Finite Group Databasebeta

← #196125

Subgroups — #196125

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic, complemented
2C22×#440C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic, complemented
4#17448C211yesmaximal, characteristic
5#17364C211yesmaximal, characteristic
6#9094C214yesmaximal, complemented
7#9090C214yesmaximal, complemented
8#7919C211yesmaximal, characteristic, complemented
9#7861C211yesmaximal, characteristic, complemented
10C23×SD16C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13#2984—22no—
14#2984—22no—
15#2983—22no—
16#2983C2212yescomplemented
17#2982C2212yescomplemented
18Q8×M16—22no—
19Q8×M16—22no—
20C2×#440—22no—
21C2×#440—22no—
22C2×#440—22no—
23C2×#440—22no—
24C2×#440C412yes—
25C2×#440C412yes—
26C2×#440C2212yes—
27C2×#440C2212yes—
28#2920—22no—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32#2920C2211yescharacteristic
33C22×#107C411yescharacteristic
34C22×#103C411yescharacteristic
35C22×#102C411yescharacteristic
36C22×#97C2211yescharacteristic
37#1491—22no—
38#1490—22no—
39#1487—22no—
40#1486—22no—
41#1479C2212yescomplemented
42#1479C2212yescomplemented
43#1478C2214yes—
44#1466—22no—
45#1462—22no—
46#1460C2214yes—
47#1281C2214yescomplemented
48#1275C2214yescomplemented
49C23×Q8C2×C411yescharacteristic
50C23×D4C2×C411yescharacteristic
51C24×C4—21no—
52C22×SD16—22no—
53C22×SD16—22no—
54C22×SD16—28no—
55C22×SD16C2×C412yes—
56C22×SD16C2×C412yes—
57C23×C8C2×C411yescharacteristic
58C23×C8C2×C411yescharacteristic
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D412yes—
68C2×C4×Q8D412yes—
69C2×C4×Q8D412yes—
70C2×C4×Q8D412yes—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2×C412yes—
84C2×C4×D4C2312yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#446—28no—
92#446—28no—
93#446—24no—
94#446—24no—
95C8×Q8—24no—
96C8×Q8—24no—
97#440—24no—
98#440—24no—
99#440—24no—
100#440—24no—
101#440—24no—
102#440—24no—
103#440—24no—
104#440—24no—
105#432—24no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432—24no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#432D412yes—
115#431—24no—
116#431—21no—
117#431—24no—
118#431—21no—
119#431—21no—
120#431D412yes—
121#431C2312yes—
122#431C2312yes—
123#431C2312yes—
124#431C2312yes—
125C2×#107—22no—
126C2×#107—22no—
127C2×#107C2×C412yes—
128C2×#107C2×C412yes—
129#423—24no—
130#423—21no—
131#423—21no—
132#423—21no—
133#423—21no—
134C2×#103—22no—
135C2×#103—22no—
136C2×#103C2×C412yes—
137C2×#103C2×C412yes—
138C2×#102—22no—
139C2×#102—22no—
140C2×#102C2×C412yes—
141C2×#102C2×C412yes—
142#407—21no—
143#407—21no—
144#407—21no—
145#407D412yes—
146C4×M16—24no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—21no—
150C4×M16D412yes—
151C2×#97—21no—
152C2×#97—21no—
153C2×#97—21no—
154C2×#97—21no—
155C2×#97C2×C412yes—
156C2×#97C2312yes—
157Q8⋊C8—24no—
158Q8⋊C8—24no—
159D4⋊C8—28no—
160#323—28no—
161C25—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—24no—
166C22×Q8—42no—
167C22×Q8C22⋊C412yes—
168C22×Q8C22⋊C412yes—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—42no—
178C22×D4C22×C412yes—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4—41no—
187C23×C4—41no—
188C23×C4—84no—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4111yescharacteristic
192C2×SD16—216no—
193C2×SD16—216no—
194C2×SD16—24no—
195C2×SD16—24no—
196C2×SD16—24no—
197C2×SD16—24no—
198C2×SD16—24no—
199C2×SD16M1618yescomplemented
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—41no—
203C2×M16—41no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—24no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—42no—
210C22×C8C22×C412yes—
211C22×C8C22×C412yes—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8C22⋊C412yes—
229C4×Q8C22⋊C412yes—
230C4×Q8C22⋊C412yes—
231C4×Q8C22⋊C412yes—
232C4×Q8C2×D412yes—
233C4×Q8C2×D412yes—
234C4×Q8C2×D412yes—
235C4×Q8C2×D412yes—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—41no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—41no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C22⋊C412yes—
269C2×C4⋊C4C22⋊C412yes—
270C2×C4⋊C4C22×C412yes—
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42#4111yescharacteristic
298C2×C42C2411yescharacteristic
299#107—24no—
300#107—24no—
301C4⋊C8—28no—
302C4⋊C8—28no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311#103—24no—
312#103—24no—
313#102—24no—
314#102—24no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—41no—
321C22⋊C8—41no—
322C22⋊C8C2×D414yes—
323#97—28no—
324#97—22no—
325#97—22no—
326#97—22no—
327#97—22no—
328#97—22no—
329#97—22no—
330#97—41no—
331#97—41no—
332C4×C8—28no—
333C4×C8—22no—
334C4×C8—22no—
335C4×C8—22no—
336C4×C8C2×D414yes—
337C24—22no—
338C24—44no—
339C24—41no—
340C24—41no—
341C24—82no—
342C24—82no—
343C24C4×D411yescharacteristic
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—44no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8C2×C22⋊C412yes—
359C2×Q8C2×C22⋊C412yes—
360C2×Q8C2×M1614yes—
361C2×D4—24no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4C2×M1614yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—82no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—82no—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4C4×D411yescharacteristic
410C22×C4C4×D411yescharacteristic
411C22×C4#12011yescharacteristic
412C22×C4#12011yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12111yescharacteristic
416C22×C4#12111yescharacteristic
417C22×C4#12311yescharacteristic
418C22×C4#12311yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2×#4111yescharacteristic
422SD16—216no—
423SD16—216no—
424M16—44no—
425M16—44no—
426M16—82no—
427M16—82no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8C2×M1614yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4C2×C22⋊C412yes—
458C4⋊C4C2×C22⋊C412yes—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42C2×C22⋊C411yescharacteristic
472C42C2×C22⋊C411yescharacteristic
473C42C4×D411yescharacteristic
474C42C4×D411yescharacteristic
475C42#12011yescharacteristic
476C42#12011yescharacteristic
477C42#12111yescharacteristic
478C42#12111yescharacteristic
479C42#12111yescharacteristic
480C42#12111yescharacteristic
481C42#12311yescharacteristic
482C42#12311yescharacteristic
483C42C22×D411yescharacteristic
484C42C22×D411yescharacteristic
485C42C22×D411yescharacteristic
486C42C2×#4111yescharacteristic
487C23—22no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—816no—
499C23—82no—
500C23—82no—
501C23—82no—
502C23#38611yescharacteristic
503C23#38611yescharacteristic
504C23C2×C4×D411yescharacteristic
505Q8—22no—
506Q8—24no—
507Q8—22no—
508Q8—22no—
509Q8—44no—
510Q8—44no—
511Q8—42no—
512Q8#40714yes—
513D4—24no—
514D4—24no—
515D4—44no—
516D4—44no—
517D4—42no—
518D4—42no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4#38611yescharacteristic
559C2×C4#38611yescharacteristic
560C2×C4#38611yescharacteristic
561C2×C4#38611yescharacteristic
562C2×C4#38611yescharacteristic
563C2×C4#38611yescharacteristic
564C2×C4C22×C22⋊C411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12111yescharacteristic
569C2×C4C2×#12311yescharacteristic
570C2×C4C22×M1612yes—
571C2×C4#56812yes—
572C8—24no—
573C8—44no—
574C8—44no—
575C8—42no—
576C8—82no—
577C8—82no—
578C22—28no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—44no—
586C22—84no—
587C22—88no—
588C22—88no—
589C22—84no—
590C22—82no—
591C22—82no—
592C22#190312yes—
593C22#190512yes—
594C22C2×#38611yescharacteristic
595C22D4×M1612yes—
596C4—21no—
597C4—21no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—24no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4C2×#40711yescharacteristic
612C4#287911yescharacteristic
613C4D4×M1612yes—
614C2—22no—
615C2—22no—
616C2—22no—
617C2—84no—
618C2#2089811yescharacteristic
619C2C2×#190311yescharacteristic
620C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60248C2semidirect
3. #17449C2semidirect
6. #9094C2semidirect
7. #9090C2semidirect
8. #7919C2semidirect
9. #7861C2semidirect
16. #2983C22semidirect
17. #2982C22semidirect
41. #1479C22semidirect
42. #1479C22semidirect
47. #1281C22semidirect
48. #1275C22semidirect
199. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
#602482
C22×#4402
#174492
#174482
#173642
#90942
#90902
#79192
#78612