Finite Group Databasebeta

← #4421689

Subgroups — #4421689

Order 512.

3964 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#551C211yesmaximal, characteristic
2C2×#2483C212yesmaximal, complemented
3C2×#2622C211yesmaximal, characteristic
4#16657C214yesmaximal, complemented
5#16364C214yesmaximal, complemented
6#16032C212yesmaximal, complemented
7#16024C212yesmaximal, complemented
8#16024C212yesmaximal, complemented
9#15982C212yesmaximal, complemented
10#14892C214yesmaximal, complemented
11#14823C214yesmaximal, complemented
12#14657C211yesmaximal, characteristic
13#14341C212yesmaximal, complemented
14C2×#551—28no—
15C2×#551—216no—
16C22×#126C2212yes—
17C22×#124C2211yescharacteristic
18C22×#123C2212yes—
19C22×#123C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×#120C2211yescharacteristic
22C22×C4×D4C2212yes—
23C22×#117C2212yes—
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2212yes—
26#2622—28no—
27#2483—216no—
28C2×#397C2214yes—
29C2×#397C2214yes—
30C2×#397C2214yes—
31C2×#394C2214yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2214yes—
40C2×#379C2211yescharacteristic
41C2×#379C2212yes—
42#1446C2214yescomplemented
43#1444C2214yescomplemented
44#1444C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1441C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2214yes—
52#1440C2214yes—
53#1440C2212yes—
54#1440C2212yes—
55#1440C2212yes—
56#1440C2212yes—
57#1440C2214yes—
58#1440C2211yescharacteristic
59#1440C2211yescharacteristic
60#1440C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1438C2214yes—
64#1438C2212yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yescomplemented
71C26C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79#551—2128no—
80C2×#126—28no—
81C2×#126—28no—
82C2×#126—28no—
83C2×#124—24no—
84C2×#124—28no—
85C2×#123—28no—
86C2×#123—24no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×#120—28no—
94C2×#120D418yes—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×#117—28no—
99C2×#117—28no—
100C2×#117—28no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4D4116yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#397—216no—
133#397—216no—
134#397—216no—
135#394—216no—
136#386—28no—
137#385—216no—
138#385—28no—
139#385—28no—
140#385—28no—
141#381—216no—
142#381—28no—
143#381—28no—
144#379—24no—
145#379—28no—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25—44no—
166C25—44no—
167C25—44no—
168C25C2×D414yes—
169C25C2×D418yes—
170C25C2411yescharacteristic
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—44no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4C2×D418yes—
183C22×D4C2×D414yes—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D418yes—
201C23×C4C2×D418yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212#126—264no—
213#124—232no—
214#123—264no—
215#123—232no—
216C22⋊Q8—264no—
217#120—232no—
218C4×D4—264no—
219#117—264no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C22⋊C4—264no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—416no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4C2×D418yes—
285C2×C22⋊C4C2×D418yes—
286C2×C22⋊C4C2×D418yes—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—48no—
327C24—48no—
328C24—416no—
329C24—48no—
330C24—416no—
331C24—416no—
332C24—88no—
333C24—88no—
334C24—88no—
335C24—88no—
336C24#120116yes—
337C24C22×D412yes—
338C24C22×D414yes—
339C24C22×D412yes—
340C24C22×D414yes—
341C24C22×D414yes—
342C24C22×D414yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—88no—
346C2×Q8—84no—
347C2×D4—216no—
348C2×D4—28no—
349C2×D4—416no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—88no—
355C22×C4—216no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—88no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—216no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C22⋊C4—216no—
437C22⋊C4—28no—
438C22⋊C4—216no—
439C22⋊C4—216no—
440C22⋊C4—216no—
441C22⋊C4—216no—
442C22⋊C4—216no—
443C22⋊C4—28no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C42—28no—
454C42—216no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—28no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—22no—
474C23—28no—
475C23—24no—
476C23—432no—
477C23—416no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—832no—
499C23—832no—
500C23—832no—
501C23C2×#12014yes—
502C23C2×#12018yes—
503C23C23×D412yes—
504C23C23×D411yescharacteristic
505C23#58514yes—
506C23#58514yes—
507C23#58514yes—
508Q8—88no—
509D4—432no—
510D4—88no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—432no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—816no—
569C2×C4—88no—
570C2×C4—816no—
571C22—28no—
572C22—28no—
573C22—28no—
574C22—28no—
575C22—24no—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—44no—
586C22—42no—
587C22—48no—
588C22—832no—
589C22—84no—
590C22—84no—
591C22C22×#12011yescharacteristic
592C22#344912yes—
593C22#344912yes—
594C22#348514yes—
595C22#348514yes—
596C22#348512yes—
597C22#348514yes—
598C22#356614yes—
599C22#356614yes—
600C22#356614yes—
601C22#356814yes—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—42no—
608C4—84no—
609C4—84no—
610C4—84no—
611C4—84no—
612C2—48no—
613C2—44no—
614C2—82no—
615C2#3508812yes—
616C2#3544611yescharacteristic
617C2#4141414yes—
618C2#4141414yes—
619C2#4877514yes—
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
2. C2×#2483C2semidirect
4. #16657C2semidirect
5. #16364C2semidirect
6. #16032C2semidirect
7. #16024C2semidirect
8. #16024C2semidirect
9. #15982C2semidirect
10. #14892C2semidirect
11. #14823C2semidirect
13. #14341C2semidirect
42. #1446C22semidirect
43. #1444C22semidirect
44. #1444C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
60. #1440C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
70. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5512
C2×#24832
C2×#26222
#166572
#163642
#160322
#160242
#159822
#148922
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