Finite Group Databasebeta

← #4380494

Subgroups — #4380494

Order 512.

3346 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×#544C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic
4C2×#2625C211yesmaximal, characteristic
5#16364C214yesmaximal
6#16104C214yesmaximal, complemented
7#16093C214yesmaximal, complemented
8#15881C212yesmaximal, complemented
9#15850C212yesmaximal, complemented
10#15783C212yesmaximal, complemented
11#15783C212yesmaximal, complemented
12#14892C214yesmaximal
13#14660C211yesmaximal, characteristic
14#14292C211yesmaximal, characteristic
15#13923C211yesmaximal, characteristic
16C2×#544—28no—
17C2×#544—28no—
18C2×#544—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21C22×#125C2212yes—
22C22×#125C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C22×#117C2212yes—
27C22×#117C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29#2625—28no—
30#2499—28no—
31#2484—28no—
32C2×#400C2212yes—
33C2×#400C2212yes—
34C2×#398C2212yes—
35C2×#398C2212yes—
36C2×#397C2214yes—
37C2×#397C2214yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#389C2212yes—
42C2×#387C2214yes—
43C2×#387C2214yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51#1446C2212yes—
52#1446C2214yes—
53#1446C2214yes—
54#1446C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1444C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1436C2214yes—
72#1436C2212yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2212yes—
78#1436C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C23×Q8C2312yes—
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87#544—2128no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×#125—28no—
95C2×#125—28no—
96C2×#125—28no—
97C2×#125—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8D418yes—
108C2×C22⋊Q8D418yes—
109C2×#117—28no—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117—28no—
114C2×#117—28no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4D418yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C42C2312yes—
142C22×C42C2312yes—
143C22×C42C2312yes—
144C22×C42C2311yescharacteristic
145C22×C42C2311yescharacteristic
146#400—28no—
147#400—28no—
148#398—28no—
149#398—28no—
150#397—216no—
151#397—216no—
152#391—28no—
153#391—24no—
154#391—24no—
155#389—28no—
156#387—216no—
157#387—216no—
158#385—28no—
159#384—24no—
160#382—24no—
161#381—24no—
162#381—28no—
163#380—24no—
164C4×C22⋊C4—24no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C25C2411yescharacteristic
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8C2×D414yes—
205C22×Q8C2×D414yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—28no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C4⋊Q8—264no—
235C4⋊Q8—264no—
236#125—264no—
237#125—264no—
238C22⋊Q8—264no—
239C22⋊Q8—232no—
240C22⋊Q8—232no—
241#117—264no—
242#117—416no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4C2×D418yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—42no—
325C2×C42—42no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351C24—22no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C2511yescharacteristic
360C2×Q8—216no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8—82no—
370C2×Q8—82no—
371C2×Q8—88no—
372C2×Q8—88no—
373C2×Q8—84no—
374C2×Q8—84no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—84no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4#120116yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C42−1+414yes—
462C22×C42−1+418yes—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—416no—
479C4⋊C4—88no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—84no—
486C42—216no—
487C42—216no—
488C42—44no—
489C42—44no—
490C42—48no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×#12014yes—
507C23C23×D411yescharacteristic
508C23C23×D411yescharacteristic
509C23C23×D411yescharacteristic
510C23C2×2−1+414yes—
511C23C2×2−1+412yes—
512C23#58512yes—
513C23#58512yes—
514C23#58512yes—
515Q8—88no—
516Q8—88no—
517Q8—816no—
518C2×C4—232no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—88no—
592C2×C4C2×#12014yes—
593C2×C4C2×#12014yes—
594C2×C4#53618yes—
595C22—24no—
596C22—24no—
597C22—24no—
598C22—44no—
599C22—44no—
600C22—48no—
601C22—44no—
602C22—44no—
603C22C22×#12011yescharacteristic
604C22C2×#53612yes—
605C22#348512yes—
606C22#348512yes—
607C22#348511yescharacteristic
608C22#348511yescharacteristic
609C22#348612yes—
610C22#348612yes—
611C22#348912yes—
612C22#349712yes—
613C22#349911yescharacteristic
614C22#350011yescharacteristic
615C22#355514yes—
616C22#356314yes—
617C22#356514yes—
618C22#356514yes—
619C4—24no—
620C4—24no—
621C4—28no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—82no—
629C4—84no—
630C4—84no—
631C4—82no—
632C4—82no—
633C2—44no—
634C2#3524211yescharacteristic
635C2#3544611yescharacteristic
636C2#3566011yescharacteristic
637C2#4264414yes—
638C2#4721112yes—
639C2#4721612yes—
640C2#5016812yes—
641C2#5016912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #16104C2semidirect
7. #16093C2semidirect
8. #15881C2semidirect
9. #15850C2semidirect
10. #15783C2semidirect
11. #15783C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#24842
C2×#24992
C2×#26252
#163642
#161042
#160932
#158812
#158502
#157832
#148922
#146602
#142922
#139232