Finite Group Databasebeta

← #195861

Subgroups — #195861

Order 512.

1449 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2#36621C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic, complemented
4C2×#1483C212yesmaximal, complemented
5#9076C212yesmaximal, complemented
6#9075C212yesmaximal, complemented
7#9074C212yesmaximal, complemented
8#9073C212yesmaximal, complemented
9#7895C212yesmaximal, complemented
10C2×#529C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432—21no—
15C2×#431C2212yes—
16C2×#431C2212yes—
17C2×#423—21no—
18C22×C4⋊C8C2212yes—
19C2×#407—21no—
20C22×C22⋊C8—21no—
21C2×C4×M16C2212yes—
22#2855C411yescharacteristic
23#2854C411yescharacteristic
24#2853—21no—
25#2852—21no—
26#2852—21no—
27#2851C411yescharacteristic
28#2851C411yescharacteristic
29#2850C411yescharacteristic
30#2850C411yescharacteristic
31#2849—21no—
32#2848C411yescharacteristic
33#2847C411yescharacteristic
34C4×C22⋊Q8C2211yescharacteristic
35C4×C22⋊Q8C2211yescharacteristic
36C4×#123—21no—
37C4×#121C2211yescharacteristic
38C4×#121C2211yescharacteristic
39C4×#120—21no—
40#2290—21no—
41#2290—21no—
42#2283C2211yescharacteristic
43#1797—21no—
44#1797—21no—
45#1797C2212yes—
46#1797C2212yes—
47C2×#336—21no—
48#1483—28no—
49C2×Q8⋊C8C2214yescomplemented
50C2×D4⋊C8C2214yescomplemented
51#1279C2214yescomplemented
52#1279C2214yescomplemented
53C24×C4C2×C411yescharacteristic
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C22×M16—22no—
57C23×C8—22no—
58#529—24no—
59#529—24no—
60C2×#126—21no—
61C2×#125C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#120—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C2×#117—21no—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92#432—28no—
93#432—28no—
94#431—28no—
95#431—28no—
96#423—28no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—24no—
100C2×C4⋊C8—24no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8C2314yes—
103#407—28no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—44no—
109C4×M16—24no—
110C4×M16D418yescomplemented
111C2×#97—22no—
112C2×C4×C8C2314yes—
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#389C2×C411yescharacteristic
119#389C2×C411yescharacteristic
120#389C2×C411yescharacteristic
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#386—21no—
146#386—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#385—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#383—21no—
158#382C2×C411yescharacteristic
159#382C2×C411yescharacteristic
160#381—21no—
161#381—21no—
162#380C2×C411yescharacteristic
163#380C2×C411yescharacteristic
164#380C2×C411yescharacteristic
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#379—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4C2311yescharacteristic
172C4×C4⋊C4C2311yescharacteristic
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178#376—21no—
179#336—22no—
180#336—22no—
181Q8⋊C8—28no—
182D4⋊C8—28no—
183C25C22⋊C411yescharacteristic
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—21no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22⋊C411yescharacteristic
206C23×C4C22⋊C411yescharacteristic
207C23×C4C22⋊C411yescharacteristic
208C23×C4C22⋊C411yescharacteristic
209C23×C4C22⋊C411yescharacteristic
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C2×M16—28no—
220C2×M16—44no—
221C2×M16—44no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8—44no—
227#126—42no—
228#126—42no—
229#125—24no—
230#124—24no—
231#123—44no—
232#123—42no—
233#123—42no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—42no—
237#121—24no—
238#121—24no—
239#121—42no—
240#120—44no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—44no—
246C4×D4—44no—
247C4×D4—42no—
248C4×D4—42no—
249#117—44no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42C22⋊C411yescharacteristic
305C2×C42C22⋊C411yescharacteristic
306C2×C42C22⋊C411yescharacteristic
307C2×C42C22⋊C411yescharacteristic
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D414yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42#4114yes—
321C2×C42C2411yescharacteristic
322C4⋊C8—28no—
323C4⋊C8—28no—
324C4⋊C8—24no—
325C4⋊C8—24no—
326C4⋊C8—44no—
327C22⋊C8—28no—
328C22⋊C8—28no—
329C22⋊C8—44no—
330C22⋊C8—44no—
331#97—28no—
332C4×C8—24no—
333C4×C8C2×D418yes—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—41no—
361C24—41no—
362C24C2×C22⋊C411yescharacteristic
363C24C2×C22⋊C411yescharacteristic
364C24C2×C22⋊C411yescharacteristic
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—42no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—42no—
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C4×D414yes—
459C22×C4#12012yes—
460C22×C4#12114yes—
461C22×C4#12312yes—
462C22×C4#13114yes—
463C22×C4AGL(1,ℤ/8)12yes—
464C22×C4#13712yes—
465C22×C4C23×C411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468M16—416no—
469C2×C8—28no—
470C2×C8—44no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8—44no—
474C2×C8—44no—
475C2×C8—88no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—21no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—21no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—42no—
528C42—41no—
529C42—41no—
530C42—42no—
531C42C4×D414yes—
532C42#12012yes—
533C42#12114yes—
534C42#12312yes—
535C42C22×D412yes—
536C42C2×#4112yes—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—28no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23#38612yes—
568C23#40812yes—
569C23#41712yes—
570C23#42112yes—
571C23#42112yes—
572C23C22×C22⋊C411yescharacteristic
573Q8—44no—
574D4—44no—
575D4—84no—
576D4—84no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4#38612yes—
650C2×C4#38612yes—
651C2×C4#38612yes—
652C2×C4#40812yes—
653C2×C4#41712yes—
654C2×C4C2×C4×D412yes—
655C2×C4C2×#12011yescharacteristic
656C2×C4C2×#12112yes—
657C2×C4C2×#12311yescharacteristic
658C2×C4C2×#13112yes—
659C2×C4#56812yes—
660C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
661C2×C4C2×#13711yescharacteristic
662C2×C4#57512yes—
663C8—48no—
664C8—84no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—416no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22#189714yes—
694C22C2×#38611yescharacteristic
695C22C2×#40811yescharacteristic
696C22#287911yescharacteristic
697C22C2×#41711yescharacteristic
698C22#289611yescharacteristic
699C22C2×#42111yescharacteristic
700C22C2×#42111yescharacteristic
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—21no—
706C4—21no—
707C4—22no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4#293314yes—
720C4#305512yes—
721C4#305612yes—
722C2—24no—
723C2—42no—
724C2—42no—
725C2—161no—
726C2#1746511yescharacteristic
727C2#1746611yescharacteristic
728C2#1749211yescharacteristic
729C2#1749411yescharacteristic
730C2#2089711yescharacteristic
731C2#2097111yescharacteristic
732C2C2×#189711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radical#2847C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. C2×#1797C2semidirect
4. C2×#1483C2semidirect
5. #9076C2semidirect
6. #9075C2semidirect
7. #9074C2semidirect
8. #9073C2semidirect
9. #7895C2semidirect
49. C2×Q8⋊C8C22semidirect
50. C2×D4⋊C8C22semidirect
51. #1279C22semidirect
52. #1279C22semidirect
110. C4×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
#366212
C2×#17972
C2×#14832
#90762
#90752
#90742
#90732
#78952