Finite Group Databasebeta

← #228142

Subgroups — #228142

Order 512.

829 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23016C211yesmaximal, characteristic, complemented
2D4×#104C211yesmaximal, characteristic, complemented
3#18516C211yesmaximal, characteristic
4#17994C211yesmaximal, characteristic, complemented
5#17507C211yesmaximal, characteristic, complemented
6#9847C211yesmaximal, characteristic, complemented
7#9590C211yesmaximal, characteristic, complemented
8#9581C211yesmaximal, characteristic, complemented
9#9233C211yesmaximal, characteristic, complemented
10#9208C211yesmaximal, characteristic, complemented
11#9053C211yesmaximal, characteristic, complemented
12#8906C211yesmaximal, characteristic, complemented
13#8902C211yesmaximal, characteristic, complemented
14#8191C211yesmaximal, characteristic, complemented
15#8174C211yesmaximal, characteristic, complemented
16D4×#41C2211yescharacteristic
17#3169C2211yescharacteristic
18#3100C2211yescharacteristic
19#3035C2211yescharacteristic
20C8×#41C2211yescharacteristic
21C2×#444—21no—
22#2935C2211yescharacteristic, complemented
23D4×M16C2211yescharacteristic
24#2906C2211yescharacteristic
25C22×#104—21no—
26C42×D4C2211yescharacteristic
27#1980C2211yescharacteristic
28#1962C2211yescharacteristic
29#1962C2211yescharacteristic, complemented
30#1950C2211yescharacteristic, complemented
31#1938—21no—
32#1896—21no—
33#1876—21no—
34#1871—21no—
35#1866—21no—
36#1806C2211yescharacteristic
37#1789—21no—
38#1767C2211yescharacteristic, complemented
39C4×#104C2211yescharacteristic
40#1744C2211yescharacteristic, complemented
41#1649C411yescharacteristic, complemented
42#1647C411yescharacteristic, complemented
43#1641C411yescharacteristic, complemented
44#1639C411yescharacteristic, complemented
45#1627C411yescharacteristic, complemented
46#1625C411yescharacteristic, complemented
47#1621C411yescharacteristic, complemented
48#1619C411yescharacteristic, complemented
49#1589—21no—
50#1581—21no—
51#1581—21no—
52C8×D8—21no—
53#1571C2211yescharacteristic, complemented
54#1567C2211yescharacteristic, complemented
55#1540C2211yescharacteristic
56#1502C2211yescharacteristic, complemented
57#1492—21no—
58#1492C2211yescharacteristic
59#1492C2211yescharacteristic, complemented
60#1486—21no—
61#1486C2211yescharacteristic
62#1480—21no—
63#1480C2211yescharacteristic
64#1476C2211yescharacteristic
65C2×D4⋊C8—21no—
66C2×D4⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C8×M16C2211yescharacteristic, complemented
69#1303C2211yescharacteristic, complemented
70#1301C2211yescharacteristic, complemented
71#1279C2211yescharacteristic, complemented
72#1279C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1276C2211yescharacteristic, complemented
75C22×#41—21no—
76C2×#135—21no—
77C2×#131—21no—
78C2×#131—21no—
79C22×M16—21no—
80#550C2×C411yescharacteristic
81D4×Q8C2×C411yescharacteristic
82#548—21no—
83#547—21no—
84#547C2×C411yescharacteristic
85#546—21no—
86#546—41no—
87D42—21no—
88D42C2×C411yescharacteristic
89#532—21no—
90#532C2311yescharacteristic
91#532C2311yescharacteristic
92#525—21no—
93#525—41no—
94C4×#41C2311yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4C2311yescharacteristic
98C22×C42—21no—
99#500C2×C411yescharacteristic, complemented
100#493C2×C411yescharacteristic, complemented
101#486C2×C411yescharacteristic, complemented
102#486C2×C411yescharacteristic, complemented
103#480—21no—
104#474C2×C411yescharacteristic
105#474C2×C411yescharacteristic
106#466—41no—
107#464—21no—
108#463—21no—
109#462C2×C411yescharacteristic, complemented
110#461C2×C411yescharacteristic, complemented
111#460C2×C411yescharacteristic, complemented
112#459C2×C411yescharacteristic
113#459C2×C411yescharacteristic
114#459C2×C411yescharacteristic, complemented
115#447—21no—
116C8×Q8—21no—
117#444—44no—
118C4×D8—21no—
119C4×D8—21no—
120C4×D8—41no—
121#436—21no—
122#436—21no—
123#435—41no—
124#435—41no—
125C8×D4—21no—
126C8×D4—21no—
127C8×D4—21no—
128C8×D4—41no—
129#431—21no—
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132C2×#108—21no—
133#424—21no—
134#424—41no—
135#423—21no—
136#423C2311yescharacteristic
137#423C2311yescharacteristic
138C2×C4⋊C8—21no—
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×#104—21no—
142C2×#104—21no—
143C2×#104—21no—
144C2×#104—21no—
145C2×#104—21no—
146C2×#104—21no—
147C2×#104—21no—
148C2×#104—44no—
149C2×#104D411yescharacteristic
150C2×#104D411yescharacteristic
151#416—21no—
152#408—21no—
153#407—21no—
154C4×M16C2311yescharacteristic
155C2×#97—21no—
156C2×C4×C8—21no—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2311yescharacteristic
161C4×C22⋊C4—21no—
162C4×C22⋊C4—41no—
163C43C2311yescharacteristic
164#341—21no—
165#339—21no—
166#339D411yescharacteristic
167#339D411yescharacteristic
168#336—21no—
169#336—41no—
170#335—21no—
171#335—21no—
172#335—41no—
173#332C2×C411yescharacteristic, complemented
174#332C2×C411yescharacteristic, complemented
175#331C2×C411yescharacteristic, complemented
176#331C2×C411yescharacteristic, complemented
177#327—41no—
178Q8⋊C8C2×C411yescharacteristic, complemented
179Q8⋊C8C2×C411yescharacteristic, complemented
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8—21no—
186D4⋊C8—21no—
187D4⋊C8—21no—
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8C2×C411yescharacteristic
190D4⋊C8C2×C411yescharacteristic
191D4⋊C8C2×C411yescharacteristic
192D4⋊C8C2×C411yescharacteristic, complemented
193D4⋊C8C2×C411yescharacteristic, complemented
194D4⋊C8D411yescharacteristic
195D4⋊C8D411yescharacteristic
196D4⋊C8D411yescharacteristic
197D4⋊C8D411yescharacteristic
198#322—21no—
199C82—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41—21no—
203C2×#41—21no—
204C2×#41—41no—
205C2×#41—41no—
206C2×#41—42no—
207C2×#41—42no—
208C2×#41—81no—
209C2×#41—81no—
210C2×#41C2×D411yescharacteristic
211C22×Q8—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—41no—
215C22×D4#4111yescharacteristic
216C23×C4—21no—
217C23×C4—41no—
218C23×C4—41no—
219#135—42no—
220#135—42no—
221C2×Q16—21no—
222C2×SD16—21no—
223C2×SD16—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8—41no—
228#131—42no—
229#131—42no—
230#131—81no—
231#131—81no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—41no—
235C2×M16—41no—
236C2×M16—41no—
237C2×M16—42no—
238C2×M16—81no—
239C2×M16C2×D411yescharacteristic
240C2×M16C2×D411yescharacteristic
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—41no—
246C22×C8—41no—
247C22×C8—41no—
248C22×C8—41no—
249C4⋊Q8—21no—
250C4⋊Q8C22×C411yescharacteristic
251C4⋊Q8C22×C411yescharacteristic
252#127—21no—
253#127C22×C411yescharacteristic
254#127C22×C411yescharacteristic
255#124—21no—
256#124—21no—
257#124—41no—
258#123—41no—
259#123—41no—
260#123—41no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8—41no—
264C22⋊Q8—41no—
265#121—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272#121—21no—
273#121—41no—
274#121—41no—
275#121—41no—
276#120—41no—
277#120—41no—
278#120—41no—
279C4×Q8C22×C411yescharacteristic
280C4×Q8#4111yescharacteristic
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—41no—
292C4×D4—41no—
293C4×D4—41no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4C22×C411yescharacteristic
297C4×D4C22×C411yescharacteristic
298C4×D4C22×C411yescharacteristic
299C4×D4C2×D411yescharacteristic
300C4×D4C2×D411yescharacteristic
301C4×D4#4111yescharacteristic
302#117—21no—
303#117C2×D411yescharacteristic
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4#4111yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—41no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—42no—
315C2×C42—41no—
316C2×C42—81no—
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2411yescharacteristic
320#108—42no—
321#107—21no—
322#107—21no—
323#107—41no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—41no—
332C4⋊C8—41no—
333C4⋊C8—41no—
334C4⋊C8C22×C411yescharacteristic
335C4⋊C8C22×C411yescharacteristic
336C4⋊C8C22×C411yescharacteristic
337C4⋊C8C22×C411yescharacteristic
338C4⋊C8C2×D411yescharacteristic
339C4⋊C8C2×D411yescharacteristic
340#104—22no—
341#104—42no—
342#104—42no—
343#104—42no—
344#104—42no—
345#104—81no—
346#104—81no—
347#104—81no—
348#104D814yescomplemented
349#103—21no—
350#103—21no—
351#102—21no—
352#102—21no—
353#102—21no—
354#102—21no—
355#102—21no—
356#102—21no—
357#102—41no—
358#102—41no—
359C22⋊C8—21no—
360C22⋊C8—21no—
361C22⋊C8—41no—
362C22⋊C8—41no—
363C22⋊C8—41no—
364C22⋊C8—41no—
365C22⋊C8—41no—
366#97—21no—
367#97—42no—
368C4×C8—21no—
369C4×C8—21no—
370C4×C8—21no—
371C4×C8—21no—
372C4×C8—21no—
373C4×C8—41no—
374C4×C8C22×C411yescharacteristic
375C4×C8C22×C411yescharacteristic
376C4×C8C2×D411yescharacteristic
377C4×C8C2×D411yescharacteristic
378#95—21no—
379#95—41no—
380C24—21no—
381C24—41no—
382#41—21no—
383#41—21no—
384#41—42no—
385#41—42no—
386#41—41no—
387#41—41no—
388#41—41no—
389#41—41no—
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—82no—
395#41—82no—
396#41—81no—
397#41—82no—
398#41—82no—
399#41—81no—
400#41C2×D812yes—
401#41#13512yes—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—41no—
405C2×Q8—42no—
406C2×Q8—81no—
407C2×Q8C4×D411yescharacteristic
408C2×D4—21no—
409C2×D4—22no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—41no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—81no—
428C2×D4—81no—
429C2×D4—82no—
430C2×D4—82no—
431C2×D4C4×D411yescharacteristic
432C2×D4C4×D411yescharacteristic
433C2×D4C4×D411yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—44no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4H(5,𝔽2)11yescharacteristic
462Q16—42no—
463SD16—42no—
464SD16—42no—
465D8—42no—
466D8—42no—
467D8—81no—
468D8—82no—
469M16—21no—
470M16—21no—
471M16—21no—
472M16—41no—
473M16—41no—
474M16—41no—
475M16—41no—
476M16—82no—
477M16—82no—
478M16—82no—
479M16C2×D812yes—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—82no—
500C2×C8—82no—
501C2×C8—81no—
502C2×C8—81no—
503C2×C8—82no—
504C2×C8—82no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4C4×D411yescharacteristic
517C4⋊C4C4×D411yescharacteristic
518C4⋊C4C4×D411yescharacteristic
519C4⋊C4C4×D411yescharacteristic
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—44no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—82no—
547C42—81no—
548C42C2×D812yes—
549C42C23×C411yescharacteristic
550C42C22×D411yescharacteristic
551C42C2×#4111yescharacteristic
552C42H(5,𝔽2)11yescharacteristic
553C23—21no—
554C23—21no—
555C23—21no—
556C23—42no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—81no—
562C23—82no—
563C23—82no—
564C23—161no—
565C23#54611yescharacteristic
566Q8—21no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—82no—
572Q8C4×D812yes—
573D4—22no—
574D4—22no—
575D4—21no—
576D4—21no—
577D4—21no—
578D4—21no—
579D4—42no—
580D4—41no—
581D4—41no—
582D4—41no—
583D4—41no—
584D4—41no—
585D4—41no—
586D4—41no—
587D4—41no—
588D4—82no—
589D4—81no—
590D4—81no—
591D4—82no—
592D4—82no—
593D4—82no—
594D4—81no—
595D4—81no—
596D4—81no—
597D4—81no—
598D4—161no—
599D4C4×D812yes—
600D4#44412yescomplemented
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—81no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—88no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4C2×C4×D411yescharacteristic
644C2×C4C2×C4×D411yescharacteristic
645C2×C4#51811yescharacteristic
646C2×C4D4211yescharacteristic
647C2×C4#54611yescharacteristic
648C2×C4#55111yescharacteristic
649C2×C4C22×D811yescharacteristic
650C2×C4C2×#13511yescharacteristic
651C2×C4#57611yescharacteristic
652C2×C4#57711yescharacteristic
653C8—22no—
654C8—41no—
655C8—41no—
656C8—41no—
657C8—41no—
658C8—41no—
659C8—41no—
660C8—41no—
661C8—41no—
662C8—41no—
663C8—41no—
664C8—41no—
665C8—81no—
666C8—81no—
667C8—81no—
668C22—21no—
669C22—21no—
670C22—22no—
671C22—22no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—82no—
680C22—82no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22—164no—
688C22#235811yescharacteristic
689C22#329411yescharacteristic
690C22#329711yescharacteristic
691C4—22no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—42no—
696C4—41no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—81no—
701C4—82no—
702C4—81no—
703C4—81no—
704C4C2×C4×D811yescharacteristic
705C4#294611yescharacteristic
706C4C2×#44411yescharacteristic
707C4#295611yescharacteristic
708C4D4×D811yescharacteristic
709C4#328211yescharacteristic
710C2—21no—
711C2—21no—
712C2—42no—
713C2—42no—
714C2—81no—
715C2—161no—
716C2—161no—
717C2#2287011yescharacteristic
718C2#2288611yescharacteristic
719C2#2289811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#18516C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23016C2semidirect
2. D4×#104C2semidirect
4. #17994C2semidirect
5. #17507C2semidirect
6. #9847C2semidirect
7. #9590C2semidirect
8. #9581C2semidirect
9. #9233C2semidirect
10. #9208C2semidirect
11. #9053C2semidirect
12. #8906C2semidirect
13. #8902C2semidirect
14. #8191C2semidirect
15. #8174C2semidirect
22. #2935C22semidirect
29. #1962C22semidirect
30. #1950C22semidirect
38. #1767C22semidirect
40. #1744C22semidirect
41. #1649C4semidirect
42. #1647C4semidirect
43. #1641C4semidirect
44. #1639C4semidirect
45. #1627C4semidirect
46. #1625C4semidirect
47. #1621C4semidirect
48. #1619C4semidirect
53. #1571C22semidirect
54. #1567C22semidirect
56. #1502C22semidirect
59. #1492C22semidirect
68. C8×M16C22semidirect
69. #1303C22semidirect
70. #1301C22semidirect
71. #1279C22semidirect
72. #1279C22semidirect
73. #1278C22semidirect
74. #1276C22semidirect
99. #500C2×C4semidirect
100. #493C2×C4semidirect
101. #486C2×C4semidirect
102. #486C2×C4semidirect
109. #462C2×C4semidirect
110. #461C2×C4semidirect
111. #460C2×C4semidirect
114. #459C2×C4semidirect
173. #332C2×C4semidirect
174. #332C2×C4semidirect
175. #331C2×C4semidirect
176. #331C2×C4semidirect
178. Q8⋊C8C2×C4semidirect
179. Q8⋊C8C2×C4semidirect
192. D4⋊C8C2×C4semidirect
193. D4⋊C8C2×C4semidirect
348. #104D8semidirect
600. D4#444semidirect

Maximal subgroups

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

subgroupindex
#230162
D4×#1042
#185162
#179942
#175072
#98472
#95902
#95812
#92332
#92082
#90532
#89062
#89022
#81912
#81742