Finite Group Databasebeta

← #481408

Subgroups — #481408

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61430C211yesmaximal, characteristic
2#33479C211yesmaximal, characteristic, complemented
3#17829C211yesmaximal, characteristic, complemented
4#11675C211yesmaximal, characteristic, complemented
5#11674C211yesmaximal, characteristic, complemented
6#11673C211yesmaximal, characteristic, complemented
7#11662C211yesmaximal, characteristic, complemented
8#11660C211yesmaximal, characteristic, complemented
9#11659C211yesmaximal, characteristic, complemented
10#11649C211yesmaximal, characteristic, complemented
11#11648C211yesmaximal, characteristic, complemented
12#11647C211yesmaximal, characteristic, complemented
13#11594C211yesmaximal, characteristic, complemented
14#11592C211yesmaximal, characteristic, complemented
15#11591C211yesmaximal, characteristic, complemented
16#3480—21no—
17#3396C411yescharacteristic
18#3383C411yescharacteristic
19#3365—21no—
20#3342C411yescharacteristic
21#3339C411yescharacteristic
22#3339C411yescharacteristic
23#3304—21no—
24#3301—21no—
25#3301—21no—
26#3287C411yescharacteristic
27#3285C411yescharacteristic
28#3285C411yescharacteristic
29#3169C2211yescharacteristic
30#3168C2211yescharacteristic
31#3168C2211yescharacteristic
32#3103—21no—
33#3102—21no—
34#3102—21no—
35C8×#41C2211yescharacteristic
36C4×#135C2211yescharacteristic
37C4×#135C2211yescharacteristic
38C2×C4×Q16C2211yescharacteristic
39#2925—21no—
40#2183—21no—
41#2169—21no—
42#2168—21no—
43#2168—21no—
44#2162C2211yescharacteristic
45#2162C2211yescharacteristic
46#2162C2211yescharacteristic
47C4×M32C2211yescharacteristic
48C4×M32C2211yescharacteristic
49C2×C4×C16C2211yescharacteristic
50#1572—21no—
51#1572—21no—
52#1570—21no—
53#1568C2211yescharacteristic
54#1568C2211yescharacteristic
55#1567C2211yescharacteristic
56#1564C2211yescharacteristic
57#1564C2211yescharacteristic
58C2×#335C2211yescharacteristic
59#1332C2211yescharacteristic
60#1332C2211yescharacteristic
61#1332C2211yescharacteristic
62#1332C2211yescharacteristic
63#1332C2211yescharacteristic
64#1332C2211yescharacteristic
65#1332C2211yescharacteristic
66#1332C2211yescharacteristic
67#1331C2211yescharacteristic
68#1331C2211yescharacteristic
69#1331C2211yescharacteristic
70#1331C2211yescharacteristic
71#1331C2211yescharacteristic
72#1331C2211yescharacteristic
73#1331C2211yescharacteristic
74#1331C2211yescharacteristic
75C2×#135C2×C411yescharacteristic
76C2×#135C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78#559—21no—
79#559—41no—
80#557—21no—
81#556—21no—
82#556—41no—
83#553—21no—
84#553—21no—
85#553—41no—
86#553—41no—
87#550—21no—
88#548—21no—
89#548—21no—
90#532—21no—
91#532—41no—
92#529—21no—
93#529—21no—
94#529—41no—
95#529—41no—
96C4×#41—21no—
97C4×#41—21no—
98C4×#41—41no—
99C4×#41D411yescharacteristic
100C2×C4×Q8—21no—
101C2×M32—21no—
102C2×M32—21no—
103C22×C16—21no—
104#500C2×C411yescharacteristic
105#499C2×C411yescharacteristic
106#499C2×C411yescharacteristic
107#495C2×C411yescharacteristic
108#495C2×C411yescharacteristic
109#493C2×C411yescharacteristic
110#488—21no—
111#488—21no—
112#486—21no—
113#485—21no—
114#485—21no—
115#482—21no—
116#482—21no—
117#481—21no—
118#481—21no—
119#479—21no—
120#479—21no—
121#478—21no—
122#478—21no—
123#475—21no—
124#475—21no—
125#467C2×C411yescharacteristic
126#467C2×C411yescharacteristic
127#466—21no—
128#466C2×C411yescharacteristic
129#466C2×C411yescharacteristic
130#466C2×C411yescharacteristic
131#466C2×C411yescharacteristic
132#465—21no—
133#465—21no—
134#464—21no—
135#464—21no—
136#463—21no—
137#463—21no—
138#460—21no—
139#460—21no—
140#452—21no—
141#452—21no—
142#449—21no—
143#449—21no—
144#448—21no—
145#448—21no—
146C8×Q8C2×C411yescharacteristic
147C8×Q8D411yescharacteristic
148C4×Q16—21no—
149C4×Q16—21no—
150C4×Q16C2×C411yescharacteristic
151C4×Q16C2×C411yescharacteristic
152C4×Q16C2×C411yescharacteristic
153C4×Q16C2×C411yescharacteristic
154C4×Q16C2311yescharacteristic
155C4×Q16C2311yescharacteristic
156C4×SD16—21no—
157C4×SD16—21no—
158C4×SD16—21no—
159C4×SD16—21no—
160C4×SD16—21no—
161C4×SD16—21no—
162C4×D8—21no—
163C4×D8C2×C411yescharacteristic
164C4×D8C2×C411yescharacteristic
165C4×D8C2311yescharacteristic
166C4×D8C2311yescharacteristic
167C8×D4C2×C411yescharacteristic
168C8×D4C2×C411yescharacteristic
169C8×D4C2×C411yescharacteristic
170C8×D4D411yescharacteristic
171C8×D4D411yescharacteristic
172C8×D4D411yescharacteristic
173#433—41no—
174#433—41no—
175#432—21no—
176#432—21no—
177#431—21no—
178#431—21no—
179#431D411yescharacteristic
180#431D411yescharacteristic
181#431D411yescharacteristic
182#427C2×C411yescharacteristic
183#427C2×C411yescharacteristic
184C2×#107C2×C411yescharacteristic
185#423—21no—
186C2×C4⋊C8—21no—
187#416—21no—
188#416—21no—
189C2×#103—21no—
190#408—41no—
191#405—41no—
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C2×C4×C8C2311yescharacteristic
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C22⋊C16—21no—
199C22⋊C16—21no—
200C22⋊C16—21no—
201#346—21no—
202#346—21no—
203C4×C16—21no—
204C4×C16C2311yescharacteristic
205C4×C16C2311yescharacteristic
206C4×C16C2311yescharacteristic
207C4×C16C2311yescharacteristic
208#335—21no—
209#335—21no—
210#335C2311yescharacteristic
211#335C2311yescharacteristic
212#335C2311yescharacteristic
213#335C2311yescharacteristic
214#334—21no—
215#334—21no—
216#334—21no—
217#334—21no—
218C2×#41—21no—
219C2×#41—21no—
220C2×#41—41no—
221C2×#41D811yescharacteristic
222C22×Q8—21no—
223#135—22no—
224#135—22no—
225#135—41no—
226#135—41no—
227C2×Q16—22no—
228C2×Q16—21no—
229C2×Q16—41no—
230C2×Q16C22×C411yescharacteristic
231C2×Q16C22×C411yescharacteristic
232C2×SD16—21no—
233C2×SD16—21no—
234C2×SD16—21no—
235C2×SD16—21no—
236C2×D8—21no—
237C2×D8C22×C411yescharacteristic
238C2×D8C22×C411yescharacteristic
239C2×M16—41no—
240C22×C8—41no—
241C22×C8C22×C411yescharacteristic
242C22×C8C22×C411yescharacteristic
243C22×C8C22×C411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C4⋊Q8—21no—
248C4⋊Q8—41no—
249#127—21no—
250#127—41no—
251#126—41no—
252#126—41no—
253#126—41no—
254#126—41no—
255#126—41no—
256#126—41no—
257#125—21no—
258#125—21no—
259#125—41no—
260#125—41no—
261#124—21no—
262#124—21no—
263#124—41no—
264#124—41no—
265#124—41no—
266#124—41no—
267#124—41no—
268#123—41no—
269#123—41no—
270#123—41no—
271#123—41no—
272#123—41no—
273#123—41no—
274C22⋊Q8—21no—
275C22⋊Q8—21no—
276C22⋊Q8—41no—
277C22⋊Q8—41no—
278#121—21no—
279#121—21no—
280#121—21no—
281#121—21no—
282#121—41no—
283#121—41no—
284#121—41no—
285#121—41no—
286#121—41no—
287#121—41no—
288#121—41no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—41no—
296C4×Q8—41no—
297C4×Q8—41no—
298C4×Q8—41no—
299C4×Q8C22⋊C411yescharacteristic
300C4×Q8SD1611yescharacteristic
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—41no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314C4×D4—41no—
315C4×D4—41no—
316C4×D4—41no—
317C4×D4—41no—
318C4×D4C22⋊C411yescharacteristic
319C4×D4C22⋊C411yescharacteristic
320C4×D4C22⋊C411yescharacteristic
321C4×D4SD1611yescharacteristic
322C4×D4SD1611yescharacteristic
323C4×D4SD1611yescharacteristic
324#117—21no—
325#117—21no—
326#117—41no—
327#117—41no—
328#117—41no—
329#117—41no—
330#117D811yescharacteristic
331#117D811yescharacteristic
332#117D811yescharacteristic
333C2×C4⋊C4—21no—
334C2×C4⋊C4—41no—
335C2×C42—41no—
336C2×C42—41no—
337C2×C42—41no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341M32—42no—
342M32—42no—
343C2×C16—21no—
344C2×C16—21no—
345C2×C16—21no—
346C2×C16—21no—
347C2×C16—42no—
348#107—21no—
349#107—21no—
350#107C22×C411yescharacteristic
351#107C22×C411yescharacteristic
352#107C22×C411yescharacteristic
353#107C22×C411yescharacteristic
354#106—21no—
355#106—21no—
356#106—21no—
357#106—21no—
358C4⋊C8—21no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—21no—
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#103—21no—
377#103—21no—
378#102—21no—
379#102—21no—
380#102—21no—
381#102—21no—
382#102—21no—
383#102—21no—
384#102—21no—
385#102—21no—
386C22⋊C8—41no—
387C22⋊C8—41no—
388C22⋊C8—41no—
389C22⋊C8—41no—
390C22⋊C8C22⋊C411yescharacteristic
391C22⋊C8C22⋊C411yescharacteristic
392C22⋊C8C22⋊C411yescharacteristic
393C22⋊C8C22⋊C411yescharacteristic
394C22⋊C8C22⋊C411yescharacteristic
395C22⋊C8C22⋊C411yescharacteristic
396#97—41no—
397#97—41no—
398C4×C8—41no—
399C4×C8—41no—
400C4×C8C22×C411yescharacteristic
401C4×C8C22×C411yescharacteristic
402C4×C8C22×C411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2×D411yescharacteristic
408C4×C8C2×D411yescharacteristic
409C4×C8C2411yescharacteristic
410#41—41no—
411#41—41no—
412#41—42no—
413#41—42no—
414#41—81no—
415#41—81no—
416#41—82no—
417#41—82no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—42no—
421C2×Q8—41no—
422C2×Q8—41no—
423C2×Q8—81no—
424C2×Q8#10211yescharacteristic
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4#10211yescharacteristic
436C2×D4#10211yescharacteristic
437C2×D4#10211yescharacteristic
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×D811yescharacteristic
450C22×C4C2×D811yescharacteristic
451C22×C4C2×D811yescharacteristic
452C22×C4C2×SD1611yescharacteristic
453C22×C4C2×SD1611yescharacteristic
454C22×C4C2×SD1611yescharacteristic
455Q16—21no—
456Q16—21no—
457Q16—21no—
458Q16—42no—
459Q16#13112yes—
460SD16—42no—
461SD16—42no—
462SD16—81no—
463SD16—81no—
464D8—21no—
465D8—21no—
466D8—21no—
467D8—21no—
468D8—81no—
469M16—82no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—82no—
484C2×C8C2×C22⋊C411yescharacteristic
485C2×C8C2×C22⋊C411yescharacteristic
486C2×C8C2×C22⋊C411yescharacteristic
487C2×C8C2×C22⋊C411yescharacteristic
488C2×C8C2×C22⋊C411yescharacteristic
489C2×C8C2×C22⋊C411yescharacteristic
490C2×C8C23×C411yescharacteristic
491C2×C8C22×D411yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4#10211yescharacteristic
511C4⋊C4#10211yescharacteristic
512C4⋊C4#10211yescharacteristic
513C4⋊C4#10211yescharacteristic
514C4⋊C4#10211yescharacteristic
515C4⋊C4#10211yescharacteristic
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4#10211yescharacteristic
533C22⋊C4#10211yescharacteristic
534C22⋊C4#10211yescharacteristic
535C22⋊C4#10211yescharacteristic
536C22⋊C4#10211yescharacteristic
537C22⋊C4#10211yescharacteristic
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42C2×C22⋊C411yescharacteristic
547C42C2×C22⋊C411yescharacteristic
548C42C2×C22⋊C411yescharacteristic
549C42C2×D811yescharacteristic
550C42C2×D811yescharacteristic
551C42C2×D811yescharacteristic
552C42C2×SD1611yescharacteristic
553C42C2×SD1611yescharacteristic
554C42C2×SD1611yescharacteristic
555C42C22×D411yescharacteristic
556C16—22no—
557C16—41no—
558C16—41no—
559C16—41no—
560C23—41no—
561C23—41no—
562C23C2×#10211yescharacteristic
563C23C2×#10211yescharacteristic
564C23C2×#10211yescharacteristic
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—82no—
571Q8—82no—
572D4—41no—
573D4—41no—
574D4—41no—
575D4—41no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—82no—
580D4—82no—
581D4—82no—
582D4—82no—
583D4—82no—
584D4—161no—
585D4—161no—
586D4—161no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—161no—
616C2×C4—161no—
617C2×C4—161no—
618C2×C4C2×#10211yescharacteristic
619C2×C4C2×#10211yescharacteristic
620C2×C4C2×#10211yescharacteristic
621C2×C4C2×#10211yescharacteristic
622C2×C4C2×#10211yescharacteristic
623C2×C4C2×#10211yescharacteristic
624C2×C4C2×#10211yescharacteristic
625C2×C4C2×#10211yescharacteristic
626C2×C4C2×#10211yescharacteristic
627C2×C4C22×C22⋊C411yescharacteristic
628C2×C4C22×D811yescharacteristic
629C2×C4C22×SD1611yescharacteristic
630C8—21no—
631C8—21no—
632C8—21no—
633C8—21no—
634C8—21no—
635C8—21no—
636C8—21no—
637C8—81no—
638C8—81no—
639C8—81no—
640C8—81no—
641C8C2×#13111yescharacteristic
642C8#56811yescharacteristic
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22C22×#10211yescharacteristic
659C4—21no—
660C4—21no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—42no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—81no—
675C4#288011yescharacteristic
676C4#290111yescharacteristic
677C4#341411yescharacteristic
678C4#341411yescharacteristic
679C2—41no—
680C2—41no—
681C2—41no—
682C2—161no—
683C2—161no—
684C2#1750711yescharacteristic
685C2#3338011yescharacteristic
686C2#3338511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#61430C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #33479C2semidirect
3. #17829C2semidirect
4. #11675C2semidirect
5. #11674C2semidirect
6. #11673C2semidirect
7. #11662C2semidirect
8. #11660C2semidirect
9. #11659C2semidirect
10. #11649C2semidirect
11. #11648C2semidirect
12. #11647C2semidirect
13. #11594C2semidirect
14. #11592C2semidirect
15. #11591C2semidirect

Maximal subgroups

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

subgroupindex
#614302
#334792
#178292
#116752
#116742
#116732
#116622
#116602
#116592
#116492
#116482
#116472
#115942
#115922
#115912