Finite Group Databasebeta

← #6495525

Subgroups — #6495525

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2#36639C211yesmaximal, characteristic
3#33314C211yesmaximal, characteristic, complemented
4#33293C211yesmaximal, characteristic, complemented
5#33238C211yesmaximal, characteristic, complemented
6#33230C211yesmaximal, characteristic, complemented
7#29854C214yesmaximal, complemented
8#29198C214yesmaximal, complemented
9#28655C212yesmaximal, complemented
10#28653C212yesmaximal, complemented
11#27895C214yesmaximal, complemented
12#25674C212yesmaximal, complemented
13#25671C212yesmaximal
14#22224C214yesmaximal, complemented
15C4×#432C211yesmaximal, characteristic, complemented
16C2×#533—21no—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#500C2212yes—
20C2×#498C2212yes—
21C2×#489C2214yes—
22C2×#487C2212yes—
23C2×#486C2212yes—
24C2×#432C2212yes—
25#2868C2211yescharacteristic
26#2868C2211yescharacteristic
27#2862—21no—
28#2862C2211yescharacteristic
29#2862C2211yescharacteristic
30#2861—21no—
31#2861—21no—
32#2859—21no—
33#2859C2212yes—
34#2858C2211yescharacteristic
35#2858C2211yescharacteristic
36#2857—21no—
37#2857—21no—
38#2857C2212yes—
39#2857C2212yes—
40#2850—21no—
41#2850—21no—
42#2850C2212yes—
43#2850C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8—21no—
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×#124—21no—
49C4×#124—21no—
50C4×#124C2212yes—
51C4×#124C2212yes—
52#2297—21no—
53#2297—21no—
54#2285—21no—
55#2285C2212yes—
56C2×C43C2211yescharacteristic
57#2067C2218yes—
58#2066C2218yes—
59#2036C2218yes—
60#2034C2218yes—
61#1993C2214yes—
62#1990C2212yes—
63#1990C2212yes—
64#1959C2214yes—
65#1959C2214yes—
66#1959C2214yes—
67#1957C2212yes—
68#1956C2212yes—
69#1927C2218yes—
70#1925C2218yes—
71#1864C2218yes—
72#1863C2218yes—
73#1847C2214yes—
74#1847C2214yes—
75#1836C2214yes—
76#1767C2212yes—
77C4×C4⋊C8C2212yes—
78#1749C2212yes—
79C4×C22⋊C8C2212yes—
80#1745C2212yes—
81C4×#97C2211yescharacteristic
82C4×#97C2211yescharacteristic
83C22×#41—21no—
84#533—44no—
85#533—48no—
86#532—44no—
87C2×C4⋊Q8—21no—
88C2×C4⋊Q8—21no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2312yes—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#125—22no—
98C2×#125—21no—
99C2×#124—21no—
100C2×#124—21no—
101C2×#124—21no—
102C2×#124C2312yes—
103C2×#124C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—22no—
106C2×C22⋊Q8—22no—
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121C2312yes—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8—21no—
113C2×C4×Q8C2312yes—
114C2×C4×D4—21no—
115C2×C4×D4C2312yes—
116C2×#117—22no—
117C2×#117—21no—
118C22×C42D411yescharacteristic
119C22×C42D411yescharacteristic
120C22×C42D411yescharacteristic
121C22×C42D411yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#500—44no—
126#498—44no—
127#489—216no—
128#487—44no—
129#486—44no—
130#432—28no—
131C2×#107C2318yes—
132C2×#106C2318yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2318yes—
135C2×#103C2318yes—
136C2×#102C2318yes—
137C2×#102C2318yes—
138C2×C22⋊C8C2314yes—
139C2×#97C2312yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142C2×#97C2312yes—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#391—22no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—21no—
153#391—21no—
154#391—21no—
155#391C2312yes—
156#391C2312yes—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390—21no—
162#390—21no—
163#390C2312yes—
164#390C2312yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389C2314yes—
172#389C2314yes—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#386—22no—
182#386—22no—
183#386—22no—
184#386—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#384C2312yes—
193#382—21no—
194#382—21no—
195#382—22no—
196#382—22no—
197#382C2312yes—
198#382C2312yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380C2314yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4C2314yes—
207C4×C22⋊C4—22no—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C43D411yescharacteristic
212C43D411yescharacteristic
213C43D411yescharacteristic
214C43D411yescharacteristic
215C43C2311yescharacteristic
216C43C2311yescharacteristic
217C43C2311yescharacteristic
218C43C2311yescharacteristic
219#340C2318yes—
220#340C2318yes—
221#336C2314yes—
222#336C2314yes—
223C2×#41—44no—
224C2×#41—44no—
225C2×#41—44no—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—21no—
230C22×Q8C2412yes—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4C2412yes—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8C2414yes—
244C22×C8C2414yes—
245C4⋊Q8—28no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255#127—24no—
256#125—44no—
257#125—48no—
258#124—28no—
259#124—28no—
260#124—44no—
261#124—44no—
262#124—44no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265C22⋊Q8—48no—
266C22⋊Q8—48no—
267#121—44no—
268#121—44no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—48no—
272C4×Q8—48no—
273C4×D4—44no—
274C4×D4—44no—
275#117—44no—
276#117—48no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
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⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2414yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344C2×C42C2411yescharacteristic
345C2×C42C2411yescharacteristic
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348#107—48no—
349#106—48no—
350C4⋊C8—28no—
351#103—216no—
352#103—48no—
353#102—216no—
354#102—48no—
355C22⋊C8—28no—
356#97—24no—
357#97—24no—
358#97—24no—
359#97#4118yescomplemented
360C4×C8—24no—
361C4×C8—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374C24—22no—
375C24C22×D411yescharacteristic
376#41—816no—
377C2×Q8—24no—
378C2×Q8—24no—
379C2×Q8—24no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—24no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—42no—
403C2×D4—42no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4#13714yes—
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C2511yescharacteristic
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8C2×#4118yes—
453C4⋊C4—28no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—44no—
513C42—48no—
514C42AGL(1,ℤ/8)14yes—
515C42C2×#4114yes—
516C42H(5,𝔽2)14yes—
517C422−1+414yes—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×#13712yes—
532C23C23×D411yescharacteristic
533Q8—48no—
534Q8—88no—
535Q8—84no—
536Q8—88no—
537Q8—84no—
538D4—48no—
539D4—88no—
540D4—84no—
541D4—84no—
542C2×C4—28no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—88no—
591C2×C4#53212yes—
592C2×C4#53212yes—
593C2×C4#53512yes—
594C2×C4#53512yes—
595C2×C4#53612yes—
596C2×C4#53612yes—
597C2×C4C2×AGL(1,ℤ/8)12yes—
598C2×C4C2×AGL(1,ℤ/8)12yes—
599C2×C4C2×AGL(1,ℤ/8)12yes—
600C2×C4C2×#13712yes—
601C2×C4C2×#13712yes—
602C2×C4#57512yes—
603C2×C4#57512yes—
604C2×C4C22×#4112yes—
605C2×C4C2×H(5,𝔽2)12yes—
606C2×C4C2×2−1+412yes—
607C8—28no—
608C8—44no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22C2×#53211yescharacteristic
624C22C2×#53511yescharacteristic
625C22C2×#53611yescharacteristic
626C22C22×AGL(1,ℤ/8)11yescharacteristic
627C22C22×#13711yescharacteristic
628C22C2×#57511yescharacteristic
629C22C2×#57511yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4#321414yes—
647C4#321514yes—
648C4#351114yes—
649C2—42no—
650C2—162no—
651C2#3529811yescharacteristic
652C2#6070011yescharacteristic
653C2#6070111yescharacteristic
654C2C2×#321511yescharacteristic
655C2C2×#321411yescharacteristic
656C2#6086011yescharacteristic
657C2#6086211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33314C2semidirect
4. #33293C2semidirect
5. #33238C2semidirect
6. #33230C2semidirect
7. #29854C2semidirect
8. #29198C2semidirect
9. #28655C2semidirect
10. #28653C2semidirect
11. #27895C2semidirect
12. #25674C2semidirect
14. #22224C2semidirect
15. C4×#432C2semidirect
359. #97#41semidirect

Maximal subgroups

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

subgroupindex
#366452
#366392
#333142
#332932
#332382
#332302
#298542
#291982
#286552
#286532
#278952
#256742
#256712
#222242
C4×#4322