Finite Group Databasebeta

← #204619

Subgroups — #204619

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic, complemented
3C2×#1511C212yesmaximal, complemented
4C2×#1510C212yesmaximal
5#14685C211yesmaximal, characteristic, complemented
6#9682C214yesmaximal, complemented
7#9681C214yesmaximal, complemented
8C22×#124C411yescharacteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#407C2212yes—
13C2×C4×M16—21no—
14#2850—21no—
15#2850—21no—
16#2850—21no—
17#2850—21no—
18#2850C412yescomplemented
19#2850C412yescomplemented
20#2850C2212yes—
21#2850C2212yes—
22C2×#391C411yescharacteristic, complemented
23C2×#390C411yescharacteristic, complemented
24C2×#388—21no—
25C2×#388—21no—
26C2×#386—21no—
27C2×#386—21no—
28C2×#382C411yescharacteristic, complemented
29C2×#381C2212yes—
30C2×#380—21no—
31#1749—21no—
32#1749C2212yescomplemented
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yescomplemented
35#1728—21no—
36#1511—28no—
37#1510—28no—
38C2×#328C2218yescomplemented
39C2×#327C2218yes—
40C4⋊#95—21no—
41C4⋊#95C2212yescomplemented
42#1434—21no—
43#1434C2212yescomplemented
44C22⋊#95—21no—
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—44no—
55C2×#124C2×C412yes—
56C2×#124C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2312yes—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—24no—
66C2×#121—24no—
67C2×#121C2×C412yes—
68C2×#121C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C42—44no—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89C22×C42D411yescharacteristic
90C22×C42D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#407—24no—
94#407D418yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2314yes—
97C4×M16—44no—
98C2×#97—22no—
99C2×C4×C8—22no—
100#391—21no—
101#391—21no—
102#391C2×C412yes—
103#391C2×C412yes—
104#390—21no—
105#390—21no—
106#390C2×C412yes—
107#390C2×C412yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#386—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#382—21no—
125#382—21no—
126#382C2×C412yes—
127#382C2×C412yes—
128#381—24no—
129#381—24no—
130#380—44no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95C2314yes—
140#336—22no—
141#336—22no—
142#328—216no—
143#327—216no—
144C25—21no—
145C25C2×D411yescharacteristic
146C22×Q8—21no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C22×C412yes—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—24no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4C22×C412yes—
162C23×C4—24no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D414yes—
192C23×C4#4114yes—
193C23×C4C2411yescharacteristic
194C2×M16—28no—
195C2×M16—44no—
196C2×M16—44no—
197C22×C8—24no—
198#124—44no—
199#124—44no—
200#124—44no—
201#124—44no—
202C22⋊Q8—28no—
203C22⋊Q8—24no—
204C22⋊Q8—24no—
205C22⋊Q8—44no—
206C22⋊Q8—42no—
207C22⋊Q8—42no—
208#121—28no—
209#121—24no—
210#121—24no—
211#121—44no—
212#121—42no—
213#121—42no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C22⋊C8—28no—
277C22⋊C8—24no—
278C22⋊C8C2×D418yes—
279#97—44no—
280C4×C8—44no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—24no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24#12412yes—
312C24#12712yes—
313C24C22×D411yescharacteristic
314C2×Q8—24no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—88no—
321C2×D4—24no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—48no—
386C22×C4—41no—
387C22×C4—48no—
388C22×C4—41no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4#9914yes—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C4×D414yes—
419C22×C4C4×D414yes—
420C22×C4#12114yes—
421C22×C4#12114yes—
422C22×C4#12412yes—
423C22×C4#12712yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D412yes—
427C22×C4C2×#4112yes—
428M16—416no—
429C2×C8—28no—
430C2×C8—44no—
431C2×C8—44no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—88no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—88no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—24no—
480C23—21no—
481C23—21no—
482C23—24no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—24no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—84no—
506C23—84no—
507C23#39012yes—
508C23#39012yes—
509C23C2×#9912yes—
510C23#41012yes—
511C23C22×C22⋊C411yescharacteristic
512C23C2×#12411yescharacteristic
513C23C2×#12711yescharacteristic
514C23#57512yes—
515C23#57512yes—
516Q8—84no—
517Q8—84no—
518D4—84no—
519D4—84no—
520D4—88no—
521D4—84no—
522D4—84no—
523C2×C4—28no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—84no—
589C2×C4—84no—
590C2×C4#39012yes—
591C2×C4#39012yes—
592C2×C4C2×#9912yes—
593C2×C4C2×#9912yes—
594C2×C4#42112yes—
595C2×C4#42112yes—
596C2×C4C2×C4×D412yes—
597C2×C4C2×#12112yes—
598C8—48no—
599C22—21no—
600C22—22no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—22no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—48no—
617C22—48no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—168no—
630C22#196414yes—
631C22C2×#39011yescharacteristic
632C22C22×#9911yescharacteristic
633C22C2×#41011yescharacteristic
634C22#289611yescharacteristic
635C22#289611yescharacteristic
636C22C2×#42111yescharacteristic
637C22C2×#42111yescharacteristic
638C22#314012yes—
639C22#315312yes—
640C4—22no—
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4—164no—
656C2—22no—
657C2—24no—
658C2—22no—
659C2—42no—
660C2—162no—
661C2#1755411yescharacteristic
662C2#1755911yescharacteristic
663C2#1758011yescharacteristic
664C2#1758711yescharacteristic
665C2#2120411yescharacteristic
666C2C2×#196411yescharacteristic
667C2#2125211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#407C2semidirect
3. C2×#1511C2semidirect
5. #14685C2semidirect
6. #9682C2semidirect
7. #9681C2semidirect
8. C22×#124C4semidirect
18. #2850C4semidirect
19. #2850C4semidirect
22. C2×#391C4semidirect
23. C2×#390C4semidirect
28. C2×#382C4semidirect
32. #1749C22semidirect
34. C4×C22⋊C8C22semidirect
38. C2×#328C22semidirect
41. C4⋊#95C22semidirect
43. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C4×#4072
C2×#15112
C2×#15102
#146852
#96822
#96812