Finite Group Databasebeta

← #764239

Subgroups — #764239

Order 512.

5230 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic, complemented
2C2×#2561C212yesmaximal, complemented
3C22×#394C211yesmaximal, characteristic
4#15613C218yesmaximal, complemented
5#15275C218yesmaximal, complemented
6#14934C212yesmaximal, complemented
7#14717C212yesmaximal, complemented
8#14139C212yesmaximal, complemented
9#13869C212yesmaximal, complemented
10#13867C212yesmaximal
11#13775C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C2×#543—216no—
14C2×#543—28no—
15C22×C22⋊Q8C2212yes—
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2214yescomplemented
18C22×C22⋊Q8C2214yescomplemented
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2561—216no—
26C2×#397C2214yescomplemented
27C2×#394—28no—
28C2×#394C2218yes—
29C2×#393C2214yescomplemented
30C2×#393C2214yescomplemented
31C2×#391C2214yescomplemented
32C2×#388C2218yescomplemented
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#381C2214yes—
36C2×#379C2214yes—
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2211yescharacteristic
40#1446C2214yescomplemented
41#1442C2214yescomplemented
42#1442C2218yescomplemented
43#1441C2218yescomplemented
44#1440C2218yescomplemented
45#1439C2214yescomplemented
46#1439C2218yescomplemented
47#1438C2218yes—
48#1437C2218yescomplemented
49C4⋊#95C2214yescomplemented
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2212yes—
53C22⋊#95C2214yescomplemented
54C22⋊#95C2214yescomplemented
55C26C2311yescharacteristic
56C26C2311yescharacteristic
57C23×Q8C2312yes—
58C23×Q8C2312yes—
59C23×D4—24no—
60C23×D4—28no—
61C23×D4—44no—
62C23×D4—44no—
63C23×D4C2314yes—
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71#543—2128no—
72C2×C22⋊Q8—216no—
73C2×C22⋊Q8—28no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—216no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—216no—
80C2×C22⋊Q8—216no—
81C2×C22⋊Q8—216no—
82C2×C22⋊Q8D4116yescomplemented
83C22×C4⋊C4—24no—
84C22×C4⋊C4—44no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—216no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—44no—
99C22×C22⋊C4D418yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2314yes—
112#397—216no—
113#394—232no—
114#393—216no—
115#393—48no—
116#391—48no—
117#388—232no—
118#386—48no—
119#386—48no—
120#381—216no—
121#379—28no—
122#379—216no—
123C4×C22⋊C4—28no—
124C2×#95—24no—
125C2×#95C2314yes—
126C2×#95C2318yes—
127C2×#95C2314yes—
128C2×#95C2318yes—
129C2×#95C2318yes—
130C2×#95C2318yes—
131C2×#95C2314yes—
132C2×#95C2318yes—
133C2×#95C2318yes—
134C2×#95C2314yes—
135C25—24no—
136C25—24no—
137C25—22no—
138C25—24no—
139C25—24no—
140C25—24no—
141C25—24no—
142C25—42no—
143C25—42no—
144C25—42no—
145C25—42no—
146C25—42no—
147C25—42no—
148C25—42no—
149C25C2×D414yes—
150C25#4118yes—
151C25C2412yes—
152C25C2412yes—
153C25C2411yescharacteristic
154C22×Q8—28no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8C2×D418yescomplemented
163C22×D4—28no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—416no—
171C22×D4—464no—
172C22×D4—416no—
173C22×D4—416no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2×D418yes—
201C23×C4C2×D414yes—
202C23×C4C2418yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C22⋊Q8—2128no—
211C22⋊Q8—2128no—
212C22⋊Q8—264no—
213C22⋊Q8—264no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—416no—
227C2×C4⋊C4C2×D4116yescomplemented
228C2×C4⋊C4C2×D418yescomplemented
229C2×C22⋊C4—264no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—232no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—216no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—216no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—416no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4C2×D4116yes—
266C2×C22⋊C4C2×D4116yes—
267C2×C42—28no—
268C2×C42—44no—
269C2×C42—44no—
270#95—28no—
271#95—28no—
272#95—216no—
273#95—216no—
274#95—216no—
275#95—44no—
276#95—48no—
277#95—48no—
278#95—48no—
279#95—44no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—28no—
291C24—28no—
292C24—232no—
293C24—28no—
294C24—24no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—22no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—44no—
303C24—44no—
304C24—416no—
305C24—44no—
306C24—44no—
307C24—416no—
308C24—416no—
309C24—416no—
310C24—416no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—416no—
318C24—88no—
319C24—88no—
320C24—88no—
321C24—88no—
322C24—88no—
323C24—88no—
324C24—88no—
325C24#124116yes—
326C24C22×D414yes—
327C24C22×D412yes—
328C24C22×D414yes—
329C24C2×#4114yes—
330C24C2×#4114yes—
331C24C2×#4114yes—
332C24H(5,𝔽2)14yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—416no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×Q8—88no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—816no—
344C2×D4—232no—
345C2×D4—416no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—416no—
349C2×D4—416no—
350C2×D4—4128no—
351C2×D4—416no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—28no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—416no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—416no—
403C22×C4—816no—
404C22×C4—816no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4#120116yes—
412C22×C4C22×D418yes—
413C22×C4C22×D418yes—
414C22×C4C22×D414yes—
415C22×C4C22×D414yes—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4H(5,𝔽2)14yes—
419C4⋊C4—216no—
420C4⋊C4—232no—
421C4⋊C4—232no—
422C4⋊C4—216no—
423C4⋊C4—416no—
424C22⋊C4—232no—
425C22⋊C4—232no—
426C22⋊C4—232no—
427C22⋊C4—432no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—816no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C42—88no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—28no—
446C23—24no—
447C23—24no—
448C23—28no—
449C23—28no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—232no—
459C23—24no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—416no—
464C23—416no—
465C23—416no—
466C23—416no—
467C23—416no—
468C23—48no—
469C23—48no—
470C23—48no—
471C23—48no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—48no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—48no—
482C23—48no—
483C23—44no—
484C23—48no—
485C23—864no—
486C23—88no—
487C23—88no—
488C23—88no—
489C23—88no—
490C23—864no—
491C23—88no—
492C23—88no—
493C23—88no—
494C23—88no—
495C23C2×#12014yes—
496C23C2×#12418yes—
497C23C2×#12414yes—
498C23#53614yes—
499C23#548116yes—
500C23C23×D411yescharacteristic
501C23C23×D412yes—
502C23C22×#4112yes—
503C23C2×H(5,𝔽2)12yes—
504C23C2×H(5,𝔽2)12yes—
505C23C2×2−1+412yes—
506C23#58512yes—
507C23#58512yes—
508Q8—816no—
509Q8—816no—
510D4—464no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—216no—
522C2×C4—28no—
523C2×C4—216no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—48no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—88no—
552C2×C4—816no—
553C2×C4—816no—
554C2×C4—88no—
555C2×C4—88no—
556C2×C4C2×#12018yes—
557C22—232no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—24no—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—416no—
568C22—416no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—416no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—832no—
578C22—832no—
579C22—88no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22#243618yes—
585C22C22×#12011yescharacteristic
586C22C22×#12411yescharacteristic
587C22C2×#53611yescharacteristic
588C22C2×#54814yes—
589C22#348411yescharacteristic
590C22#348411yescharacteristic
591C22#348511yescharacteristic
592C22#348511yescharacteristic
593C22#348814yes—
594C22#349014yes—
595C22#349114yes—
596C22#349614yes—
597C22#353514yes—
598C22#356114yes—
599C4—28no—
600C4—48no—
601C4—44no—
602C4—44no—
603C4—48no—
604C4—88no—
605C4—82no—
606C4—82no—
607C2—28no—
608C2—48no—
609C2—84no—
610C2C2×#243611yescharacteristic
611C2#3534711yescharacteristic
612C2#3536711yescharacteristic
613C2#3545511yescharacteristic
614C2#3551211yescharacteristic
615C2#3672011yescharacteristic
616C2#3696611yescharacteristic
617C2#4302418yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#543C2semidirect
2. C2×#2561C2semidirect
4. #15613C2semidirect
5. #15275C2semidirect
6. #14934C2semidirect
7. #14717C2semidirect
8. #14139C2semidirect
9. #13869C2semidirect
11. #13775C2semidirect
17. C22×C22⋊Q8C22semidirect
18. C22×C22⋊Q8C22semidirect
26. C2×#397C22semidirect
29. C2×#393C22semidirect
30. C2×#393C22semidirect
31. C2×#391C22semidirect
32. C2×#388C22semidirect
40. #1446C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1441C22semidirect
44. #1440C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
48. #1437C22semidirect
49. C4⋊#95C22semidirect
53. C22⋊#95C22semidirect
54. C22⋊#95C22semidirect
82. C2×C22⋊Q8D4semidirect
162. C22×Q8C2×D4semidirect
227. C2×C4⋊C4C2×D4semidirect
228. C2×C4⋊C4C2×D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#25612
C22×#3942
#156132
#152752
#149342
#147172
#141392
#138692
#138672
#137752