Finite Group Databasebeta

← #205525

Subgroups — #205525

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60782C211yesmaximal, characteristic
2#20535C211yesmaximal, characteristic, complemented
3#17803C211yesmaximal, characteristic, complemented
4#17546C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6#17495C211yesmaximal, characteristic, complemented
7#17488C211yesmaximal, characteristic, complemented
8#9754C212yesmaximal, complemented
9#9753C212yesmaximal, complemented
10#9751C212yesmaximal, complemented
11#9750C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3212—22no—
15#3211—21no—
16#3211—21no—
17#3210C412yescomplemented
18#3209C411yescharacteristic, complemented
19#3208C411yescharacteristic, complemented
20#3207C412yescomplemented
21#3206C411yescharacteristic, complemented
22#3205C411yescharacteristic, complemented
23#3147C2212yes—
24#3131C2212yes—
25#3035—22no—
26#3002—21no—
27#3002—21no—
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1771—22no—
34#1770C2212yescomplemented
35C8×C4⋊C4C2212yescomplemented
36#1749—21no—
37C4×C22⋊C8—21no—
38#1572—22no—
39#1571—21no—
40#1570—21no—
41#1564C2212yescomplemented
42#1564C2212yescomplemented
43#1509—22no—
44#1504—22no—
45#1503C2211yescharacteristic, complemented
46#1503C2211yescharacteristic, complemented
47#1502—22no—
48#1502C2212yes—
49#1502C2212yes—
50#1502C2211yescharacteristic, complemented
51#1502C2211yescharacteristic, complemented
52C2×#327—22no—
53#1496—22no—
54#1494—22no—
55#1490—22no—
56#1488—22no—
57#1481C2212yescomplemented
58#1480C2212yes—
59#1480C2212yes—
60#1480C2212yescomplemented
61C22×#41—21no—
62C2×#135—22no—
63C2×#135—22no—
64#533C2×C411yescharacteristic
65#533C2×C411yescharacteristic
66#533C2311yescharacteristic
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532—21no—
72#532—21no—
73#532C2×C412yes—
74#532C2×C411yescharacteristic
75#532C2×C411yescharacteristic
76#532C2312yes—
77#532C2311yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#127C2×C411yescharacteristic
80#529—21no—
81#529—21no—
82#529—21no—
83#529—21no—
84#529—21no—
85#529—21no—
86C2×#124—21no—
87#525—21no—
88#525—21no—
89#525—21no—
90C2×#121—21no—
91C2×#121—21no—
92C4×#41—21no—
93C4×#41—21no—
94C4×#41—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C22×C4⋊C4—21no—
98C22×C42—21no—
99C22×C42C2311yescharacteristic
100#495—22no—
101#495—22no—
102#494C2×C412yescomplemented
103#493C2×C412yescomplemented
104#492C2×C412yescomplemented
105#492C2×C412yescomplemented
106#488—22no—
107#488—22no—
108#487C2×C412yescomplemented
109#487C2×C412yescomplemented
110#486C2×C412yescomplemented
111#486C2×C412yescomplemented
112#464—22no—
113#464—22no—
114#463—22no—
115#463—22no—
116#462—22no—
117#461—22no—
118#460—22no—
119#459—22no—
120#452—22no—
121#452—22no—
122#451—21no—
123#451—21no—
124#450—21no—
125#450—21no—
126#449—22no—
127#449—22no—
128#448—21no—
129#448—21no—
130#447—21no—
131#447—21no—
132#432—21no—
133#432—21no—
134#432—21no—
135#432—21no—
136#432D412yes—
137#432D412yes—
138#431—21no—
139#431—21no—
140#431—21no—
141#431—21no—
142#431C2×C411yescharacteristic
143#431C2×C411yescharacteristic
144#431C2×C412yes—
145#431D411yescharacteristic
146#431D411yescharacteristic
147#431D412yes—
148#431C2312yes—
149#431C2311yescharacteristic
150#431C2311yescharacteristic
151#423—21no—
152C2×C4⋊C8—21no—
153C2×C4⋊C8D412yes—
154C2×C4⋊C8C2312yes—
155#416—22no—
156#416—22no—
157#407—21no—
158C2×C22⋊C8—21no—
159C2×C22⋊C8—21no—
160C2×C22⋊C8D411yescharacteristic
161C2×C22⋊C8D411yescharacteristic
162C4×M16—21no—
163C2×#97—21no—
164C2×#97—22no—
165C2×C4×C8—21no—
166C2×C4×C8—22no—
167C2×C4×C8C2312yes—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4C2311yescharacteristic
172C4×C4⋊C4C2311yescharacteristic
173C2×#95D412yes—
174#336—22no—
175#336—22no—
176#335—22no—
177#335—22no—
178#334—22no—
179#334—22no—
180#328—22no—
181#328—22no—
182#327—24no—
183#327—24no—
184#327—22no—
185#327—22no—
186Q8⋊C8—24no—
187D4⋊C8—24no—
188D4⋊C8—24no—
189D4⋊C8—24no—
190C2×#41—22no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41—22no—
195C2×#41—41no—
196C2×#41—41no—
197C2×#41—82no—
198C2×#41—82no—
199C22×Q8—21no—
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4—41no—
204C23×C4—22no—
205C23×C4—41no—
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4#4111yescharacteristic
209#135—82no—
210#135—82no—
211#135—82no—
212#135—82no—
213C2×Q16—22no—
214C2×Q16—22no—
215C2×SD16—22no—
216C2×SD16—22no—
217C2×SD16—22no—
218C2×SD16—22no—
219C2×D8—22no—
220C2×D8—22no—
221C2×M16—41no—
222C22×C8—22no—
223C22×C8—41no—
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8C22×C412yes—
230C4⋊Q8C22×C411yescharacteristic
231C4⋊Q8C22×C411yescharacteristic
232C4⋊Q8C22×C411yescharacteristic
233#127—21no—
234#127—21no—
235#127—21no—
236#127C22×C412yes—
237#127C22×C411yescharacteristic
238#126—41no—
239#126—41no—
240#125—21no—
241#125—21no—
242#125—21no—
243#124—22no—
244#124—21no—
245#124—21no—
246#124—22no—
247#124—21no—
248#124—41no—
249#124—41no—
250#123—41no—
251#123—41no—
252#123—41no—
253C22⋊Q8—21no—
254C22⋊Q8—21no—
255C22⋊Q8—21no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259#121—21no—
260#121—21no—
261#121—22no—
262#121—22no—
263#121—21no—
264#121—21no—
265#121—22no—
266#121—21no—
267#121—22no—
268#121—21no—
269#121—41no—
270#121—41no—
271#121—41no—
272#121—41no—
273#120—41no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8—21no—
279C4×Q8—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—22no—
287C4×D4—21no—
288C4×D4—22no—
289C4×D4—21no—
290C4×D4—41no—
291C4×D4—41no—
292C4×D4—41no—
293C4×D4—41no—
294C4×D4—41no—
295C4×D4—41no—
296C4×D4—41no—
297C4×D4—41no—
298C4×D4—41no—
299C4×D4—41no—
300#117—21no—
301#117—21no—
302#117—21no—
303#117—41no—
304#117—41no—
305#117—41no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—44no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4—41no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—41no—
318C2×C42—82no—
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42#4111yescharacteristic
328C2×C42#4111yescharacteristic
329C2×C42#4111yescharacteristic
330C2×C42C2411yescharacteristic
331C4⋊C8—22no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—22no—
335C4⋊C8—41no—
336C4⋊C8—41no—
337C4⋊C8C22⋊C412yes—
338C4⋊C8C22⋊C412yes—
339C4⋊C8C22⋊C412yes—
340C4⋊C8C22⋊C412yes—
341C4⋊C8C2×D414yes—
342#103—22no—
343#103—22no—
344#103—22no—
345#103—22no—
346#102—22no—
347#102—22no—
348#102—22no—
349#102—22no—
350C22⋊C8—22no—
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8—21no—
354C22⋊C8—21no—
355C22⋊C8—41no—
356C22⋊C8—41no—
357C22⋊C8C22⋊C411yescharacteristic
358C22⋊C8C22⋊C411yescharacteristic
359C22⋊C8C22⋊C411yescharacteristic
360C22⋊C8C22⋊C411yescharacteristic
361C22⋊C8C22⋊C412yes—
362C22⋊C8C22⋊C412yes—
363#97—22no—
364#97—22no—
365#97—41no—
366#97—82no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—22no—
370C4×C8—41no—
371C4×C8—82no—
372C4×C8C22×C412yes—
373C4×C8C22×C412yes—
374C4×C8C2×D412yes—
375C4×C8C2×D412yes—
376#95—22no—
377#95C2×D412yes—
378#95C2×D412yes—
379C24—41no—
380C24#12111yescharacteristic
381#41—42no—
382#41—42no—
383#41—42no—
384#41—42no—
385#41—42no—
386#41—42no—
387#41—82no—
388#41—82no—
389#41—82no—
390#41—82no—
391#41—82no—
392#41—84no—
393#41—82no—
394C2×Q8—22no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—41no—
399C2×Q8—82no—
400C2×D4—22no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—41no—
407C2×D4—42no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4—82no—
417C2×D4—82no—
418C2×D4—82no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—84no—
434C22×C4—82no—
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C4×D411yescharacteristic
438C22×C4C4×D411yescharacteristic
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4#12111yescharacteristic
442C22×C4#12111yescharacteristic
443C22×C4#12111yescharacteristic
444C22×C4#12111yescharacteristic
445C22×C4#12111yescharacteristic
446C22×C4#12411yescharacteristic
447C22×C4#12411yescharacteristic
448C22×C4#12711yescharacteristic
449C22×C4#12711yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C2×#4111yescharacteristic
453Q16—82no—
454Q16—82no—
455SD16—82no—
456SD16—82no—
457SD16—82no—
458SD16—82no—
459D8—82no—
460D8—82no—
461M16—82no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—24no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—82no—
473C2×C8C2×C22⋊C412yes—
474C2×C8C2×C22⋊C412yes—
475C2×C8C2×C22⋊C412yes—
476C2×C8C2×C22⋊C412yes—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C42—41no—
510C42—42no—
511C42—42no—
512C42—41no—
513C42—41no—
514C42—82no—
515C42C2×C22⋊C411yescharacteristic
516C42C2×C22⋊C411yescharacteristic
517C42C4×D411yescharacteristic
518C42C4×D411yescharacteristic
519C42C4×D411yescharacteristic
520C42C4×D411yescharacteristic
521C42#12111yescharacteristic
522C42#12111yescharacteristic
523C42#12411yescharacteristic
524C42#12411yescharacteristic
525C42#12711yescharacteristic
526C42#12711yescharacteristic
527C42C23×C411yescharacteristic
528C42C22×D411yescharacteristic
529C42C22×D411yescharacteristic
530C42C2×#4111yescharacteristic
531C23—22no—
532C23—21no—
533C23—21no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—82no—
538C23—82no—
539C23—82no—
540C23#39011yescharacteristic
541C23#39011yescharacteristic
542C23C2×#12111yescharacteristic
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—81no—
548Q8—82no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—44no—
553D4—42no—
554D4—82no—
555D4—81no—
556D4—82no—
557D4—82no—
558D4—82no—
559D4—84no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—81no—
564D4—82no—
565D4—81no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—88no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4#39011yescharacteristic
603C2×C4#39011yescharacteristic
604C2×C4#39011yescharacteristic
605C2×C4#39011yescharacteristic
606C2×C4#39011yescharacteristic
607C2×C4#39011yescharacteristic
608C2×C4C22×C22⋊C411yescharacteristic
609C2×C4C2×C4×D411yescharacteristic
610C2×C4C2×C4×D411yescharacteristic
611C2×C4C2×#12111yescharacteristic
612C2×C4C2×#12411yescharacteristic
613C2×C4C2×#12711yescharacteristic
614C2×C4#57612yes—
615C2×C4#57712yes—
616C8—22no—
617C8—22no—
618C8—22no—
619C8—22no—
620C8—44no—
621C8—82no—
622C8—81no—
623C8—81no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—42no—
632C22—42no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—82no—
637C22—81no—
638C22—82no—
639C22—164no—
640C22#197612yes—
641C22C2×#39011yescharacteristic
642C22#304111yescharacteristic
643C22#304311yescharacteristic
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—44no—
654C4—82no—
655C4—82no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4#289712yes—
660C4#295612yes—
661C4#304512yes—
662C2—21no—
663C2—22no—
664C2—21no—
665C2—41no—
666C2—161no—
667C2—162no—
668C2#2124111yescharacteristic
669C2C2×#197611yescharacteristic
670C2#2126311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60782C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20535C2semidirect
3. #17803C2semidirect
4. #17546C2semidirect
5. C2×#1502C2semidirect
6. #17495C2semidirect
7. #17488C2semidirect
8. #9754C2semidirect
9. #9753C2semidirect
10. #9751C2semidirect
11. #9750C2semidirect
17. #3210C4semidirect
18. #3209C4semidirect
19. #3208C4semidirect
20. #3207C4semidirect
21. #3206C4semidirect
22. #3205C4semidirect
34. #1770C22semidirect
35. C8×C4⋊C4C22semidirect
41. #1564C22semidirect
42. #1564C22semidirect
45. #1503C22semidirect
46. #1503C22semidirect
50. #1502C22semidirect
51. #1502C22semidirect
57. #1481C22semidirect
60. #1480C22semidirect
102. #494C2×C4semidirect
103. #493C2×C4semidirect
104. #492C2×C4semidirect
105. #492C2×C4semidirect
108. #487C2×C4semidirect
109. #487C2×C4semidirect
110. #486C2×C4semidirect
111. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#607822
#205352
#178032
#175462
C2×#15022
#174952
#174882
#97542
#97532
#97512
#97502