Finite Group Databasebeta

← #483274

Subgroups — #483274

Order 512.

661 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
1#62475C211yesmaximal, characteristic
2#33503C211yesmaximal, characteristic, complemented
3#17834C211yesmaximal, characteristic, complemented
4#11980C211yesmaximal, characteristic, complemented
5#11979C211yesmaximal, characteristic, complemented
6#11978C211yesmaximal, characteristic, complemented
7#11970C211yesmaximal, characteristic, complemented
8#11969C211yesmaximal, characteristic, complemented
9#11967C211yesmaximal, characteristic, complemented
10#11946C211yesmaximal, characteristic, complemented
11#11945C211yesmaximal, characteristic, complemented
12#11944C211yesmaximal, characteristic, complemented
13#11936C211yesmaximal, characteristic, complemented
14#11935C211yesmaximal, characteristic, complemented
15C2×#1363C211yesmaximal, characteristic, complemented
16#3515—21no—
17#3383C411yescharacteristic
18#3381—21no—
19#3380C411yescharacteristic
20#3339C411yescharacteristic
21#3339C411yescharacteristic
22#3338C411yescharacteristic
23#3337—21no—
24#3337—21no—
25#3336C411yescharacteristic
26#3336C411yescharacteristic
27#3335—21no—
28#3334C411yescharacteristic
29#3233—21no—
30#3232—21no—
31#3232—21no—
32#3168C2211yescharacteristic
33#3168C2211yescharacteristic
34#3168C2211yescharacteristic
35#3160C2211yescharacteristic
36C2×#499C2211yescharacteristic
37C2×#499C2211yescharacteristic
38#2990—21no—
39C8×#41C2211yescharacteristic
40#2183—21no—
41#2169—21no—
42#2168—21no—
43#2168—21no—
44#2163C2211yescharacteristic
45#2163C2211yescharacteristic
46#2162C2211yescharacteristic
47#2150C2211yescharacteristic
48#2150C2211yescharacteristic
49C2×C4⋊C16C2211yescharacteristic
50#1574—21no—
51#1573—21no—
52#1573—21no—
53#1568C2211yescharacteristic
54#1568C2211yescharacteristic
55#1566C2211yescharacteristic
56#1564C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1363C2211yescharacteristic
60#1363C2211yescharacteristic
61#1363C2211yescharacteristic
62#1363C2211yescharacteristic
63#1363C2211yescharacteristic
64#1363C2211yescharacteristic
65#1363C2211yescharacteristic
66#1363C2211yescharacteristic
67#1363C2211yescharacteristic
68#1363C2211yescharacteristic
69#1363C2211yescharacteristic
70#1363C2211yescharacteristic
71#1360C2211yescharacteristic
72#1360C2211yescharacteristic
73#1360C2211yescharacteristic
74#1360C2211yescharacteristic
75#557—21no—
76#554—21no—
77#554—21no—
78#553—21no—
79#553—21no—
80#552—21no—
81#552—21no—
82#552—41no—
83#552—41no—
84#552—41no—
85#547—21no—
86#547—41no—
87#541—41no—
88#539—41no—
89#539—41no—
90#533—21no—
91#529—21no—
92#529—21no—
93#529—21no—
94C2×#125—21no—
95C2×#125—21no—
96#520—41no—
97C4×#41D411yescharacteristic
98C2×M32—21no—
99C2×M32—21no—
100C22×C16—21no—
101#500C2×C411yescharacteristic
102#500C2×C411yescharacteristic
103#499—21no—
104#499—21no—
105#499—21no—
106#499—21no—
107#499—21no—
108#499C2×C411yescharacteristic
109#499C2×C411yescharacteristic
110#499C2×C411yescharacteristic
111#499C2×C411yescharacteristic
112#499C2311yescharacteristic
113#499C2311yescharacteristic
114#499C2311yescharacteristic
115#498—21no—
116#495C2311yescharacteristic
117#488—21no—
118#488—21no—
119#488—21no—
120#485—21no—
121#485—21no—
122#484—21no—
123#484—21no—
124#483—21no—
125#483—21no—
126#482—21no—
127#482—21no—
128#481—21no—
129#481—21no—
130#480—21no—
131#480—21no—
132#479—21no—
133#479—21no—
134#479—21no—
135#479—21no—
136#478—21no—
137#478—21no—
138#478—21no—
139#478—21no—
140#477—21no—
141#476—21no—
142#475—21no—
143#474—21no—
144#467C2×C411yescharacteristic
145#467C2×C411yescharacteristic
146#467C2×C411yescharacteristic
147#466C2×C411yescharacteristic
148#466C2×C411yescharacteristic
149#466C2×C411yescharacteristic
150#465—21no—
151#465—21no—
152#465—21no—
153#446—41no—
154C8×Q8C2×C411yescharacteristic
155C8×Q8D411yescharacteristic
156C4×Q16C2×C411yescharacteristic
157C4×Q16C2×C411yescharacteristic
158C4×Q16C2×C411yescharacteristic
159C4×SD16—21no—
160C4×SD16—21no—
161C4×SD16—21no—
162C4×D8C2×C411yescharacteristic
163C4×D8C2×C411yescharacteristic
164C4×D8C2×C411yescharacteristic
165#435—41no—
166#435—41no—
167C8×D4—41no—
168C8×D4C2×C411yescharacteristic
169C8×D4C2×C411yescharacteristic
170C8×D4C2×C411yescharacteristic
171C8×D4D411yescharacteristic
172C8×D4D411yescharacteristic
173C8×D4D411yescharacteristic
174#432—21no—
175#432—21no—
176#431—21no—
177#431D411yescharacteristic
178#431D411yescharacteristic
179#431D411yescharacteristic
180#427—21no—
181#427C2×C411yescharacteristic
182#427C2×C411yescharacteristic
183C2×#107C2×C411yescharacteristic
184C2×#107C2×C411yescharacteristic
185C2×#107C2×C411yescharacteristic
186C2×#107C2×C411yescharacteristic
187C2×#106—21no—
188C2×#106—21no—
189#423—21no—
190C2×C4⋊C8—21no—
191C2×C4⋊C8—21no—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C4⋊C16C2311yescharacteristic
199C4⋊C16C2311yescharacteristic
200C4⋊C16C2311yescharacteristic
201C4⋊C16C2311yescharacteristic
202C22⋊C16—21no—
203C22⋊C16—21no—
204C22⋊C16—21no—
205#346—21no—
206#346—21no—
207C4×C16—21no—
208#335—21no—
209#335—21no—
210#335—21no—
211#334—21no—
212#334—21no—
213#334—21no—
214#334C2311yescharacteristic
215#334C2311yescharacteristic
216#334C2311yescharacteristic
217#334C2311yescharacteristic
218C2×#41—41no—
219C2×#41SD1611yescharacteristic
220C2×Q16C22×C411yescharacteristic
221C2×SD16—21no—
222C2×D8C22×C411yescharacteristic
223C2×M16—41no—
224C22×C8—41no—
225C22×C8—41no—
226C22×C8C22×C411yescharacteristic
227C22×C8C22×C411yescharacteristic
228C22×C8C22×C411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—41no—
235#126—41no—
236#126—41no—
237#126—41no—
238#126—41no—
239#126—41no—
240#126—41no—
241#126—41no—
242#125—21no—
243#125—21no—
244#125—21no—
245#125—21no—
246#125—21no—
247#125—21no—
248#125—21no—
249#125—41no—
250#125—41no—
251#125—41no—
252#125—41no—
253#125—41no—
254#125—41no—
255#125—41no—
256#125—41no—
257#125—41no—
258#124—21no—
259#123—41no—
260#123—41no—
261#123—41no—
262#123—41no—
263#123—41no—
264#123—41no—
265#123—41no—
266#123—41no—
267#123—41no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271C22⋊Q8—41no—
272C22⋊Q8—41no—
273C22⋊Q8—41no—
274C22⋊Q8—41no—
275C22⋊Q8—41no—
276C22⋊Q8—41no—
277C22⋊Q8—41no—
278C22⋊Q8—41no—
279#121—21no—
280#121—21no—
281#121—21no—
282#121—41no—
283#121—41no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×Q8—21no—
287C4×Q8—41no—
288C4×Q8—41no—
289C4×Q8—41no—
290C4×Q8C22⋊C411yescharacteristic
291C4×Q8D811yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4—21no—
295C4×D4—41no—
296C4×D4—41no—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4C22⋊C411yescharacteristic
302C4×D4C22⋊C411yescharacteristic
303C4×D4C22⋊C411yescharacteristic
304C4×D4D811yescharacteristic
305C4×D4D811yescharacteristic
306C4×D4D811yescharacteristic
307#117—21no—
308#117—21no—
309#117—41no—
310#117—41no—
311#117—41no—
312#117—41no—
313#117—41no—
314#117SD1611yescharacteristic
315#117SD1611yescharacteristic
316#117SD1611yescharacteristic
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—41no—
322C2×C4⋊C4—41no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4—41no—
325C2×C4⋊C4—41no—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329M32—81no—
330M32—81no—
331C2×C16—21no—
332C2×C16—21no—
333C2×C16—21no—
334C2×C16—21no—
335C2×C16—81no—
336#107—21no—
337#107—21no—
338#107—21no—
339#107—21no—
340#107—21no—
341#107—21no—
342#107C22×C411yescharacteristic
343#107C22×C411yescharacteristic
344#107C22×C411yescharacteristic
345#107C22×C411yescharacteristic
346#107C22×C411yescharacteristic
347#107C22×C411yescharacteristic
348#106—21no—
349#106—21no—
350#106—21no—
351#106—21no—
352#106—21no—
353#106—21no—
354#106—21no—
355#106—21no—
356#106—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—41no—
362C4⋊C8—41no—
363C4⋊C8—41no—
364C4⋊C8C22⋊C411yescharacteristic
365C4⋊C8C22⋊C411yescharacteristic
366C4⋊C8C22⋊C411yescharacteristic
367C4⋊C8C22⋊C411yescharacteristic
368C4⋊C8C22⋊C411yescharacteristic
369C4⋊C8C22⋊C411yescharacteristic
370#103—21no—
371#103—21no—
372#103—21no—
373#103—21no—
374#103—21no—
375#103—21no—
376#102—21no—
377#102—21no—
378#102—21no—
379#102—21no—
380#102—21no—
381#102—21no—
382C22⋊C8—41no—
383C22⋊C8—41no—
384C22⋊C8—41no—
385C22⋊C8C22⋊C411yescharacteristic
386C22⋊C8C22⋊C411yescharacteristic
387C22⋊C8C22⋊C411yescharacteristic
388C22⋊C8C22⋊C411yescharacteristic
389C22⋊C8C22⋊C411yescharacteristic
390C22⋊C8C22⋊C411yescharacteristic
391#97—41no—
392#97—41no—
393C4×C8—41no—
394C4×C8C22×C411yescharacteristic
395C4×C8C22×C411yescharacteristic
396C4×C8C22×C411yescharacteristic
397C4×C8C2×D411yescharacteristic
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D411yescharacteristic
400C4×C8C2×D411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C4×C8C2×D411yescharacteristic
403C4×C8C2411yescharacteristic
404#41—42no—
405#41—84no—
406C2×Q8—21no—
407C2×Q8—41no—
408C2×Q8—41no—
409C2×Q8#10211yescharacteristic
410C2×D4—21no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4#10211yescharacteristic
415C2×D4#10211yescharacteristic
416C2×D4#10211yescharacteristic
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×D811yescharacteristic
428C22×C4C2×D811yescharacteristic
429C22×C4C2×D811yescharacteristic
430C22×C4C2×SD1611yescharacteristic
431C22×C4C2×SD1611yescharacteristic
432C22×C4C2×SD1611yescharacteristic
433Q16—41no—
434SD16—41no—
435SD16—41no—
436D8—41no—
437M16—161no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—161no—
452C2×C8—161no—
453C2×C8C2×C22⋊C411yescharacteristic
454C2×C8C2×C22⋊C411yescharacteristic
455C2×C8C2×C22⋊C411yescharacteristic
456C2×C8C2×C22⋊C411yescharacteristic
457C2×C8C2×C22⋊C411yescharacteristic
458C2×C8C2×C22⋊C411yescharacteristic
459C2×C8C23×C411yescharacteristic
460C2×C8C22×D411yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4#10211yescharacteristic
494C4⋊C4#10211yescharacteristic
495C4⋊C4#10211yescharacteristic
496C4⋊C4#10211yescharacteristic
497C4⋊C4#10211yescharacteristic
498C4⋊C4#10211yescharacteristic
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4#10211yescharacteristic
513C22⋊C4#10211yescharacteristic
514C22⋊C4#10211yescharacteristic
515C22⋊C4#10211yescharacteristic
516C22⋊C4#10211yescharacteristic
517C22⋊C4#10211yescharacteristic
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42C2×C22⋊C411yescharacteristic
524C42C2×C22⋊C411yescharacteristic
525C42C2×C22⋊C411yescharacteristic
526C42C2×D811yescharacteristic
527C42C2×D811yescharacteristic
528C42C2×D811yescharacteristic
529C42C2×SD1611yescharacteristic
530C42C2×SD1611yescharacteristic
531C42C2×SD1611yescharacteristic
532C42C22×D411yescharacteristic
533C16—41no—
534C16—41no—
535C16—41no—
536C16—41no—
537C23—41no—
538C23C2×#10211yescharacteristic
539C23C2×#10211yescharacteristic
540C23C2×#10211yescharacteristic
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—82no—
545Q8—161no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—82no—
552D4—82no—
553D4—161no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—161no—
577C2×C4—161no—
578C2×C4—161no—
579C2×C4—161no—
580C2×C4C2×#10211yescharacteristic
581C2×C4C2×#10211yescharacteristic
582C2×C4C2×#10211yescharacteristic
583C2×C4C2×#10211yescharacteristic
584C2×C4C2×#10211yescharacteristic
585C2×C4C2×#10211yescharacteristic
586C2×C4C2×#10211yescharacteristic
587C2×C4C2×#10211yescharacteristic
588C2×C4C2×#10211yescharacteristic
589C2×C4C22×C22⋊C411yescharacteristic
590C2×C4C22×D811yescharacteristic
591C2×C4C22×SD1611yescharacteristic
592C8—21no—
593C8—21no—
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—21no—
599C8—81no—
600C8—81no—
601C8—81no—
602C8—81no—
603C8#57711yescharacteristic
604C8#57811yescharacteristic
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22#214312yes—
617C22C22×#10211yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4#288911yescharacteristic
637C4#289811yescharacteristic
638C2—22no—
639C2—41no—
640C2—41no—
641C2—161no—
642C2#1772111yescharacteristic
643C2C2×#214311yescharacteristic
644C2#3338811yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#62475C2
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33503C2semidirect
3. #17834C2semidirect
4. #11980C2semidirect
5. #11979C2semidirect
6. #11978C2semidirect
7. #11970C2semidirect
8. #11969C2semidirect
9. #11967C2semidirect
10. #11946C2semidirect
11. #11945C2semidirect
12. #11944C2semidirect
13. #11936C2semidirect
14. #11935C2semidirect
15. C2×#1363C2semidirect

Maximal subgroups

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

subgroupindex
#624752
#335032
#178342
#119802
#119792
#119782
#119702
#119692
#119672
#119462
#119452
#119442
#119362
#119352
C2×#13632