Finite Group Databasebeta

← #395925

Subgroups — #395925

Order 512.

1814 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊C8C211yesmaximal, characteristic
2C2×#2066C212yesmaximal, complemented
3C2×#1892C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5C2×#1807C211yesmaximal, characteristic
6#14702C211yesmaximal, characteristic
7#14697C211yesmaximal, characteristic
8#11231C212yesmaximal, complemented
9#11227C212yesmaximal, complemented
10#11077C212yesmaximal, complemented
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C22×#106C2212yes—
17C22×C4⋊C8—28no—
18C22×C4⋊C8—24no—
19C22×C4⋊C8C2212yes—
20C22×C4⋊C8C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C2×#391C2212yes—
25C2×#389C2212yes—
26C2×#385—21no—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30#2066—28no—
31#1892—24no—
32#1892—24no—
33#1807—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40#1436—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C22⋊#95—21no—
45C22⋊#95—21no—
46C22⋊#95—21no—
47C23×Q8C2311yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4C2311yescharacteristic
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—24no—
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8—42no—
60C23×C8D414yes—
61C23×C8C2311yescharacteristic
62C23×C8C2312yes—
63C23×C8C2311yescharacteristic
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8Q814yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—24no—
84C22×C42—24no—
85C22×C42D412yes—
86C22×C42C2312yes—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—42no—
90C2×C4⋊C8—232no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8D418yes—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104#391—24no—
105#391—24no—
106#389—24no—
107#389—24no—
108#385—44no—
109#380—22no—
110#380—22no—
111#380—22no—
112#380—22no—
113#380—22no—
114#380D414yes—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—24no—
130#340—42no—
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—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—42no—
161C23×C4SD1614yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—28no—
176C22×C8—28no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—28no—
188C22×C8—44no—
189C22×C8—48no—
190C22×C8—48no—
191C22×C8—48no—
192C22×C8—48no—
193C22×C8—44no—
194C22×C8—44no—
195C22×C8C2×D414yes—
196C22×C8C2×D412yes—
197C22×C8C2×D414yes—
198C22×C8C2×D412yes—
199C22⋊Q8—28no—
200C22⋊Q8—28no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C42—28no—
248C2×C42—216no—
249C2×C42—28no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42C2×D414yes—
258C2×C42#4114yes—
259#106—48no—
260C4⋊C8—216no—
261C4⋊C8—216no—
262#103—48no—
263#103—44no—
264#103—44no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24#12112yes—
280C24#12112yes—
281C24C22⋊Q812yes—
282C24C2×SD1612yes—
283C24#13512yes—
284C24#13712yes—
285C24#13712yes—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C22×C4—28no—
306C22×C4—216no—
307C22×C4—28no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—216no—
330C22×C4—28no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4#12114yes—
376C22×C4#12114yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12114yes—
380C22×C4C22⋊Q814yes—
381C22×C4C22⋊Q814yes—
382C22×C4C22⋊Q812yes—
383C22×C4#12614yes—
384C22×C4C2×SD1614yes—
385C22×C4#13514yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4112yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C42−1+412yes—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—24no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—416no—
407C2×C8—416no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8#12014yes—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—48no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—42no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—88no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C42—216no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—24no—
444C23—24no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—22no—
456C23—24no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23#39614yes—
461C23#46114yes—
462C23#46414yes—
463C23#46514yes—
464C23#47014yes—
465C23#47514yes—
466C23#47914yes—
467C23C2×#12111yescharacteristic
468C23C2×#12112yes—
469C23C2×#12111yescharacteristic
470C23C2×C22⋊Q811yescharacteristic
471C23#53712yes—
472C23C22×SD1611yescharacteristic
473C23C2×#13511yescharacteristic
474C23C2×#13512yes—
475C23C2×#13711yescharacteristic
476C23C2×#13711yescharacteristic
477C23#57512yes—
478Q8—42no—
479Q8—42no—
480Q8—44no—
481Q8—81no—
482Q8—81no—
483Q8—82no—
484C2×C4—232no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—416no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4#39614yes—
544C2×C4#46518yes—
545C2×C4C2×#12011yescharacteristic
546C2×C4C2×#12111yescharacteristic
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#53912yes—
555C2×C4D4212yes—
556C2×C4#54612yes—
557C2×C4#54712yes—
558C2×C4#54812yes—
559C2×C4D4×Q812yes—
560C2×C4#55212yes—
561C2×C4#55312yes—
562C2×C4#55412yes—
563C8—24no—
564C8—42no—
565C8—44no—
566C22—28no—
567C22—28no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—28no—
575C22—28no—
576C22#206618yescomplemented
577C22C2×#39611yescharacteristic
578C22#241612yes—
579C22#241712yes—
580C22#261811yescharacteristic
581C22#284711yescharacteristic
582C22C2×#46111yescharacteristic
583C22C2×#46411yescharacteristic
584C22#303512yes—
585C22C2×#46512yes—
586C22C2×#46511yescharacteristic
587C22#305012yes—
588C22#305011yescharacteristic
589C22C2×#47011yescharacteristic
590C22#305411yescharacteristic
591C22#305612yes—
592C22#305612yes—
593C22C2×#47511yescharacteristic
594C22C2×#47911yescharacteristic
595C22#307612yes—
596C22#324612yes—
597C22#324812yes—
598C22#324812yes—
599C22#325012yes—
600C4—24no—
601C4—28no—
602C4—44no—
603C4—82no—
604C4—82no—
605C4—84no—
606C4—161no—
607C4—161no—
608C4—162no—
609C4#240911yescharacteristic
610C4#241011yescharacteristic
611C4#241311yescharacteristic
612C4#244912yes—
613C4#248112yes—
614C4#257911yescharacteristic
615C2—28no—
616C2#1472911yescharacteristic
617C2C2×#206611yescharacteristic
618C2#2360911yescharacteristic
619C2#2475112yes—
620C2#2475512yes—
621C2#2483812yes—
622C2#2484012yes—
623C2#2977311yescharacteristic
624C2#2977411yescharacteristic
625C2#3292911yescharacteristic
626C2#3293711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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. C2×#2066C2semidirect
8. #11231C2semidirect
9. #11227C2semidirect
10. #11077C2semidirect
576. C22#2066semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C82
C2×#20662
C2×#18922
C2×#18072
#147022
#146972
#112312
#112272
#110772
#107212