Finite Group Databasebeta

← #481416

Subgroups — #481416

Order 512.

749 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#61431C211yesmaximal, characteristic
2#33484C211yesmaximal, characteristic, complemented
3#17827C211yesmaximal, characteristic, complemented
4#11682C211yesmaximal, characteristic, complemented
5#11678C211yesmaximal, characteristic, complemented
6#11677C211yesmaximal, characteristic, complemented
7#11667C211yesmaximal, characteristic, complemented
8#11665C211yesmaximal, characteristic, complemented
9#11663C211yesmaximal, characteristic, complemented
10#11656C211yesmaximal, characteristic, complemented
11#11652C211yesmaximal, characteristic, complemented
12#11651C211yesmaximal, characteristic, complemented
13#11599C211yesmaximal, characteristic, complemented
14#11597C211yesmaximal, characteristic, complemented
15C2×#1333C211yesmaximal, characteristic, complemented
16#3481—21no—
17#3396C411yescharacteristic
18#3380C411yescharacteristic
19#3361—21no—
20#3339C411yescharacteristic
21#3339C411yescharacteristic
22#3338C411yescharacteristic
23#3297—21no—
24#3297—21no—
25#3295—21no—
26#3280C411yescharacteristic
27#3280C411yescharacteristic
28D4×D8C411yescharacteristic
29#3169C2211yescharacteristic
30#3168C2211yescharacteristic
31#3168C2211yescharacteristic
32#3106—21no—
33#3105—21no—
34#3105—21no—
35C8×#41C2211yescharacteristic
36C4×#135C2211yescharacteristic
37C2×C4×D8C2211yescharacteristic
38C2×C4×D8C2211yescharacteristic
39#2925—21no—
40#2183—21no—
41#2168—21no—
42#2168—21no—
43D4×C16—21no—
44#2163C2211yescharacteristic, complemented
45#2163C2211yescharacteristic, complemented
46#2163C2211yescharacteristic, complemented
47C4×M32C2211yescharacteristic, complemented
48C2×#346C2211yescharacteristic, complemented
49C2×#346C2211yescharacteristic, complemented
50#1572—21no—
51#1572—21no—
52#1571—21no—
53#1568C2211yescharacteristic, complemented
54#1568C2211yescharacteristic, complemented
55#1566C2211yescharacteristic, complemented
56#1564C2211yescharacteristic, complemented
57#1564C2211yescharacteristic, complemented
58C2×#334C2211yescharacteristic, complemented
59#1334C2211yescharacteristic, complemented
60#1334C2211yescharacteristic, complemented
61#1334C2211yescharacteristic, complemented
62#1334C2211yescharacteristic, complemented
63#1333C2211yescharacteristic
64#1333C2211yescharacteristic
65#1333C2211yescharacteristic
66#1333C2211yescharacteristic
67#1333C2211yescharacteristic, complemented
68#1333C2211yescharacteristic, complemented
69#1333C2211yescharacteristic, complemented
70#1333C2211yescharacteristic, complemented
71#1333C2211yescharacteristic, complemented
72#1333C2211yescharacteristic, complemented
73#1333C2211yescharacteristic, complemented
74#1333C2211yescharacteristic, complemented
75C2×#135C2×C411yescharacteristic
76C22×D8C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78#559—21no—
79#554—21no—
80#554—21no—
81#553—21no—
82#553—21no—
83#553—41no—
84#551—41no—
85#551—41no—
86#546—21no—
87#546—21no—
88#546—41no—
89D42—21no—
90#532—21no—
91#529—21no—
92#529—21no—
93#529—41no—
94#525—41no—
95#525—41no—
96C4×#41—21no—
97C4×#41D411yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—41no—
101C2×M32—21no—
102C22×C16—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#484—21no—
114#484—21no—
115#482—21no—
116#482—21no—
117#481—21no—
118#481—21no—
119#478—21no—
120#478—21no—
121#478—21no—
122#478—21no—
123#477—21no—
124#476—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#463—21no—
135#463—21no—
136#463—21no—
137#463—21no—
138#460—21no—
139#459—21no—
140#450—21no—
141#450—21no—
142#449—21no—
143#449—21no—
144#447—21no—
145#447—21no—
146C8×Q8C2×C411yescharacteristic
147C8×Q8D411yescharacteristic
148C4×Q16C2×C411yescharacteristic
149C4×Q16C2311yescharacteristic
150C4×SD16—21no—
151C4×SD16—21no—
152C4×SD16—21no—
153C4×SD16—21no—
154C4×D8—21no—
155C4×D8—21no—
156C4×D8—21no—
157C4×D8—21no—
158C4×D8—21no—
159C4×D8C2×C411yescharacteristic
160C4×D8C2×C411yescharacteristic
161C4×D8C2×C411yescharacteristic
162C4×D8C2×C411yescharacteristic
163C4×D8C2×C411yescharacteristic
164C4×D8C2311yescharacteristic
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
183C2×#107C2×C411yescharacteristic
184C2×#107C2×C411yescharacteristic
185#423—21no—
186C2×C4⋊C8—21no—
187#416—21no—
188C2×#102—21no—
189C2×#102—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—
203#346C2311yescharacteristic
204#346C2311yescharacteristic
205#346C2311yescharacteristic
206#346C2311yescharacteristic
207C4×C16—21no—
208#335—21no—
209#335—21no—
210#335—21no—
211#335—21no—
212#334—21no—
213#334—21no—
214#334C2311yescharacteristic
215#334C2311yescharacteristic
216#334C2311yescharacteristic
217#334C2311yescharacteristic
218C2×#41—21no—
219C2×#41—41no—
220C2×#41D811yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—41no—
224C23×C4—41no—
225C23×C4—41no—
226#135—22no—
227#135—41no—
228C2×Q16C22×C411yescharacteristic
229C2×SD16—21no—
230C2×SD16—21no—
231C2×SD16—21no—
232C2×D8—22no—
233C2×D8—22no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—41no—
238C2×D8—41no—
239C2×D8C22×C411yescharacteristic
240C2×D8C22×C411yescharacteristic
241C2×D8C22×C411yescharacteristic
242C2×M16—41no—
243C22×C8—41no—
244C22×C8C22×C411yescharacteristic
245C22×C8C22×C411yescharacteristic
246C22×C8C22×C411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C4⋊Q8—21no—
251#127—21no—
252#126—41no—
253#126—41no—
254#126—41no—
255#126—41no—
256#125—21no—
257#125—21no—
258#125—41no—
259#124—21no—
260#124—21no—
261#124—41no—
262#124—41no—
263#123—41no—
264#123—41no—
265#123—41no—
266#123—41no—
267#123—41no—
268#123—41no—
269#123—41no—
270#123—41no—
271C22⋊Q8—21no—
272C22⋊Q8—21no—
273C22⋊Q8—41no—
274C22⋊Q8—41no—
275C22⋊Q8—41no—
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#120—41no—
289#120—41no—
290#120—41no—
291#120—41no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8C22⋊C411yescharacteristic
295C4×Q8SD1611yescharacteristic
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4—41no—
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×D4C22⋊C411yescharacteristic
318C4×D4C22⋊C411yescharacteristic
319C4×D4C22⋊C411yescharacteristic
320C4×D4SD1611yescharacteristic
321C4×D4SD1611yescharacteristic
322C4×D4SD1611yescharacteristic
323#117—21no—
324#117—41no—
325#117—41no—
326#117—41no—
327#117D811yescharacteristic
328#117D811yescharacteristic
329#117D811yescharacteristic
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—41no—
333C2×C22⋊C4—41no—
334C2×C22⋊C4—41no—
335C2×C22⋊C4—41no—
336C2×C22⋊C4—41no—
337C2×C42—41no—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341M32—42no—
342C2×C16—21no—
343C2×C16—21no—
344C2×C16—21no—
345C2×C16—21no—
346C2×C16—42no—
347C2×C16—42no—
348#107—21no—
349#107—21no—
350#107—21no—
351#107C22×C411yescharacteristic
352#107C22×C411yescharacteristic
353#107C22×C411yescharacteristic
354#107C22×C411yescharacteristic
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#102—21no—
375#102—21no—
376#102—21no—
377#102—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
410C24—41no—
411C24—41no—
412#41—42no—
413#41—42no—
414#41—81no—
415#41—162no—
416C2×Q8—21no—
417C2×Q8—41no—
418C2×Q8#10211yescharacteristic
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—81no—
434C2×D4—81no—
435C2×D4—84no—
436C2×D4#10211yescharacteristic
437C2×D4#10211yescharacteristic
438C2×D4#10211yescharacteristic
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—162no—
451C22×C4—162no—
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×D811yescharacteristic
456C22×C4C2×D811yescharacteristic
457C22×C4C2×D811yescharacteristic
458C22×C4C2×SD1611yescharacteristic
459C22×C4C2×SD1611yescharacteristic
460C22×C4C2×SD1611yescharacteristic
461Q16—21no—
462Q16—21no—
463SD16—42no—
464SD16—81no—
465SD16—81no—
466D8—21no—
467D8—21no—
468D8—21no—
469D8—21no—
470D8—21no—
471D8—42no—
472D8—42no—
473D8—81no—
474D8#13112yes—
475M16—82no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—82no—
490C2×C8C2×C22⋊C411yescharacteristic
491C2×C8C2×C22⋊C411yescharacteristic
492C2×C8C2×C22⋊C411yescharacteristic
493C2×C8C2×C22⋊C411yescharacteristic
494C2×C8C2×C22⋊C411yescharacteristic
495C2×C8C2×C22⋊C411yescharacteristic
496C2×C8C23×C411yescharacteristic
497C2×C8C22×D411yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
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—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4#10211yescharacteristic
514C4⋊C4#10211yescharacteristic
515C4⋊C4#10211yescharacteristic
516C4⋊C4#10211yescharacteristic
517C4⋊C4#10211yescharacteristic
518C4⋊C4#10211yescharacteristic
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—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4#10211yescharacteristic
539C22⋊C4#10211yescharacteristic
540C22⋊C4#10211yescharacteristic
541C22⋊C4#10211yescharacteristic
542C22⋊C4#10211yescharacteristic
543C22⋊C4#10211yescharacteristic
544C42—41no—
545C42—41no—
546C42—41no—
547C42C2×C22⋊C411yescharacteristic
548C42C2×C22⋊C411yescharacteristic
549C42C2×C22⋊C411yescharacteristic
550C42C2×D811yescharacteristic
551C42C2×D811yescharacteristic
552C42C2×D811yescharacteristic
553C42C2×SD1611yescharacteristic
554C42C2×SD1611yescharacteristic
555C42C2×SD1611yescharacteristic
556C42C22×D411yescharacteristic
557C16—22no—
558C16—41no—
559C16—41no—
560C16—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—82no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—161no—
569C23—161no—
570C23C2×#10211yescharacteristic
571C23C2×#10211yescharacteristic
572C23C2×#10211yescharacteristic
573Q8—41no—
574Q8—41no—
575Q8—41no—
576Q8—161no—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—82no—
587D4—82no—
588D4—82no—
589D4—82no—
590D4—82no—
591D4—82no—
592D4—161no—
593D4—161no—
594D4—161no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—161no—
618C2×C4—161no—
619C2×C4—161no—
620C2×C4—161no—
621C2×C4—161no—
622C2×C4—161no—
623C2×C4C2×#10211yescharacteristic
624C2×C4C2×#10211yescharacteristic
625C2×C4C2×#10211yescharacteristic
626C2×C4C2×#10211yescharacteristic
627C2×C4C2×#10211yescharacteristic
628C2×C4C2×#10211yescharacteristic
629C2×C4C2×#10211yescharacteristic
630C2×C4C2×#10211yescharacteristic
631C2×C4C2×#10211yescharacteristic
632C2×C4C22×C22⋊C411yescharacteristic
633C2×C4C22×D811yescharacteristic
634C2×C4C22×SD1611yescharacteristic
635C8—21no—
636C8—21no—
637C8—21no—
638C8—21no—
639C8—21no—
640C8—21no—
641C8—21no—
642C8—81no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8C2×#13111yescharacteristic
647C8#56811yescharacteristic
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—164no—
666C22—164no—
667C22#214512yes—
668C22C22×#10211yescharacteristic
669C4—21no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4#288011yescharacteristic
686C4#290111yescharacteristic
687C2—22no—
688C2—41no—
689C2—41no—
690C2—82no—
691C2—161no—
692C2—161no—
693C2#1750711yescharacteristic
694C2C2×#214511yescharacteristic
695C2#3339211yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radical#61431C2
LCSG ▷ C2×C8 ▷ 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. #33484C2semidirect
3. #17827C2semidirect
4. #11682C2semidirect
5. #11678C2semidirect
6. #11677C2semidirect
7. #11667C2semidirect
8. #11665C2semidirect
9. #11663C2semidirect
10. #11656C2semidirect
11. #11652C2semidirect
12. #11651C2semidirect
13. #11599C2semidirect
14. #11597C2semidirect
15. C2×#1333C2semidirect
44. #2163C22semidirect
45. #2163C22semidirect
46. #2163C22semidirect
47. C4×M32C22semidirect
48. C2×#346C22semidirect
49. C2×#346C22semidirect
53. #1568C22semidirect
54. #1568C22semidirect
55. #1566C22semidirect
56. #1564C22semidirect
57. #1564C22semidirect
58. C2×#334C22semidirect
59. #1334C22semidirect
60. #1334C22semidirect
61. #1334C22semidirect
62. #1334C22semidirect
67. #1333C22semidirect
68. #1333C22semidirect
69. #1333C22semidirect
70. #1333C22semidirect
71. #1333C22semidirect
72. #1333C22semidirect
73. #1333C22semidirect
74. #1333C22semidirect

Maximal subgroups

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

subgroupindex
#614312
#334842
#178272
#116822
#116782
#116772
#116672
#116652
#116632
#116562
#116522
#116512
#115992
#115972
C2×#13332