Finite Group Databasebeta

← #696714

Subgroups — #696714

Order 512.

3712 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic, complemented
3C2×#2487C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#16674C214yesmaximal, complemented
6#16032C214yesmaximal
7#16024C214yesmaximal, complemented
8#15236C212yesmaximal, complemented
9#15230C212yesmaximal, complemented
10#14884C214yesmaximal, complemented
11#14258C211yesmaximal, characteristic, complemented
12#13976C212yesmaximal, complemented
13#13939C211yesmaximal, characteristic, complemented
14#13879C212yesmaximal, complemented
15#13839C211yesmaximal, characteristic
16C2×#556—28no—
17C2×#556—28no—
18C2×#556—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2212yes—
21C22×#126C2212yes—
22C22×#124C2212yes—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32#2503—28no—
33#2487—28no—
34C2×#400C2214yes—
35C2×#400C2212yes—
36C2×#399C2212yes—
37#2317—28no—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#386C2212yes—
43C2×#385C2214yes—
44C2×#385C2212yes—
45C2×#384C2212yes—
46C2×#382C2212yes—
47C2×#382C2212yes—
48C2×#381C2214yes—
49C2×#380C2212yes—
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53#1446C2218yes—
54#1444C2218yes—
55#1442C2218yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2218yes—
59#1440C2212yes—
60#1440C2218yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2218yes—
65#1438C2212yes—
66#1436C2218yes—
67#1436C2218yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79#556—2128no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×#126—28no—
84C2×#126—28no—
85C2×#126—28no—
86C2×#126—28no—
87C2×#126—28no—
88C2×#126—28no—
89C2×#124—28no—
90C2×#124—28no—
91C2×#124—28no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8D418yes—
98C2×C4×Q8—24no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8—24no—
102C2×C4×Q8—28no—
103C2×C4×Q8—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4D418yes—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—24no—
110C2×#117—24no—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117—28no—
114C2×#117D418yes—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2318yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2314yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149#400—28no—
150#400—216no—
151#399—28no—
152#391—28no—
153#391—28no—
154#389—216no—
155#389—28no—
156#386—28no—
157#385—216no—
158#385—28no—
159#384—28no—
160#382—28no—
161#382—28no—
162#381—216no—
163#380—28no—
164#380—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—28no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C2×#95C2314yes—
178C2×#95C2318yes—
179C2×#95C2318yes—
180C2×#95C2318yes—
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—28no—
191C22×Q8—44no—
192C22×Q8C2×D414yes—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—42no—
197C22×D4C2×D414yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—48no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C4⋊Q8—232no—
226#126—264no—
227#126—264no—
228#124—264no—
229C22⋊Q8—232no—
230C22⋊Q8—232no—
231C4×Q8—264no—
232C4×Q8—232no—
233C4×D4—232no—
234#117—232no—
235#117—232no—
236#117—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
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—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
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—216no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—48no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4C2×D418yes—
304C2×C22⋊C4C2×D414yes—
305C2×C22⋊C4C2×D414yes—
306C2×C22⋊C4C2×D418yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42C2×D418yes—
323C2×C42C2×D414yes—
324C2×C42#4118yes—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—216no—
329#95—216no—
330#95—216no—
331#95—28no—
332#95—28no—
333#95—216no—
334#95—28no—
335#95—48no—
336#95—48no—
337#95—44no—
338#95—48no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C2511yescharacteristic
356C2×Q8—216no—
357C2×Q8—28no—
358C2×Q8—216no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4#120116yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4118yes—
449C22×C4C2×#4114yes—
450C22×C42−1+414yes—
451C22×C42−1+414yes—
452C4⋊C4—216no—
453C4⋊C4—216no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—816no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—28no—
471C22⋊C4—216no—
472C22⋊C4—28no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—88no—
477C42—216no—
478C42—216no—
479C42—216no—
480C42—28no—
481C42—216no—
482C42—48no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C2×#12014yes—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C22×#4112yes—
519C23C2×2−1+412yes—
520C23C2×2−1+412yes—
521C23#58512yes—
522C23#58514yes—
523Q8—432no—
524Q8—416no—
525D4—416no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—232no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—416no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—48no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—816no—
589C2×C4C2×#12014yes—
590C2×C4C2×#12014yes—
591C2×C4#52514yes—
592C2×C4#53614yes—
593C2×C4#53614yes—
594C2×C4#54718yes—
595C2×C4#54718yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—48no—
603C22—44no—
604C22—44no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22C22×#12011yescharacteristic
613C22C2×#52511yescharacteristic
614C22C2×#53611yescharacteristic
615C22C2×#53611yescharacteristic
616C22#344911yescharacteristic
617C22C2×#54712yes—
618C22C2×#54712yes—
619C22#348712yes—
620C22#348912yes—
621C22#348912yes—
622C22#349314yes—
623C22#350012yes—
624C22#350012yes—
625C22#350714yes—
626C22#355514yes—
627C22#356514yes—
628C4—24no—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—44no—
634C4—48no—
635C4—44no—
636C4—44no—
637C4—84no—
638C4—84no—
639C4—84no—
640C4#240618yes—
641C2—44no—
642C2—82no—
643C2C2×#240611yescharacteristic
644C2#3530511yescharacteristic
645C2#3567611yescharacteristic
646C2#3575912yes—
647C2#3787012yes—
648C2#3949012yes—
649C2#4291512yes—
650C2#4526114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2503C2semidirect
4. C2×#2317C2semidirect
5. #16674C2semidirect
7. #16024C2semidirect
8. #15236C2semidirect
9. #15230C2semidirect
10. #14884C2semidirect
11. #14258C2semidirect
12. #13976C2semidirect
13. #13939C2semidirect
14. #13879C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#25032
C2×#24872
C2×#23172
#166742
#160322
#160242
#152362
#152302
#148842
#142582
#139762
#139392
#138792
#138392