Finite Group Databasebeta

← #354244

Subgroups — #354244

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#137C211yesmaximal, characteristic
2C4×#520C211yesmaximal, characteristic
3C2×#1882C212yesmaximal
4C4×#419C211yesmaximal, characteristic, complemented
5C2×#1759C211yesmaximal, characteristic
6#13786C211yesmaximal, characteristic, complemented
7#10957C212yesmaximal, complemented
8#10899C212yesmaximal, complemented
9#10781C212yesmaximal
10C4×#343C211yesmaximal, characteristic, complemented
11#10701C211yesmaximal, characteristic, complemented
12C22×#137C2211yescharacteristic
13C2×#520C2212yes—
14C2×C4×#41C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C4×#137—24no—
17C4×#137C418yescomplemented
18C2×#441—21no—
19C2×#440—21no—
20C2×C4×Q16—21no—
21C2×C4×SD16—21no—
22C2×#428C2211yescharacteristic
23C22×#104C2212yes—
24C2×#417C2211yescharacteristic
25C2×#419C2212yes—
26C2×#419C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×#391C2212yes—
30#2276—21no—
31#2275—22no—
32#2275—21no—
33#2274—22no—
34#2274C2212yes—
35C42×Q8—21no—
36C42×D4—22no—
37C42×D4—21no—
38C4×#117—21no—
39C4×#117C2211yescharacteristic
40C2×C4×C4⋊C4—21no—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×#376C2211yescharacteristic
43#1882C4116yescomplemented
44#1763—21no—
45#1763—21no—
46C4×#103—22no—
47#1759—24no—
48C2×#343C2212yes—
49C2×#343C2212yes—
50#1736C2212yes—
51C2×#339C2212yes—
52C2×#339C2212yes—
53C2×#339C2212yes—
54#1436—22no—
55C4⋊#95C2212yes—
56#1434—22no—
57C22×#41C2311yescharacteristic
58C23×Q8C2311yescharacteristic
59C2×#137—22no—
60C2×#137—24no—
61C2×#137C2×C418yes—
62C2×#137C2×C414yes—
63C22×Q16—21no—
64C22×SD16—21no—
65C22×M16C2311yescharacteristic
66C22×M16C2312yes—
67C22×M16C2311yescharacteristic
68#520—24no—
69#520—24no—
70#520—24no—
71#520C2×C418yes—
72C4×#41—22no—
73C4×#41—22no—
74C4×#41C2×C414yes—
75C4×#41D414yes—
76C2×C4×Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—22no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×Q8—22no—
84C2×C4×Q8C2×C414yescomplemented
85C2×C4×Q8D411yescharacteristic
86C2×C4×Q8D411yescharacteristic
87C2×C4×D4—22no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4—24no—
91C2×C4×D4D411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×#117—22no—
94C2×#117—21no—
95C2×#117C2311yescharacteristic
96C2×#117C2312yes—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—21no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C42—21no—
102C22×C42—21no—
103C22×C42—22no—
104C22×C42D412yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#441—28no—
108#440—28no—
109C4×Q16—28no—
110C4×SD16—28no—
111#428—22no—
112#428C2×C414yescomplemented
113C2×#107—21no—
114C2×#106—21no—
115C2×#104—24no—
116C2×#104—24no—
117C2×#104C2×C418yes—
118#419—22no—
119#419—24no—
120#419C2×C418yescomplemented
121#419C2×C414yescomplemented
122#417—22no—
123#417C2×C414yes—
124C2×#103—21no—
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—22no—
128C2×#103—22no—
129C2×#102—21no—
130C4×M16—22no—
131C4×M16—22no—
132C4×M16—22no—
133C4×M16C2×C414yescomplemented
134C2×#97—21no—
135C2×#97—21no—
136C2×C4×C8—21no—
137C2×C4×C8—21no—
138#391—24no—
139#391C2×C418yescomplemented
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4D411yescharacteristic
153C4×C4⋊C4D411yescharacteristic
154C4×C4⋊C4D412yes—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—24no—
162#376—22no—
163#376—21no—
164#376—22no—
165#376—22no—
166C2×#95—22no—
167C2×#95—24no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—24no—
171C2×#95D412yes—
172C43—22no—
173C43—21no—
174#343—24no—
175#343C2×C418yescomplemented
176#340—22no—
177#340—22no—
178#339—24no—
179#339—24no—
180#339C2×C418yescomplemented
181C2×#41—21no—
182C2×#41—21no—
183C2×#41—22no—
184C2×#41—22no—
185C2×#41C22×C412yes—
186C2×#41C22×C414yes—
187C2×#41C2×D412yes—
188C2×#41#4114yes—
189C22×Q8—21no—
190C22×Q8—22no—
191C22×Q8—21no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—21no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8C22×C412yes—
198C22×Q8C22×C414yes—
199C22×Q8C2×D411yescharacteristic
200C22×D4—21no—
201C22×D4C2×D411yescharacteristic
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—24no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—22no—
209C23×C4C2×D414yes—
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2411yescharacteristic
213#137—28no—
214#137—216no—
215#137C42116yescomplemented
216C2×Q16—24no—
217C2×Q16—28no—
218C2×SD16—24no—
219C2×SD16—28no—
220C2×M16—21no—
221C2×M16—22no—
222C2×M16—21no—
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—22no—
229C2×M16—22no—
230C2×M16C22×C414yes—
231C2×M16C22×C412yes—
232C2×M16C22×C414yes—
233C2×M16C22×C412yes—
234C22×C8—21no—
235C22×C8—22no—
236C22×C8—21no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8C22⋊C414yes—
250C4×Q8C22⋊C414yes—
251C4×D4—216no—
252C4×D4—28no—
253C4×D4—28no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—216no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4C22⋊C414yes—
262C4×D4C22⋊C414yes—
263#117—28no—
264#117—21no—
265#117—21no—
266#117—22no—
267#117—22no—
268#117—28no—
269#117—24no—
270#117—24no—
271#117C22×C412yes—
272#117C22×C414yes—
273#117C2×D412yes—
274#117#4114yes—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C22×C412yes—
294C2×C4⋊C4C22×C414yes—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4#4114yes—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C42—24no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—21no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—41no—
326C2×C42—41no—
327C2×C42—42no—
328C2×C42—84no—
329C2×C42—82no—
330C2×C42C22×C414yes—
331C2×C42C22×C412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42#4112yes—
338C2×C42#4112yes—
339#107—24no—
340#106—24no—
341#104—216no—
342#103—24no—
343#103—24no—
344#103—24no—
345#103—28no—
346#103—28no—
347#102—24no—
348#97—24no—
349#97—42no—
350C4×C8—24no—
351C4×C8—42no—
352#95—24no—
353#95—24no—
354#95—216no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—42no—
361#95—42no—
362#95—84no—
363C24—21no—
364C24C22×D411yescharacteristic
365#41—22no—
366#41—24no—
367#41—22no—
368#41—24no—
369#41—24no—
370#41—24no—
371#41—24no—
372#41C2×C4214yes—
373#41#11714yes—
374#41C4×D418yes—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—42no—
390C2×Q8—42no—
391C2×Q8—42no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8C2×C4214yescomplemented
395C2×Q8C2×C22⋊C412yes—
396C2×Q8C4×D414yes—
397C2×D4—22no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—22no—
403C2×D4C2×C22⋊C412yes—
404C2×D4C4×D414yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C4×D418yes—
447C22×C4C4×D412yes—
448C22×C4C4×D412yes—
449C22×C4#12011yescharacteristic
450C22×C4#12011yescharacteristic
451C22×C4#12011yescharacteristic
452C22×C4#12011yescharacteristic
453C22×C4#12112yes—
454C22×C4#12112yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C2×#4112yes—
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4C2×#4111yescharacteristic
464C22×C4C2×#4112yes—
465Q16—216no—
466SD16—216no—
467M16—22no—
468M16—24no—
469M16—24no—
470M16—22no—
471M16—22no—
472M16—24no—
473M16—42no—
474M16—42no—
475M16—42no—
476M16C2×C4214yescomplemented
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—24no—
481C2×C8—42no—
482C2×C8—42no—
483C2×C8—42no—
484C4⋊C4—216no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4C2×C22⋊C412yes—
502C4⋊C4C2×C22⋊C412yes—
503C4⋊C4C4×D414yes—
504C4⋊C4C4×D414yes—
505C22⋊C4—216no—
506C22⋊C4—22no—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—24no—
512C42—216no—
513C42—22no—
514C42—24no—
515C42—22no—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—24no—
520C42—24no—
521C42—42no—
522C42—44no—
523C42—42no—
524C42—42no—
525C42—42no—
526C42—44no—
527C42—44no—
528C42—82no—
529C42—82no—
530C42—84no—
531C42—84no—
532C42C2×C22⋊C412yes—
533C42C2×C22⋊C412yes—
534C42C4×D412yes—
535C42C4×D412yes—
536C42#12112yes—
537C42#12112yes—
538C42#12312yes—
539C42#12312yes—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23C2×C4×D411yescharacteristic
549C23C2×C4×D412yes—
550C23C2×#12011yescharacteristic
551C23C2×#12111yescharacteristic
552C23#52512yes—
553Q8—22no—
554Q8—24no—
555Q8—24no—
556Q8—24no—
557Q8—42no—
558Q8—42no—
559Q8—44no—
560Q8—44no—
561Q8C4×C22⋊C414yes—
562D4—24no—
563D4—24no—
564D4—28no—
565D4—22no—
566D4—24no—
567D4C4×C22⋊C414yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—816no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4#38612yes—
620C2×C4#38612yes—
621C2×C4#38612yes—
622C2×C4#38612yes—
623C2×C4C22×C4211yescharacteristic
624C2×C4C22×C22⋊C411yescharacteristic
625C2×C4C2×#11711yescharacteristic
626C2×C4C2×C4×D412yes—
627C2×C4C2×C4×D412yes—
628C2×C4C2×C4×D411yescharacteristic
629C2×C4C2×C4×D412yes—
630C2×C4C4×#4111yescharacteristic
631C2×C4C4×#4112yes—
632C2×C4C4×#4111yescharacteristic
633C2×C4C4×#4112yes—
634C2×C4C2×#12111yescharacteristic
635C2×C4C2×#12311yescharacteristic
636C2×C4#52512yes—
637C2×C4#52512yes—
638C2×C4#52912yes—
639C2×C4#52912yes—
640C2×C4#53212yes—
641C8—24no—
642C8—42no—
643C8—44no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22C42×D411yescharacteristic
658C22C2×#38611yescharacteristic
659C22#229011yescharacteristic
660C22C4×#12012yes—
661C22C4×#12112yes—
662C22#301911yescharacteristic
663C22#302011yescharacteristic
664C22#302212yes—
665C4—24no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—42no—
672C4—42no—
673C4—48no—
674C4—44no—
675C4—44no—
676C4—84no—
677C4—82no—
678C4C2×C4×C22⋊C411yescharacteristic
679C4C42×D411yescharacteristic
680C4#229011yescharacteristic
681C4#229711yescharacteristic
682C4C4×#12112yes—
683C4C4×#12312yes—
684C2—22no—
685C2—21no—
686C2—21no—
687C2—44no—
688C2C4×#38611yescharacteristic
689C2#2094311yescharacteristic
690C2#2094411yescharacteristic
691C2#2147412yes—
692C2#2147612yes—
subgroupG/H
center, socleC23C2×#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C4×#419C2semidirect
6. #13786C2semidirect
7. #10957C2semidirect
8. #10899C2semidirect
10. C4×#343C2semidirect
11. #10701C2semidirect
17. C4×#137C4semidirect
43. #1882C4semidirect
84. C2×C4×Q8C2×C4semidirect
112. #428C2×C4semidirect
120. #419C2×C4semidirect
121. #419C2×C4semidirect
133. C4×M16C2×C4semidirect
139. #391C2×C4semidirect
175. #343C2×C4semidirect
180. #339C2×C4semidirect
215. #137C42semidirect
394. C2×Q8C2×C42semidirect
476. M16C2×C42semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1372
C4×#5202
C2×#18822
C4×#4192
C2×#17592
#137862
#109572
#108992
#107812
C4×#3432
#107012