Finite Group Databasebeta

← #617365

Subgroups — #617365

Order 512.

6138 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#525C211yesmaximal, characteristic
2C2×#2848C212yesmaximal, complemented
3C2×#2542C212yesmaximal, complemented
4C2×#2621C211yesmaximal, characteristic, complemented
5C2×#2616C212yesmaximal, complemented
6#15430C214yesmaximal, complemented
7#15384C214yesmaximal, complemented
8#15340C214yesmaximal, complemented
9#15283C214yesmaximal, complemented
10#15177C214yesmaximal, complemented
11#13905C212yesmaximal, complemented
12#13777C211yesmaximal, characteristic, complemented
13C23×#41C2211yescharacteristic
14C25×C4C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C2×#525—216no—
17C2×#525—28no—
18C22×#123C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2214yescomplemented
23C22×#121C2212yes—
24C22×#120C2212yes—
25C22×C4×D4C2212yes—
26C22×C4×D4C2212yes—
27C22×#117C2211yescharacteristic
28#2848—216no—
29#2621—28no—
30#2616—216no—
31#2542—216no—
32C2×#399C2214yes—
33C2×#399C2214yescomplemented
34C2×#398C2214yescomplemented
35C2×#397C2214yes—
36C2×#395C2214yescomplemented
37C2×#394C2218yes—
38C2×#394C2214yescomplemented
39C2×#391C2212yescomplemented
40C2×#390C2214yes—
41C2×#385C2214yes—
42C2×#385C2218yes—
43C2×#384C2214yescomplemented
44C2×#382C2211yescharacteristic, complemented
45C2×#380C2214yes—
46C2×#380C2212yes—
47C2×#380C2214yescomplemented
48C2×#379C2214yes—
49C2×C4×C4⋊C4C2212yescomplemented
50#1445C2214yescomplemented
51#1444C2218yescomplemented
52#1439C2218yes—
53#1439C2218yescomplemented
54#1438C2218yes—
55#1438C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yescomplemented
59#1436C2214yescomplemented
60C4⋊#95C2212yescomplemented
61C26C2311yescharacteristic
62C22×#41—28no—
63C22×#41—24no—
64C22×#41—28no—
65C22×#41—44no—
66C23×Q8C2312yes—
67C23×Q8C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4—28no—
73C24×C4—24no—
74C24×C4—28no—
75C24×C4—44no—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—28no—
83#525—2128no—
84C2×#123—28no—
85C2×#123—28no—
86C2×#123—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—216no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—48no—
98C2×C22⋊Q8—48no—
99C2×#121—28no—
100C2×#121—28no—
101C2×#121—28no—
102C2×#120—28no—
103C2×#120—28no—
104C2×#120—28no—
105C2×C4×D4—28no—
106C2×C4×D4—28no—
107C2×C4×D4—28no—
108C2×C4×D4—28no—
109C2×C4×D4—28no—
110C2×C4×D4D4116yes—
111C2×#117—28no—
112C2×#117D418yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2318yes—
118C22×C4⋊C4C2318yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2318yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2318yes—
132C22×C22⋊C4C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136#399—216no—
137#399—216no—
138#398—216no—
139#397—216no—
140#395—216no—
141#394—232no—
142#394—216no—
143#391—28no—
144#390—216no—
145#385—216no—
146#385—232no—
147#384—216no—
148#382—24no—
149#380—216no—
150#380—216no—
151#380D4116yescomplemented
152#379—216no—
153C4×C4⋊C4—28no—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C25—24no—
162C25—22no—
163C25—24no—
164C25—28no—
165C25—42no—
166C25—44no—
167C25C2412yes—
168C25C2411yescharacteristic
169C25C2412yes—
170C2×#41—232no—
171C2×#41—464no—
172C2×#41—416no—
173C2×#41—416no—
174C22×Q8—24no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—28no—
182C22×Q8—42no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4C2×D418yes—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—232no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—464no—
223C23×C4—416no—
224C23×C4—416no—
225C23×C4C2×D418yes—
226C23×C4C2×D418yes—
227C23×C4C2×D414yes—
228C23×C4C2×D418yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2418yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C4⋊Q8—416no—
239#123—264no—
240C22⋊Q8—264no—
241C22⋊Q8—464no—
242C22⋊Q8—464no—
243C22⋊Q8—432no—
244#121—264no—
245#120—264no—
246C4×D4—264no—
247C4×D4—264no—
248#117—232no—
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—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—216no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—48no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D4116yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—216no—
295C2×C22⋊C4—216no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—216no—
298C2×C22⋊C4—48no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4C2×D4116yes—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—48no—
311C2×C42C2×D414yes—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314#95—28no—
315#95—216no—
316#95—216no—
317#95—28no—
318#95—216no—
319#95—48no—
320#95C2×D4116yes—
321C24—24no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—216no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—44no—
348C24—48no—
349C24—48no—
350C24—416no—
351C24—48no—
352C24—44no—
353C24—48no—
354C24C22×D414yes—
355C24C22×D412yes—
356C24C22×D414yes—
357C24C22×D414yes—
358C242−1+418yes—
359C24C2511yescharacteristic
360#41—4128no—
361C2×Q8—28no—
362C2×Q8—416no—
363C2×Q8—44no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—416no—
369C2×Q8—416no—
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—28no—
373C2×D4—216no—
374C2×D4—216no—
375C2×D4—48no—
376C2×D4—416no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—44no—
382C2×D4—416no—
383C2×D4—416no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—232no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—232no—
419C22×C4—232no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—4128no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4#127116yes—
445C22×C4C22×D414yes—
446C22×C4C22×D4116yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4H(5,𝔽2)14yes—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—216no—
461C4⋊C4—232no—
462C4⋊C4—432no—
463C4⋊C4—432no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—416no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—416no—
473C22⋊C4—432no—
474C22⋊C4—432no—
475C22⋊C4—432no—
476C22⋊C4—416no—
477C22⋊C4—816no—
478C42—28no—
479C42—416no—
480C42#120116yes—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—416no—
504C23—464no—
505C23—432no—
506C23—432no—
507C23—464no—
508C23—432no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—44no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C2×#12714yes—
542C23#53618yes—
543C23C23×D411yescharacteristic
544C23C23×D412yes—
545C23C23×D412yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+414yes—
548C23#58514yes—
549Q8—416no—
550Q8—816no—
551D4—432no—
552D4—416no—
553D4—432no—
554D4—432no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—232no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—216no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—216no—
570C2×C4—216no—
571C2×C4—28no—
572C2×C4—432no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—48no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—432no—
588C2×C4—432no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4C2×#12018yes—
600C2×C4C2×#12014yes—
601C2×C4C2×#12714yes—
602C2×C4C2×#12714yes—
603C2×C4#53514yes—
604C2×C4D42116yes—
605C22—216no—
606C22—28no—
607C22—24no—
608C22—28no—
609C22—464no—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22C22×#12011yescharacteristic
631C22C22×#12711yescharacteristic
632C22C2×#53511yescharacteristic
633C22C2×#53612yes—
634C22#344912yes—
635C22C2×D4214yes—
636C22#348512yes—
637C22#348612yes—
638C22#348914yes—
639C22#348914yes—
640C22#348914yes—
641C22#353812yes—
642C22#353912yes—
643C22#354114yes—
644C4—24no—
645C4—28no—
646C4—416no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—84no—
651C4—84no—
652C4#243318yes—
653C2—48no—
654C2—44no—
655C2—84no—
656C2C2×#243311yescharacteristic
657C2#3530512yes—
658C2#3563412yes—
659C2#3737512yes—
660C2#3765114yes—
661C2#3765614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#525C2
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×#2848C2semidirect
3. C2×#2542C2semidirect
4. C2×#2621C2semidirect
5. C2×#2616C2semidirect
6. #15430C2semidirect
7. #15384C2semidirect
8. #15340C2semidirect
9. #15283C2semidirect
10. #15177C2semidirect
11. #13905C2semidirect
12. #13777C2semidirect
15. C22×C4⋊Q8C22semidirect
22. C22×C22⋊Q8C22semidirect
33. C2×#399C22semidirect
34. C2×#398C22semidirect
36. C2×#395C22semidirect
38. C2×#394C22semidirect
39. C2×#391C22semidirect
43. C2×#384C22semidirect
44. C2×#382C22semidirect
47. C2×#380C22semidirect
49. C2×C4×C4⋊C4C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
53. #1439C22semidirect
58. #1437C22semidirect
59. #1436C22semidirect
60. C4⋊#95C22semidirect
151. #380D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5252
C2×#28482
C2×#25422
C2×#26212
C2×#26162
#154302
#153842
#153402
#152832
#151772
#139052
#137772