Finite Group Databasebeta

← #5955458

Subgroups — #5955458

Order 512.

5776 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
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2322C212yesmaximal, complemented
3C2×C4×#120C211yesmaximal, characteristic
4#15073C211yesmaximal, characteristic, complemented
5#15069C211yesmaximal, characteristic, complemented
6#15001C214yesmaximal, complemented
7#14918C212yesmaximal, complemented
8#14911C212yesmaximal
9#14874C211yesmaximal, characteristic, complemented
10#14873C211yesmaximal, characteristic, complemented
11#14765C214yesmaximal, complemented
12#14763C214yesmaximal, complemented
13#14723C214yesmaximal, complemented
14#13776C212yesmaximal, complemented
15C4×#379C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2212yes—
19C22×C22⋊Q8—28no—
20C22×C22⋊Q8—216no—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2214yes—
23C22×#120C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25C22×C4×D4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C22⋊C4C2212yes—
29C2×#395C2214yescomplemented
30C2×#393C2214yescomplemented
31#2322—216no—
32C4×#120—28no—
33C2×#389C2214yescomplemented
34C2×#388C2214yescomplemented
35C2×#387C2214yescomplemented
36C2×#386C2212yes—
37C2×#386C2211yescharacteristic
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#385C2211yescharacteristic
41C2×#385C2212yes—
42C2×#385C2214yescomplemented
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#379C2212yes—
47C2×#379C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2214yes—
52#1446C2214yescomplemented
53#1446C2214yescomplemented
54#1444C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2218yes—
58#1439C2218yes—
59#1439C2218yes—
60#1439C2214yescomplemented
61#1438C2218yes—
62#1436C2218yescomplemented
63#1436C2218yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2212yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×Q8—28no—
71C23×Q8—24no—
72C23×Q8—48no—
73C23×Q8C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2312yes—
76C24×C4—28no—
77C24×C4—48no—
78C24×C4D418yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×C22⋊Q8—2128no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—232no—
93C2×C22⋊Q8—232no—
94C2×#120—24no—
95C2×#120—24no—
96C2×#120—24no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—48no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4D4116yescomplemented
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—28no—
115C22×C22⋊C4—28no—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2318yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2314yes—
133#395—48no—
134#393—48no—
135#389—216no—
136#388—216no—
137#387—216no—
138#386—28no—
139#386—24no—
140#386—28no—
141#385—28no—
142#385—24no—
143#385—28no—
144#385—216no—
145#381—28no—
146#381—28no—
147#381—28no—
148#379—24no—
149#379—28no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—48no—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C25—24no—
169C25—28no—
170C25—22no—
171C25—22no—
172C25—24no—
173C25—44no—
174C25C2×D414yes—
175C25C2411yescharacteristic
176C25C2412yes—
177C25C2411yescharacteristic
178C22×Q8—232no—
179C22×Q8—216no—
180C22×Q8—28no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—48no—
184C22×Q8—464no—
185C22×Q8—432no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—44no—
195C22×D4#4114yes—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—232no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—432no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—48no—
218C23×C4—44no—
219C23×C4—464no—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4C2×D418yes—
224C23×C4C2×D418yes—
225C23×C4C2×D414yes—
226C23×C4C2×D418yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239#123—264no—
240C22⋊Q8—2128no—
241C22⋊Q8—2128no—
242#120—232no—
243C4×D4—432no—
244C4×D4—416no—
245C2×C4⋊C4—232no—
246C2×C4⋊C4—264no—
247C2×C4⋊C4—216no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4C2×D4116yes—
263C2×C4⋊C4C2×D4116yes—
264C2×C22⋊C4—264no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—216no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4#4114yes—
299C2×C22⋊C4#4118yes—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—44no—
305C2×C42—48no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—216no—
315#95—216no—
316#95—28no—
317#95—28no—
318#95—48no—
319#95—48no—
320C24—24no—
321C24—24no—
322C24—232no—
323C24—28no—
324C24—216no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—28no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—21no—
338C24—24no—
339C24—24no—
340C24—48no—
341C24—432no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—48no—
346C24—48no—
347C24C22×D414yes—
348C24C22×D412yes—
349C24C22×D414yes—
350C24C2×#4112yes—
351C24C2×#4112yes—
352C24C2×#4114yes—
353C24C2×#4112yes—
354C242−1+418yes—
355C24C2511yescharacteristic
356C2×Q8—232no—
357C2×Q8—432no—
358C2×Q8—432no—
359C2×Q8—4128no—
360C2×Q8—416no—
361C2×Q8—416no—
362C2×D4—216no—
363C2×D4—28no—
364C2×D4—216no—
365C2×D4—416no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—44no—
371C2×D4—48no—
372C22×C4—232no—
373C22×C4—28no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—24no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—432no—
430C22×C4—432no—
431C22×C4—4128no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—88no—
441C22×C4#120116yes—
442C22×C4C22×D418yes—
443C22×C4C22×D418yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)18yes—
453C4⋊C4—232no—
454C4⋊C4—232no—
455C4⋊C4—232no—
456C4⋊C4—232no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—88no—
460C22⋊C4—264no—
461C22⋊C4—216no—
462C22⋊C4—232no—
463C22⋊C4—28no—
464C22⋊C4—216no—
465C22⋊C4—432no—
466C22⋊C4—48no—
467C22⋊C4—44no—
468C22⋊C4—48no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C42—48no—
475C42—44no—
476C42—88no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—216no—
484C23—232no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—216no—
492C23—216no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—22no—
497C23—28no—
498C23—432no—
499C23—432no—
500C23—416no—
501C23—464no—
502C23—464no—
503C23—416no—
504C23—44no—
505C23—44no—
506C23—48no—
507C23—44no—
508C23—44no—
509C23—48no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—48no—
519C23C2×#12014yes—
520C23#52518yes—
521C23#52514yes—
522C23#53218yes—
523C23#53214yes—
524C23#53618yes—
525C23C23×D411yescharacteristic
526C23C23×D412yes—
527C23C22×#4111yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4112yes—
530C23C2×H(5,𝔽2)14yes—
531C23C2×2−1+414yes—
532Q8—464no—
533D4—432no—
534D4—416no—
535C2×C4—232no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—216no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—216no—
554C2×C4—28no—
555C2×C4—416no—
556C2×C4—432no—
557C2×C4—432no—
558C2×C4—432no—
559C2×C4—432no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—48no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—48no—
590C2×C4—88no—
591C2×C4C2×#12014yes—
592C2×C4C2×#12014yes—
593C2×C4#53418yes—
594C22—28no—
595C22—28no—
596C22—216no—
597C22—216no—
598C22—24no—
599C22—24no—
600C22—28no—
601C22—28no—
602C22—28no—
603C22—464no—
604C22—44no—
605C22—44no—
606C22—416no—
607C22—416no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—48no—
612C22—42no—
613C22—82no—
614C22—84no—
615C22—81no—
616C22—81no—
617C22#238818yes—
618C22C22×#12011yescharacteristic
619C22C2×#52511yescharacteristic
620C22C2×#52512yes—
621C22C2×#53211yescharacteristic
622C22C2×#53212yes—
623C22C2×#53412yes—
624C22C2×#53612yes—
625C22#348414yes—
626C22#348614yes—
627C22#352014yes—
628C22#352014yes—
629C22#352018yes—
630C4—28no—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—48no—
635C4—48no—
636C4—48no—
637C4—42no—
638C4—82no—
639C4—84no—
640C4—81no—
641C4—81no—
642C2—28no—
643C2—48no—
644C2—82no—
645C2C2×#238811yescharacteristic
646C2#3498011yescharacteristic
647C2#3508412yes—
648C2#3508512yes—
649C2#3524211yescharacteristic
650C2#3535714yes—
651C2#3539714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
2. C2×#2322C2semidirect
4. #15073C2semidirect
5. #15069C2semidirect
6. #15001C2semidirect
7. #14918C2semidirect
9. #14874C2semidirect
10. #14873C2semidirect
11. #14765C2semidirect
12. #14763C2semidirect
13. #14723C2semidirect
14. #13776C2semidirect
15. C4×#379C2semidirect
29. C2×#395C22semidirect
30. C2×#393C22semidirect
33. C2×#389C22semidirect
34. C2×#388C22semidirect
35. C2×#387C22semidirect
42. C2×#385C22semidirect
52. #1446C22semidirect
53. #1446C22semidirect
54. #1444C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
60. #1439C22semidirect
62. #1436C22semidirect
63. #1436C22semidirect
105. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#23222
C2×C4×#1202
#150732
#150692
#150012
#149182
#149112
#148742
#148732
#147652
#147632
#147232
#137762
C4×#3792