
197#380—21no—
198#380—21no—
199#380C2312yes—
200#380C2312yes—
201#380C2312yes—
202#380C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4C2312yes—
224#376—22no—
225C43C2311yescharacteristic
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#340C2312yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—21no—
236C22×Q8—22no—
237C22×Q8C2412yes—
238C22×D4—22no—
239C22×D4C2×D411yescharacteristic
240C22×D4C2411yescharacteristic
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D412yes—
245C23×C4C2×D412yes—
246C23×C4C2412yes—
247C2×SD16—24no—
248C2×SD16—24no—
249C2×SD16—42no—
250C22×C8—28no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8C2414yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C4⋊Q8—22no—
257C4⋊Q8—22no—
258C4⋊Q8—22no—
259C4⋊Q8—22no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—82no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266#121—24no—
267#121—42no—
268#121—44no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—82no—
272C4×D4—24no—
273C4×D4—22no—
274C4×D4—24no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—41no—
282C4×D4—84no—
283#117—84no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4C2×D412yes—
322C2×C22⋊C4C2×D412yes—
323C2×C22⋊C4C2×D412yes—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C2×C42C2412yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—44no—
344#107—44no—
345#106—44no—
346#106—42no—
347#106—44no—
348C4⋊C8—28no—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—22no—
352C4⋊C8—42no—
353C4⋊C8#4114yes—
354#103—24no—
355#103—42no—
356#103—42no—
357#103—44no—
358#103—44no—
359#102—28no—
360#102—42no—
361C22⋊C8—44no—
362C4×C8—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—24no—
375#95—24no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—21no—
386C24C22×D412yes—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—82no—
409C2×D4#13714yes—
410C2×D4H(5,𝔽2)12yes—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C2511yescharacteristic
450SD16—48no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—88no—
460C2×C8C2×#4114yes—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—24no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—82no—
489C4⋊C4—81no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—82no—
493C4⋊C4—82no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—42no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—84no—
506C22⋊C4—82no—
507C22⋊C4—84no—
508C42—24no—
509C42—24no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—21no—
514C42—22no—
515C42—21no—
516C42—41no—
517C42—44no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—44no—
522C42—84no—
523C42#13514yes—
524C42C2×#4112yes—
525C42H(5,𝔽2)12yes—
526C422−1+412yes—
527C422−1+412yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×#13714yes—
542C23C23×D411yescharacteristic
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—84no—
548Q8—164no—
549D4—28no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—88no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4#53712yes—
607C2×C4#53712yes—
608C2×C4#54712yes—
609C2×C4#54712yes—
610C2×C4C2×#13512yes—
611C2×C4C2×#13512yes—
612C2×C4C2×#13512yes—
613C2×C4C2×#13712yes—
614C2×C4#57712yes—
615C2×C4C22×#4111yescharacteristic
616C2×C4C2×H(5,𝔽2)11yescharacteristic
617C2×C4C2×H(5,𝔽2)11yescharacteristic
618C2×C4C2×2−1+411yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
620C2×C4#58511yescharacteristic
621C8—24no—
622C8—44no—
623C8—42no—
624C22—24no—
625C22—24no—
626C22—24no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22C2×#53711yescharacteristic
639C22C2×#54711yescharacteristic
640C22#348811yescharacteristic
641C22C22×#13511yescharacteristic
642C22C22×#13711yescharacteristic
643C22C2×#57711yescharacteristic
644C22#358511yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—84no—
660C4#324814yes—
661C4#334014yes—
662C4#351811yescharacteristic
663C4#351911yescharacteristic
664C4#354411yescharacteristic
665C4#354711yescharacteristic
666C2—44no—
667C2—82no—
668C2#3892211yescharacteristic
669C2C2×#324811yescharacteristic
670C2#6092911yescharacteristic
671C2C2×#334011yescharacteristic
672C2#6125411yescharacteristic
673C2#6194911yescharacteristic
674C2#6198011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2537C22
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. #36194C2semidirect
3. #33195C2semidirect
4. #32202C2semidirect
5. #30665C2semidirect
6. #29926C2semidirect
7. #29206C2semidirect
8. #28284C2semidirect
10. #27626C2semidirect
11. #26089C2semidirect
12. #25340C2semidirect
13. #24751C2semidirect
14. #24679C2semidirect
15. D4×C4⋊C8C2semidirect
16. #21962C2semidirect
17. C4×#461C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349432
#331952
#322022
#306652
#299262
#292062
#282842
#282232
#276262
#260892
#253402
#247512
#246792
D4×C4⋊C82
#219622
C4×#4612