Finite Group Databasebeta

← #354306

Subgroups — #354306

Order 512.

1876 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2930C211yesmaximal, characteristic
2C22×#405C211yesmaximal, characteristic, complemented
3C2×#1897C211yesmaximal, characteristic
4#20967C211yesmaximal, characteristic
5C2×#1765C211yesmaximal, characteristic, complemented
6C2×#1764C211yesmaximal, characteristic, complemented
7#20386C211yesmaximal, characteristic, complemented
8#10930C211yesmaximal, characteristic, complemented
9#10929C211yesmaximal, characteristic, complemented
10#10883C211yesmaximal, characteristic, complemented
11#10882C211yesmaximal, characteristic, complemented
12#10723C211yesmaximal, characteristic, complemented
13#10720C211yesmaximal, characteristic, complemented
14#10707C211yesmaximal, characteristic, complemented
15#10706C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C24×C8C2211yescharacteristic
19C2×#525C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2930—24no—
22#2930C418yescomplemented
23C2×#436—22no—
24C2×#435—22no—
25C2×#424C2211yescharacteristic
26C22×#104C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#408C2211yescharacteristic
31C2×#407C2211yescharacteristic
32C2×#407C2211yescharacteristic
33C2×#405—24no—
34C2×#405—24no—
35C2×#405—24no—
36C2×#405C2211yescharacteristic
37C2×#405C2211yescharacteristic
38C2×#405C2211yescharacteristic
39C22×#97C2211yescharacteristic
40C22×#97C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42C22×C4×C8C2211yescharacteristic
43#2280C2211yescharacteristic
44#1897C418yescomplemented
45#1893C414yescomplemented
46#1892C414yescomplemented
47#1765—24no—
48#1764—24no—
49#1753—22no—
50#1753—22no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54C2×#341C2211yescharacteristic
55#1735C2211yescharacteristic
56#1735C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61C2×#339C2211yescharacteristic
62C2×#339C2211yescharacteristic
63#1729C2211yescharacteristic
64#1729C2211yescharacteristic
65C22×#41C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C2×#131—22no—
68C2×#131—22no—
69C2×#131—22no—
70C2×#131—22no—
71C2×#131—22no—
72C2×#131C2×C414yes—
73C22×M16—22no—
74C22×M16—22no—
75C22×M16—21no—
76C22×M16—22no—
77C22×M16—22no—
78C22×M16C2×C414yes—
79C22×M16D414yes—
80C22×M16D412yes—
81C22×M16D412yes—
82C22×M16C2311yescharacteristic
83C22×M16C2311yescharacteristic
84C22×M16C2311yescharacteristic
85C22×M16C2311yescharacteristic
86C23×C8—22no—
87C23×C8—22no—
88C23×C8—22no—
89C23×C8—22no—
90C23×C8—22no—
91C23×C8—22no—
92C23×C8—21no—
93C23×C8C2×C414yes—
94C23×C8C2311yescharacteristic
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97C23×C8C2311yescharacteristic
98#525—22no—
99#525—22no—
100#525C2×C414yes—
101#525C2×C414yes—
102C2×#123—21no—
103C2×C22⋊Q8C2×C412yes—
104C2×#121C2×C412yes—
105C2×#120—21no—
106C2×C4×D4—22no—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117—22no—
111C2×#117—22no—
112C2×#117C2×C414yes—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#436—216no—
121#435—216no—
122#424—22no—
123#424C2×C414yes—
124C2×C4⋊C8—22no—
125C2×#104—22no—
126C2×#104—22no—
127C2×#104—22no—
128C2×#104—22no—
129C2×#104C2×C414yes—
130C2×#104C2×C414yes—
131#416—22no—
132#416—22no—
133#416—22no—
134#416—22no—
135C2×#103C2×C412yes—
136C2×#103C2×C412yes—
137C2×#102C2×C412yes—
138C2×#102C2×C412yes—
139#408—22no—
140#408C2×C414yes—
141#407—22no—
142#407—22no—
143#407C2×C414yes—
144#407C2×C414yes—
145C2×C22⋊C8—22no—
146C2×C22⋊C8—22no—
147C2×C22⋊C8—22no—
148#405—232no—
149#405—22no—
150#405—22no—
151#405—22no—
152#405—22no—
153#405—22no—
154#405C2×C414yes—
155C2×#97—22no—
156C2×#97—22no—
157C2×#97—22no—
158C2×#97—22no—
159C2×#97—21no—
160C2×#97—22no—
161C2×#97—21no—
162C2×#97—21no—
163C2×#97D414yescomplemented
164C2×C4×C8—22no—
165C2×C4×C8—22no—
166C2×C4×C8—22no—
167C2×C4×C8—22no—
168C2×C4×C8—21no—
169C2×C4×C8—22no—
170C2×C4×C8—21no—
171C2×C4×C8—21no—
172C2×C4×C8D414yescomplemented
173#381—21no—
174#380C2×C414yes—
175#379—21no—
176C4×C22⋊C4—22no—
177#341—22no—
178#341—22no—
179#341—22no—
180#341—22no—
181#340C2×C414yes—
182#340C2×C414yes—
183#339—22no—
184#339—22no—
185#339—22no—
186#339—22no—
187#339C2×C414yes—
188#339C2×C414yes—
189#336—22no—
190#336—22no—
191C25#4111yescharacteristic
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—21no—
195C2×#41—21no—
196C2×#41—21no—
197C2×#41—21no—
198C2×#41—21no—
199C2×#41—21no—
200C2×#41—21no—
201C2×#41—21no—
202C2×#41C22×C412yes—
203C2×#41C22×C412yes—
204C22×Q8C22×C411yescharacteristic
205C22×D4—21no—
206C22×D4—22no—
207C22×D4C22×C411yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C412yes—
220C23×C4C22×C412yes—
221C23×C4C22×C412yes—
222C23×C4C2×D412yes—
223C23×C4C2×D412yes—
224C23×C4#4111yescharacteristic
225C23×C4#4112yes—
226C23×C4C2411yescharacteristic
227#131—28no—
228#131—28no—
229#131—44no—
230#131—44no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—24no—
236C2×M16—28no—
237C2×M16—28no—
238C2×M16—28no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—28no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—42no—
261C2×M16—42no—
262C2×M16C22⋊C418yes—
263C2×M16C22⋊C418yes—
264C2×M16C22×C412yes—
265C2×M16C22×C412yes—
266C2×M16C22×C412yes—
267C2×M16C22×C412yes—
268C22×C8—28no—
269C22×C8—28no—
270C22×C8—28no—
271C22×C8—28no—
272C22×C8—28no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—21no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280C22×C8—21no—
281C22×C8—21no—
282C22×C8—21no—
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—28no—
286C22×C8—28no—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—21no—
291C22×C8—24no—
292C22×C8—42no—
293C22×C8—42no—
294C22×C8C22×C412yes—
295C22×C8C22×C412yes—
296C22×C8C22×C412yes—
297C22×C8C22×C412yes—
298C22×C8C2×D412yes—
299C22×C8C2×D412yes—
300C22×C8C2×D412yes—
301C22×C8C2×D412yes—
302C22×C8C2×D412yes—
303C22×C8C2×D411yescharacteristic
304C22×C8C2×D411yescharacteristic
305C22×C8C2×D412yes—
306C22×C8C2×D412yes—
307C22×C8#4112yes—
308C22×C8#4112yes—
309C22×C8#4112yes—
310C22×C8#4112yes—
311#123—24no—
312#123—24no—
313C22⋊Q8—24no—
314C22⋊Q8C4218yes—
315#121—24no—
316#121C4218yes—
317#120—24no—
318#120—24no—
319C4×D4—28no—
320C4×D4—28no—
321#117—28no—
322#117—28no—
323#117—21no—
324#117—21no—
325#117—28no—
326#117—28no—
327#117—21no—
328#117—21no—
329#117—21no—
330#117—21no—
331#117—21no—
332#117—21no—
333#117—21no—
334#117—21no—
335#117C22×C412yes—
336#117C22×C412yes—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4C22×C412yes—
342C2×C4⋊C4C22×C412yes—
343C2×C4⋊C4C22×C412yes—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42C22×C412yes—
363C2×C42C22×C412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42#4112yes—
367C2×C42#4112yes—
368C4⋊C8—28no—
369#104—28no—
370#104—28no—
371#103—24no—
372#103—24no—
373#102—24no—
374#102—24no—
375C22⋊C8—28no—
376C22⋊C8—28no—
377C22⋊C8—28no—
378#97—28no—
379#97—24no—
380#97—28no—
381#97—42no—
382#97—42no—
383C4×C8—28no—
384C4×C8—24no—
385C4×C8—28no—
386C4×C8—42no—
387C4×C8—42no—
388#95—24no—
389C24—21no—
390C24—21no—
391C24—21no—
392C24—21no—
393C24—21no—
394C24—44no—
395C24#11712yes—
396C24C4×D412yes—
397C24C4×D412yes—
398C24C2×#4111yescharacteristic
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—41no—
408#41—41no—
409#41—41no—
410#41—41no—
411#41—41no—
412#41—41no—
413#41—41no—
414#41—41no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8C2×C4212yes—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—28no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4—84no—
434C2×D4C2×C4212yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4C2×C4214yes—
482C22×C4C2×C22⋊C414yes—
483C22×C4#11712yes—
484C22×C4C4×D414yes—
485C22×C4C4×D414yes—
486C22×C4C4×D412yes—
487C22×C4C4×D412yes—
488C22×C4C23×C411yescharacteristic
489C22×C4C23×C411yescharacteristic
490C22×C4C23×C411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4111yescharacteristic
496C22×C4C2×#4111yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499C22×C4C2×#4111yescharacteristic
500C22×C4C2×#4111yescharacteristic
501C22×C4C2×#4111yescharacteristic
502M16—216no—
503M16—216no—
504M16—22no—
505M16—22no—
506M16—22no—
507M16—22no—
508M16—22no—
509M16—22no—
510M16—22no—
511M16—22no—
512M16—41no—
513M16—41no—
514M16—41no—
515M16—41no—
516M16—41no—
517M16—41no—
518M16—41no—
519M16—41no—
520M16—48no—
521C2×C8—232no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—41no—
545C2×C8—41no—
546C2×C8—41no—
547C2×C8—41no—
548C2×C8—41no—
549C2×C8—41no—
550C2×C8—41no—
551C2×C8—41no—
552C2×C8—48no—
553C2×C8C2×C22⋊C414yes—
554C2×C8C2×C22⋊C412yes—
555C2×C8C2×C22⋊C412yes—
556C2×C8C4×D414yes—
557C2×C8C4×D414yes—
558C2×C8C4×D414yes—
559C2×C8C4×D414yes—
560C2×C8#12014yes—
561C2×C8#12112yes—
562C2×C8#12112yes—
563C2×C8#12112yes—
564C2×C8#12112yes—
565C2×C8#12314yes—
566C4⋊C4—216no—
567C4⋊C4—22no—
568C4⋊C4—28no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4—22no—
572C4⋊C4—22no—
573C4⋊C4C2×C4214yes—
574C22⋊C4—216no—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—28no—
578C22⋊C4—28no—
579C22⋊C4—41no—
580C22⋊C4—41no—
581C22⋊C4—41no—
582C22⋊C4—41no—
583C22⋊C4—84no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42C4×D414yes—
597C42C4×D414yes—
598C23—24no—
599C23—24no—
600C23—24no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—41no—
616C23—41no—
617C23C4×C22⋊C414yes—
618C23C2×#11711yescharacteristic
619C23C2×C4×D411yescharacteristic
620C23C2×C4×D411yescharacteristic
621C23#52511yescharacteristic
622C23#52511yescharacteristic
623C23#52911yescharacteristic
624C23#52911yescharacteristic
625Q8—22no—
626Q8—22no—
627Q8—41no—
628Q8—41no—
629D4—22no—
630D4—22no—
631D4—42no—
632D4—42no—
633D4—42no—
634D4—42no—
635D4—41no—
636D4—41no—
637D4—88no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4C4×C22⋊C414yes—
674C2×C4C22×C4211yescharacteristic
675C2×C4C22×C22⋊C411yescharacteristic
676C2×C4C2×C4×D411yescharacteristic
677C2×C4C2×C4×D411yescharacteristic
678C2×C4C2×C4×D411yescharacteristic
679C2×C4C2×C4×D411yescharacteristic
680C2×C4C2×C4×D411yescharacteristic
681C2×C4C2×C4×D411yescharacteristic
682C2×C4C2×C4×D411yescharacteristic
683C2×C4C2×C4×D411yescharacteristic
684C2×C4C4×#4111yescharacteristic
685C2×C4C4×#4111yescharacteristic
686C2×C4C4×#4111yescharacteristic
687C2×C4C4×#4111yescharacteristic
688C2×C4C4×#4111yescharacteristic
689C2×C4C4×#4111yescharacteristic
690C2×C4C2×#12011yescharacteristic
691C2×C4C2×#12111yescharacteristic
692C2×C4C2×#12111yescharacteristic
693C2×C4C2×#12311yescharacteristic
694C2×C4#52511yescharacteristic
695C2×C4#52511yescharacteristic
696C2×C4#52511yescharacteristic
697C2×C4#52511yescharacteristic
698C2×C4#52911yescharacteristic
699C2×C4#52911yescharacteristic
700C2×C4#53211yescharacteristic
701C2×C4#53211yescharacteristic
702C8—24no—
703C8—24no—
704C8—24no—
705C8—42no—
706C8—42no—
707C8—42no—
708C8—42no—
709C8#38614yes—
710C8#38614yes—
711C22—22no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—22no—
716C22—22no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—42no—
727C22—44no—
728C22—82no—
729C22—82no—
730C22C2×C4×C22⋊C411yescharacteristic
731C22#229011yescharacteristic
732C22#229011yescharacteristic
733C22C4×#12011yescharacteristic
734C22C4×#12111yescharacteristic
735C22C4×#12111yescharacteristic
736C22C4×#12311yescharacteristic
737C22#295714yes—
738C4—24no—
739C4—44no—
740C4—44no—
741C4—42no—
742C4—44no—
743C4—82no—
744C4—82no—
745C4C42×D411yescharacteristic
746C4C42×D411yescharacteristic
747C4C2×#38611yescharacteristic
748C4#229711yescharacteristic
749C4C4×#12011yescharacteristic
750C4C4×#12111yescharacteristic
751C4C4×#12111yescharacteristic
752C4C4×#12311yescharacteristic
753C2—21no—
754C2—21no—
755C2—21no—
756C2—21no—
757C2—44no—
758C2—82no—
759C2C4×#38611yescharacteristic
760C2#2042212yes—
761C2#2154512yes—
762C2#2154712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC2×#525C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#405C2semidirect
5. C2×#1765C2semidirect
6. C2×#1764C2semidirect
7. #20386C2semidirect
8. #10930C2semidirect
9. #10929C2semidirect
10. #10883C2semidirect
11. #10882C2semidirect
12. #10723C2semidirect
13. #10720C2semidirect
14. #10707C2semidirect
15. #10706C2semidirect
22. #2930C4semidirect
44. #1897C4semidirect
45. #1893C4semidirect
46. #1892C4semidirect
163. C2×#97D4semidirect
172. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29302
C22×#4052
C2×#18972
#209672
C2×#17652
C2×#17642
#203862
#109302
#109292
#108832
#108822
#107232
#107202
#107072
#107062