Finite Group Databasebeta

← #392988

Subgroups — #392988

Order 512.

2420 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C23×C4×C8C211yesmaximal, characteristic, complemented
3C2×#1968C212yesmaximal, complemented
4C2×#1893C211yesmaximal, characteristic
5C2×#1892C211yesmaximal, characteristic, complemented
6C2×#1807C211yesmaximal, characteristic, complemented
7#14702C211yesmaximal, characteristic, complemented
8#14698C211yesmaximal, characteristic
9#10926C212yesmaximal, complemented
10#10925C212yesmaximal, complemented
11#10861C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#106C2212yes—
22C22×#106C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2211yescharacteristic
27C22×C4×C8—28no—
28C22×C4×C8—24no—
29C22×C4×C8C2212yes—
30C22×C4×C8C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#386—22no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36#1968—28no—
37#1893—24no—
38#1892—24no—
39#1807—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1437—22no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8C2311yescharacteristic
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4D412yes—
52C24×C4C2311yescharacteristic
53C22×SD16—24no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—24no—
57C22×SD16C2×C414yescomplemented
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—24no—
61C23×C8—24no—
62C23×C8—24no—
63C23×C8—24no—
64C23×C8C2×C414yes—
65C23×C8D414yes—
66C23×C8D414yes—
67C23×C8D414yes—
68C23×C8C2311yescharacteristic
69C23×C8C2312yes—
70C23×C8C2311yescharacteristic
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8C2×C414yes—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121C2×C414yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—24no—
93C22×C42—22no—
94C22×C42—22no—
95C22×C42—24no—
96C22×C42—24no—
97C22×C42D412yes—
98C22×C42C2312yes—
99#465—232no—
100C2×#106—22no—
101C2×#106—22no—
102C2×#106—24no—
103C2×#106—24no—
104C2×#106—24no—
105C2×#106C2×C414yescomplemented
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103C2×C414yescomplemented
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102C2×C414yescomplemented
118C2×C4×C8—232no—
119C2×C4×C8—24no—
120C2×C4×C8—24no—
121C2×C4×C8—24no—
122C2×C4×C8—24no—
123C2×C4×C8—24no—
124C2×C4×C8D418yescomplemented
125#391—24no—
126#391—24no—
127#390—24no—
128#390—24no—
129#386—48no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—21no—
138C2×#95—21no—
139#340—24no—
140#340—24no—
141#340—24no—
142C25—21no—
143C25C2×D411yescharacteristic
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C22×C412yes—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—21no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4C22×C412yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—28no—
172C23×C4—28no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4SD1618yescomplemented
183C23×C4C22×C412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D412yes—
188C23×C4C2×D414yes—
189C23×C4#4112yes—
190C23×C4#4112yes—
191C23×C4C2411yescharacteristic
192C2×SD16—28no—
193C2×SD16—216no—
194C2×SD16—44no—
195C2×SD16—48no—
196C2×SD16—44no—
197C22×C8—216no—
198C22×C8—28no—
199C22×C8—28no—
200C22×C8—28no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—216no—
208C22×C8—28no—
209C22×C8—216no—
210C22×C8—216no—
211C22×C8—28no—
212C22×C8C22⋊C4116yescomplemented
213C22×C8C22×C412yes—
214C22×C8C22×C412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D414yes—
218C22×C8C2×D414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8C2×D414yes—
222C22×C8#4114yes—
223C22×C8#4114yes—
224C22⋊Q8—28no—
225C22⋊Q8—28no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228#121—28no—
229#121—28no—
230#121—44no—
231#121—44no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4C22×C412yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C42—28no—
273C2×C42—216no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C2×D414yes—
282C2×C42#4114yes—
283#106—28no—
284#106—48no—
285#103—28no—
286#103—48no—
287#102—28no—
288#102—48no—
289C4×C8—216no—
290C4×C8—216no—
291#95—48no—
292#95—42no—
293#95—42no—
294#95—48no—
295#95—48no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—42no—
301C24—41no—
302C24—41no—
303C24—42no—
304C24—42no—
305C24—44no—
306C24—42no—
307C24C4×D412yes—
308C24#12112yes—
309C24#12712yes—
310C24C2×SD1614yes—
311C24#13514yes—
312C24C22×D411yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8—24no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—41no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—48no—
336C2×D4—416no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—88no—
352C2×D4—84no—
353C2×D4—84no—
354C22×C4—216no—
355C22×C4—232no—
356C22×C4—216no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—216no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—232no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4C2×C22⋊C414yes—
404C22×C4C4×D414yes—
405C22×C4C4×D412yes—
406C22×C4#12114yes—
407C22×C4#12112yes—
408C22×C4#12118yes—
409C22×C4#12414yes—
410C22×C4#12712yes—
411C22×C4C2×SD1618yes—
412C22×C4#13518yes—
413C22×C4C23×C411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4112yes—
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4111yescharacteristic
423SD16—416no—
424C2×C8—232no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—232no—
436C2×C8C2×C22⋊C414yes—
437C2×C8C2×C22⋊C414yes—
438C2×C8C4×D418yes—
439C2×C8#12014yes—
440C2×C8#12118yes—
441C2×C8#12118yes—
442C2×C8#12314yes—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—48no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—816no—
465C42—216no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—44no—
482C23—44no—
483C23—42no—
484C23—44no—
485C23—81no—
486C23—81no—
487C23—82no—
488C23—81no—
489C23—81no—
490C23—82no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—82no—
495C23—84no—
496C23—82no—
497C23#39014yes—
498C23C4×SD1618yes—
499C23#46518yes—
500C23#49214yes—
501C23#49514yes—
502C23C2×C4×D411yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×#12112yes—
505C23C2×#12711yescharacteristic
506C23#53212yes—
507C23C22×SD1612yes—
508C23C2×#13512yes—
509C23C2×#13514yes—
510Q8—42no—
511Q8—42no—
512Q8—44no—
513Q8—81no—
514Q8—82no—
515Q8—81no—
516D4—42no—
517D4—42no—
518D4—44no—
519D4—816no—
520D4—81no—
521D4—82no—
522D4—81no—
523C2×C4—264no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—84no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4#39014yes—
565C2×C4#465116yescomplemented
566C2×C4C22×C22⋊C411yescharacteristic
567C2×C4C2×C4×D412yes—
568C2×C4C2×C4×D411yescharacteristic
569C2×C4C4×#4112yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×#12112yes—
573C2×C4C2×#12112yes—
574C2×C4C2×#12311yescharacteristic
575C2×C4#52512yes—
576C2×C4#52511yescharacteristic
577C2×C4#52512yes—
578C2×C4#52511yescharacteristic
579C2×C4C2×#12411yescharacteristic
580C2×C4#52912yes—
581C2×C4#52912yes—
582C2×C4#53212yes—
583C2×C4#53212yes—
584C8—28no—
585C8—24no—
586C8#38618yescomplemented
587C22—216no—
588C22—28no—
589C22—24no—
590C22—28no—
591C22—28no—
592C22—28no—
593C22—24no—
594C22—88no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—162no—
599C22—162no—
600C22—162no—
601C22—162no—
602C22#196818yescomplemented
603C22C2×#39011yescharacteristic
604C22C4×#12112yes—
605C22#284711yescharacteristic
606C22#285411yescharacteristic
607C22#285612yes—
608C22C2×C4×SD1612yes—
609C22C4×#13512yes—
610C22C2×#46514yes—
611C22C2×#46512yes—
612C22#305014yes—
613C22#305012yes—
614C22C2×#49211yescharacteristic
615C22C2×#49511yescharacteristic
616C22#314712yes—
617C22#316714yes—
618C22#316814yes—
619C4—24no—
620C4—216no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4—162no—
625C4—162no—
626C4C2×#38611yescharacteristic
627C4#229711yescharacteristic
628C4C4×#12112yes—
629C4#284711yescharacteristic
630C4#285012yes—
631C4#285411yescharacteristic
632C2—28no—
633C2—162no—
634C2#1469811yescharacteristic
635C2C2×#196811yescharacteristic
636C2#2122411yescharacteristic
637C2C4×#46514yes—
638C2#3292912yes—
639C2#3312812yes—
640C2#3317012yes—
641C2#3317312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C23×C4×C8C2semidirect
3. C2×#1968C2semidirect
5. C2×#1892C2semidirect
6. C2×#1807C2semidirect
7. #14702C2semidirect
9. #10926C2semidirect
10. #10925C2semidirect
11. #10861C2semidirect
20. C2×#465C4semidirect
57. C22×SD16C2×C4semidirect
105. C2×#106C2×C4semidirect
111. C2×#103C2×C4semidirect
117. C2×#102C2×C4semidirect
124. C2×C4×C8D4semidirect
182. C23×C4SD16semidirect
212. C22×C8C22⋊C4semidirect
565. C2×C4#465semidirect
586. C8#386semidirect
602. C22#1968semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C23×C4×C82
C2×#19682
C2×#18932
C2×#18922
C2×#18072
#147022
#146982
#109262
#109252
#108612