Finite Group Databasebeta

← #395915

Subgroups — #395915

Order 512.

1758 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#106C211yesmaximal, characteristic
2C2×#2066C212yesmaximal, complemented
3C2×#1892C211yesmaximal, characteristic
4C2×#1872C211yesmaximal, characteristic
5C2×#1850C211yesmaximal, characteristic
6#14139C211yesmaximal, characteristic
7#13907C211yesmaximal, characteristic
8#11231C212yesmaximal, complemented
9#11220C212yesmaximal, complemented
10#11122C212yesmaximal, complemented
11#10730C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#106—28no—
17C22×#106—24no—
18C22×#106C2212yes—
19C22×#106C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#2066—28no—
31#1892—24no—
32#1872—24no—
33#1850—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1441—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8C2311yescharacteristic
48C24×C4—21no—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—24no—
54C23×C8—24no—
55C23×C8D414yes—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8Q814yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4—42no—
76C22×C4⋊C4D414yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2312yes—
87C2×#106—232no—
88C2×#106—24no—
89C2×#106—24no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106D418yes—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—42no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107#391—24no—
108#391—42no—
109#389—24no—
110#389—42no—
111#380—22no—
112#380—22no—
113#380—44no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#340—42no—
130#336—24no—
131C25C2×D411yescharacteristic
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C2×Q812yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—216no—
173C22×C8—28no—
174C22×C8—28no—
175C22×C8—28no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—216no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8C2×D414yes—
192C22×C8C2×D414yes—
193C22×C8#4114yes—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q812yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4SD1614yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258#106—216no—
259#106—216no—
260C4⋊C8—48no—
261#103—44no—
262#103—48no—
263#103—44no—
264C22⋊C8—44no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—44no—
271#95—82no—
272#95—82no—
273#95—82no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24#12112yes—
279C24#12112yes—
280C24C22⋊Q812yes—
281C24C2×SD1612yes—
282C24#13512yes—
283C24#13712yes—
284C24#13712yes—
285C24C22×D411yescharacteristic
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—48no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—84no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4#12114yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4C22⋊Q814yes—
381C22×C4C22⋊Q814yes—
382C22×C4C22⋊Q812yes—
383C22×C4#12614yes—
384C22×C4C2×SD1614yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4112yes—
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395C2×C8—232no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8#12118yes—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—88no—
434C4⋊C4#12014yes—
435C22⋊C4—28no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23#39614yes—
460C23#46114yes—
461C23#46414yes—
462C23#46514yes—
463C23#47014yes—
464C23#47514yes—
465C23#47914yes—
466C23C2×#12111yescharacteristic
467C23C2×#12111yescharacteristic
468C23C2×C22⋊Q811yescharacteristic
469C23D4212yes—
470C23#54712yes—
471C23C22×SD1611yescharacteristic
472C23C2×#13511yescharacteristic
473C23C2×#13711yescharacteristic
474C23C2×#13711yescharacteristic
475C23#57712yes—
476C23#57712yes—
477Q8—42no—
478Q8—42no—
479Q8—44no—
480Q8—81no—
481Q8—81no—
482Q8—82no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—816no—
523C2×C4—816no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4#39614yes—
544C2×C4C2×#12011yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12112yes—
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52512yes—
551C2×C4C2×#12611yescharacteristic
552C2×C4#53411yescharacteristic
553C2×C4#53611yescharacteristic
554C2×C4#53712yes—
555C2×C4#53912yes—
556C2×C4#54612yes—
557C2×C4#54812yes—
558C2×C4D4×Q812yes—
559C2×C4#55212yes—
560C2×C4#55312yes—
561C2×C4#55412yes—
562C8—28no—
563C8—81no—
564C8—81no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—28no—
572C22—28no—
573C22—28no—
574C22—28no—
575C22#206618yescomplemented
576C22C2×#39611yescharacteristic
577C22#241011yescharacteristic
578C22#241311yescharacteristic
579C22#241612yes—
580C22#244912yes—
581C22C2×#46111yescharacteristic
582C22C2×#46411yescharacteristic
583C22#304211yescharacteristic
584C22#304311yescharacteristic
585C22C2×#46511yescharacteristic
586C22C2×#47011yescharacteristic
587C22#306012yes—
588C22C2×#47511yescharacteristic
589C22C2×#47911yescharacteristic
590C22#308412yes—
591C22#326612yes—
592C22#326612yes—
593C22D4×SD1612yes—
594C22#328212yes—
595C22#333512yes—
596C22#333712yes—
597C22#334412yes—
598C22#334412yes—
599C4—24no—
600C4—44no—
601C4—44no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C4—162no—
608C4#240911yescharacteristic
609C4#241712yes—
610C4#248112yes—
611C4#257911yescharacteristic
612C4#261811yescharacteristic
613C4#284711yescharacteristic
614C2—28no—
615C2#1472911yescharacteristic
616C2C2×#206611yescharacteristic
617C2#2361011yescharacteristic
618C2#2435711yescharacteristic
619C2#2435811yescharacteristic
620C2#2457911yescharacteristic
621C2#2458011yescharacteristic
622C2#2477612yes—
623C2#2478312yes—
624C2#2634012yes—
625C2#2634612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2066C2semidirect
8. #11231C2semidirect
9. #11220C2semidirect
10. #11122C2semidirect
575. C22#2066semidirect

Maximal subgroups

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

subgroupindex
C23×#1062
C2×#20662
C2×#18922
C2×#18722
C2×#18502
#141392
#139072
#112312
#112202
#111222
#107302