Finite Group Databasebeta

← #4930671

Subgroups — #4930671

Order 512.

4762 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic, complemented
3C2×#2503C211yesmaximal, characteristic, complemented
4C2×#2616C211yesmaximal, characteristic
5#14876C214yesmaximal, complemented
6#14860C214yesmaximal, complemented
7#14823C218yesmaximal, complemented
8#14139C212yesmaximal, complemented
9#14138C212yesmaximal, complemented
10#13939C211yesmaximal, characteristic, complemented
11#13924C211yesmaximal, characteristic, complemented
12#13881C212yesmaximal, complemented
13#13879C212yesmaximal, complemented
14#13851C211yesmaximal, characteristic
15C23×#41C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C2×#520—28no—
19C2×#520—28no—
20C2×#520—28no—
21C22×C4×Q8C2212yes—
22C22×C4×D4C2212yes—
23C22×C4×D4C2214yes—
24C22×#117C2212yes—
25C22×#117C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28#2616—28no—
29#2503—28no—
30#2499—28no—
31C2×#400C2214yes—
32C2×#397C2214yes—
33C2×#395C2212yes—
34C2×#394C2212yes—
35C2×#394C2214yes—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2211yescharacteristic
39C2×#387C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2211yescharacteristic
43C2×#381C2214yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#1446C2212yes—
51#1445C2214yes—
52#1442C2218yes—
53#1441C2218yes—
54#1440C2218yes—
55#1440C2214yes—
56#1439C2218yes—
57#1438C2212yes—
58#1438C2214yes—
59#1436C2218yes—
60#1436C2218yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66C22⋊#95C2218yes—
67C22⋊#95C2218yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—24no—
71C22×#41—28no—
72C22×#41—28no—
73C22×#41D418yes—
74C23×Q8C2311yescharacteristic
75C23×Q8C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2314yes—
81C24×C4C2312yes—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88#520—2128no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×D4—28no—
93C2×C4×D4—216no—
94C2×C4×D4—216no—
95C2×C4×D4—28no—
96C2×C4×D4—216no—
97C2×C4×D4—28no—
98C2×#117—28no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—28no—
102C2×#117—28no—
103C2×#117D418yes—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4D418yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#400—216no—
136#397—216no—
137#395—28no—
138#394—28no—
139#394—216no—
140#391—24no—
141#391—24no—
142#391—28no—
143#387—216no—
144#384—24no—
145#382—24no—
146#382—28no—
147#381—216no—
148#380—24no—
149#380—28no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—28no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C25C2411yescharacteristic
170C25C2412yes—
171C2×#41—232no—
172C2×#41—232no—
173C2×#41—264no—
174C2×#41—232no—
175C2×#41—264no—
176C22×Q8—22no—
177C22×Q8—22no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8C2×D414yes—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4C2×D414yes—
197C22×D4C2×D418yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—48no—
210C23×C4—48no—
211C23×C4—48no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D418yes—
216C23×C4C2×D418yes—
217C23×C4#4118yes—
218C23×C4#4118yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C22⋊Q8—416no—
232C22⋊Q8—432no—
233C4×Q8—264no—
234C4×D4—2128no—
235C4×D4—264no—
236#117—264no—
237#117—232no—
238C2×C4⋊C4—264no—
239C2×C4⋊C4—232no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4C2×D418yes—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42C2×D418yes—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—216no—
315#95—216no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—48no—
322#95—48no—
323#95—48no—
324#95—48no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24C22×D412yes—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C22×D414yes—
342C24C2×#4114yes—
343C24C2×#4114yes—
344C24C2511yescharacteristic
345#41—2256no—
346C2×Q8—216no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—216no—
351C2×Q8—44no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×D4—216no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—28no—
367C2×D4—216no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—232no—
400C22×C4—216no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—416no—
429C22×C4—816no—
430C22×C4—816no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4#120116yes—
436C22×C4#123132yes—
437C22×C4C22×D418yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C42−1+414yes—
452C4⋊C4—232no—
453C4⋊C4—28no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C22⋊C4—232no—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—416no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—84no—
471C22⋊C4—88no—
472C42—232no—
473C42—216no—
474C42—44no—
475C42—48no—
476C42—44no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C2×#12014yes—
515C23C2×#12318yes—
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C2×2−1+412yes—
522C23#58514yes—
523C23#58512yes—
524Q8—232no—
525Q8—88no—
526D4—232no—
527D4—264no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—232no—
539C2×C4—264no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—216no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—816no—
587C2×C4—832no—
588C2×C4C2×#12014yes—
589C2×C4C2×#12014yes—
590C2×C4C2×#12318yes—
591C2×C4C2×#12318yes—
592C2×C4#52518yes—
593C2×C4#52514yes—
594C2×C4#52514yes—
595C22—24no—
596C22—28no—
597C22—24no—
598C22—28no—
599C22—28no—
600C22—24no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—48no—
612C22—44no—
613C22—48no—
614C22C22×#12011yescharacteristic
615C22C22×#12312yes—
616C22C2×#52512yes—
617C22C2×#52511yescharacteristic
618C22C2×#52511yescharacteristic
619C22#348512yes—
620C22#348912yes—
621C22#349314yes—
622C22#349514yes—
623C22#349712yes—
624C22#349911yescharacteristic
625C22#350012yes—
626C22#350012yes—
627C22#350011yescharacteristic
628C22#350714yes—
629C22#351214yes—
630C4—24no—
631C4—28no—
632C4—48no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—82no—
637C4—84no—
638C4—82no—
639C4—84no—
640C4—82no—
641C4#284918yes—
642C2—44no—
643C2—48no—
644C2#3561611yescharacteristic
645C2#3561812yes—
646C2#3561912yes—
647C2#3566011yescharacteristic
648C2#3575814yes—
649C2#3575912yes—
650C2#3608012yes—
651C2C2×#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2499C2semidirect
3. C2×#2503C2semidirect
5. #14876C2semidirect
6. #14860C2semidirect
7. #14823C2semidirect
8. #14139C2semidirect
9. #14138C2semidirect
10. #13939C2semidirect
11. #13924C2semidirect
12. #13881C2semidirect
13. #13879C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#24992
C2×#25032
C2×#26162
#148762
#148602
#148232
#141392
#141382
#139392
#139242
#138812
#138792
#138512