Finite Group Databasebeta

← #497413

Subgroups — #497413

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3280C211yesmaximal, characteristic
2C2×#2168C211yesmaximal, characteristic, complemented
3C2×#2139C211yesmaximal, characteristic, complemented
4C22×#357C211yesmaximal, characteristic, complemented
5D4×#107C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic, complemented
7#13205C211yesmaximal, characteristic, complemented
8#13061C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic, complemented
10#12993C211yesmaximal, characteristic, complemented
11#12991C211yesmaximal, characteristic, complemented
12#12811C211yesmaximal, characteristic, complemented
13#12733C211yesmaximal, characteristic, complemented
14#12731C211yesmaximal, characteristic, complemented
15#12637C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C23×D8C2211yescharacteristic
18C2×#546—21no—
19C22×M32C2211yescharacteristic, complemented
20C23×C16C2211yescharacteristic
21#3280—24no—
22#3280C418yescomplemented
23C2×#495C2211yescharacteristic
24C2×#467C2211yescharacteristic
25C2×#466C2211yescharacteristic
26C2×#463—21no—
27C2×#449—21no—
28C2×#450—21no—
29C2×C4×D8C2211yescharacteristic
30C2×C8×D4C2211yescharacteristic
31C22×#107C2211yescharacteristic
32C22×#107C2211yescharacteristic, complemented
33D4×C4⋊C4—21no—
34#2168—24no—
35C2×C4⋊C16C2211yescharacteristic, complemented
36#2139—24no—
37C2×#358C2211yescharacteristic, complemented
38C2×#358C2211yescharacteristic, complemented
39C2×#358C2211yescharacteristic, complemented
40C2×#358C2211yescharacteristic, complemented
41C2×#357—24no—
42C2×#357C2211yescharacteristic
43C2×#357C2211yescharacteristic
44C2×#357C2211yescharacteristic
45C2×#357C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×#357C2211yescharacteristic
48C2×#357C2211yescharacteristic
49C2×#357C2211yescharacteristic
50C2×C22⋊C16C2211yescharacteristic
51C2×C22⋊C16C2211yescharacteristic, complemented
52C2×#346C2211yescharacteristic, complemented
53#1947—21no—
54#1852—21no—
55#1852—21no—
56#1845C2211yescharacteristic, complemented
57#1808C2211yescharacteristic
58#1808C2211yescharacteristic, complemented
59C4×#107C2211yescharacteristic, complemented
60#1380C2211yescharacteristic, complemented
61#1380C2211yescharacteristic, complemented
62#1380C2211yescharacteristic, complemented
63#1380C2211yescharacteristic, complemented
64C22×#41—21no—
65C23×D4—21no—
66C2×#135—22no—
67C2×#135—22no—
68C2×#135—22no—
69C2×#135—41no—
70C2×#135—41no—
71C2×#135C2×C414yescomplemented
72C22×Q16C2311yescharacteristic
73C22×SD16—21no—
74C22×SD16—21no—
75C22×D8—22no—
76C22×D8—22no—
77C22×D8—21no—
78C22×D8—22no—
79C22×D8—41no—
80C22×D8—41no—
81C22×D8C2×C414yescomplemented
82C22×D8C2311yescharacteristic
83C22×D8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86#546—28no—
87#546—44no—
88C2×#124—21no—
89C2×#123—41no—
90C2×C22⋊Q8—21no—
91C2×#121—21no—
92C2×#121—41no—
93C2×#120—41no—
94C2×C4×D4—21no—
95C2×C4×D4—41no—
96C2×C4×D4D411yescharacteristic
97C22×C4⋊C4—21no—
98C22×C4⋊C4—21no—
99C22×C22⋊C4—41no—
100C2×M32—22no—
101C2×M32—22no—
102C2×M32—41no—
103C22×C16—22no—
104C22×C16—22no—
105C22×C16—41no—
106C22×C16C2311yescharacteristic
107C22×C16C2311yescharacteristic
108C22×C16C2311yescharacteristic
109C22×C16C2311yescharacteristic
110#495—22no—
111#495C2×C414yescomplemented
112#467—22no—
113#467C2×C414yescomplemented
114#466—22no—
115#466C2×C414yescomplemented
116#463—28no—
117#450—28no—
118#449—28no—
119C4×D8—22no—
120C4×D8C2×C414yescomplemented
121C8×D4C2×C414yes—
122C8×D4D414yes—
123C2×#107—22no—
124C2×#107—22no—
125C2×#107—21no—
126C2×#107—21no—
127C2×#107—21no—
128C2×#107—41no—
129C2×#107—41no—
130C2×#107C2311yescharacteristic
131C2×#107C2311yescharacteristic
132C2×#107C2311yescharacteristic
133C2×#107C2311yescharacteristic
134C2×#107C2311yescharacteristic
135C2×C4⋊C8D411yescharacteristic
136C2×#103—21no—
137C2×#102—21no—
138C2×#102—21no—
139C2×C22⋊C8D411yescharacteristic
140C2×C22⋊C8D411yescharacteristic
141C2×C4×C8C2311yescharacteristic
142#389—41no—
143#385—41no—
144#385—41no—
145#384—21no—
146#380—21no—
147#380—21no—
148C4×C4⋊C4—21no—
149C4⋊C16—22no—
150#358—22no—
151#358—22no—
152#358—41no—
153#358—41no—
154#357—22no—
155#357—22no—
156#357—22no—
157#357—22no—
158#357—22no—
159#357—41no—
160#357—41no—
161#357D414yescomplemented
162C22⋊C16—22no—
163C22⋊C16—22no—
164#346—22no—
165#340—21no—
166#340—21no—
167#340—21no—
168C25—41no—
169C2×#41—24no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—81no—
174C2×#41—81no—
175C22×Q8—21no—
176C22×D4—24no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—41no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—81no—
186C22×D4—81no—
187C22×D4D811yescharacteristic
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194#135—28no—
195#135—44no—
196#135—44no—
197#135—44no—
198C2×Q16—21no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16C22×C412yes—
204C2×SD16—24no—
205C2×SD16—24no—
206C2×SD16—42no—
207C2×SD16—42no—
208C2×SD16—81no—
209C2×SD16—81no—
210C2×D8—28no—
211C2×D8—24no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8—21no—
219C2×D8—21no—
220C2×D8—44no—
221C2×D8—44no—
222C2×D8—44no—
223C2×D8—42no—
224C2×D8—42no—
225C2×D8C22×C412yes—
226C2×D8C22×C412yes—
227C2×D8C2×D412yes—
228C2×D8#4112yes—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D412yes—
243C22×C8C2×D412yes—
244C22×C8C2411yescharacteristic
245#124—24no—
246#124—42no—
247#123—48no—
248C22⋊Q8—24no—
249C22⋊Q8—42no—
250#121—24no—
251#121—42no—
252#121—48no—
253#120—48no—
254C4×D4—24no—
255C4×D4—42no—
256C4×D4—84no—
257C4×D4C22⋊C414yes—
258C4×D4SD1614yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—81no—
272C2×C4⋊C4—81no—
273C2×C4⋊C4D811yescharacteristic
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4D811yescharacteristic
283C2×C22⋊C4D811yescharacteristic
284C2×C42—41no—
285C2×C42C2×D411yescharacteristic
286M32—44no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—21no—
293C2×C16—21no—
294C2×C16—21no—
295C2×C16—21no—
296C2×C16—21no—
297C2×C16—21no—
298C2×C16—44no—
299C2×C16C2×D412yes—
300#107—21no—
301#107—21no—
302#107—21no—
303#107—21no—
304#107—21no—
305#107—21no—
306#107—21no—
307#107—21no—
308#107—42no—
309#107—42no—
310#107—81no—
311#107C22×C412yes—
312#107C2×D412yes—
313C4⋊C8C22⋊C414yes—
314#103—24no—
315#102—24no—
316#102—24no—
317C22⋊C8C22⋊C414yes—
318C22⋊C8C22⋊C414yes—
319C4×C8C22×C412yes—
320C4×C8C2×D412yes—
321#95—41no—
322#95—41no—
323#95—41no—
324C24—41no—
325C24—41no—
326C24—44no—
327C24—82no—
328C24—82no—
329C24—82no—
330C24—82no—
331C24—161no—
332C24—161no—
333C24C2×D811yescharacteristic
334C24C2×D811yescharacteristic
335#41—48no—
336#41—84no—
337#41—84no—
338#41—84no—
339C2×Q8—22no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—44no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—82no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—82no—
370C2×D4—161no—
371C2×D4—161no—
372C2×D4#10214yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—82no—
395C22×C4—161no—
396C22×C4—161no—
397C22×C4—161no—
398C22×C4—161no—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×D811yescharacteristic
402C22×C4C2×D811yescharacteristic
403C22×C4C2×D811yescharacteristic
404C22×C4C2×D811yescharacteristic
405C22×C4C2×SD1612yes—
406C22×C4C2×SD1612yes—
407C22×C4C22×D411yescharacteristic
408Q16—22no—
409Q16—22no—
410Q16—41no—
411Q16—41no—
412SD16—48no—
413SD16—84no—
414D8—22no—
415D8—22no—
416D8—22no—
417D8—22no—
418D8—22no—
419D8—48no—
420D8—41no—
421D8—41no—
422D8C4×D414yescomplemented
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8C2×C22⋊C412yes—
438C2×C8C2×C22⋊C412yes—
439C2×C8C2×C22⋊C412yes—
440C2×C8C23×C411yescharacteristic
441C2×C8C22×D411yescharacteristic
442C2×C8C22×D411yescharacteristic
443C2×C8C2×#4111yescharacteristic
444C2×C8H(5,𝔽2)11yescharacteristic
445C2×C8H(5,𝔽2)11yescharacteristic
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—44no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—161no—
461C4⋊C4#10214yes—
462C22⋊C4—44no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4#10214yes—
470C22⋊C4#10214yes—
471C42—82no—
472C42C2×C22⋊C412yes—
473C42C2×SD1612yes—
474C16—22no—
475C16—41no—
476C16—41no—
477C16—41no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—88no—
493C23—81no—
494C23—81no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—81no—
500C23—81no—
501C23—88no—
502C23—81no—
503C23—81no—
504C23—164no—
505C23—164no—
506C23—164no—
507C23—164no—
508C23—164no—
509C23C2×#10212yes—
510C23C2×#10212yes—
511C23#50912yes—
512C23#51012yes—
513C23C22×D811yescharacteristic
514Q8—42no—
515Q8—42no—
516Q8—81no—
517Q8—81no—
518D4—28no—
519D4—24no—
520D4—42no—
521D4—42no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—88no—
526D4—88no—
527D4—81no—
528D4—81no—
529D4—164no—
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—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—88no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—164no—
566C2×C4—164no—
567C2×C4—164no—
568C2×C4C2×#10212yes—
569C2×C4C2×#10212yes—
570C2×C4C2×#10212yes—
571C2×C4C2×#10212yes—
572C2×C4C22×C22⋊C411yescharacteristic
573C2×C4D4211yescharacteristic
574C2×C4#54611yescharacteristic
575C2×C4C22×SD1611yescharacteristic
576C2×C4#57611yescharacteristic
577C2×C4#57711yescharacteristic
578C8—22no—
579C8—22no—
580C8—22no—
581C8C2×C4×D411yescharacteristic
582C8#51811yescharacteristic
583C8D4211yescharacteristic
584C8#54611yescharacteristic
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—1616no—
609C22—161no—
610C22—161no—
611C22—161no—
612C22—161no—
613C22#213914yescomplemented
614C22C22×#10211yescharacteristic
615C22D4×D811yescharacteristic
616C22#329411yescharacteristic
617C22C2×#50911yescharacteristic
618C22C2×#51011yescharacteristic
619C22#341511yescharacteristic
620C22#341611yescharacteristic
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—161no—
629C4—161no—
630C4—161no—
631C4D4×C22⋊C411yescharacteristic
632C4#289711yescharacteristic
633C4D4×SD1611yescharacteristic
634C4#329511yescharacteristic
635C2—24no—
636C2—42no—
637C2—84no—
638C2—162no—
639C2D4×#10211yescharacteristic
640C2C2×#213911yescharacteristic
641C2#3338111yescharacteristic
642C2#3383011yescharacteristic
643C2#3383211yescharacteristic
644C2#3387311yescharacteristic
645C2#3388111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3280C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2168C2semidirect
3. C2×#2139C2semidirect
4. C22×#357C2semidirect
5. D4×#107C2semidirect
6. #13254C2semidirect
7. #13205C2semidirect
8. #13061C2semidirect
9. #13060C2semidirect
10. #12993C2semidirect
11. #12991C2semidirect
12. #12811C2semidirect
13. #12733C2semidirect
14. #12731C2semidirect
15. #12637C2semidirect
19. C22×M32C22semidirect
22. #3280C4semidirect
32. C22×#107C22semidirect
35. C2×C4⋊C16C22semidirect
37. C2×#358C22semidirect
38. C2×#358C22semidirect
39. C2×#358C22semidirect
40. C2×#358C22semidirect
51. C2×C22⋊C16C22semidirect
52. C2×#346C22semidirect
56. #1845C22semidirect
58. #1808C22semidirect
59. C4×#107C22semidirect
60. #1380C22semidirect
61. #1380C22semidirect
62. #1380C22semidirect
63. #1380C22semidirect
71. C2×#135C2×C4semidirect
81. C22×D8C2×C4semidirect
111. #495C2×C4semidirect
113. #467C2×C4semidirect
115. #466C2×C4semidirect
120. C4×D8C2×C4semidirect
161. #357D4semidirect
422. D8C4×D4semidirect
613. C22#2139semidirect

Maximal subgroups

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

subgroupindex
C2×#32802
C2×#21682
C2×#21392
C22×#3572
D4×#1072
#132542
#132052
#130612
#130602
#129932
#129912
#128112
#127332
#127312
#126372