Finite Group Databasebeta

← #686188

Subgroups — #686188

Order 512.

4404 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#551C211yesmaximal, characteristic
2C2×#2853C211yesmaximal, characteristic, complemented
3C2×C4×#123C212yesmaximal, complemented
4#16032C212yesmaximal, complemented
5#15982C212yesmaximal, complemented
6#15949C212yesmaximal, complemented
7#15948C212yesmaximal
8#15072C214yesmaximal, complemented
9#15001C214yesmaximal, complemented
10#14167C211yesmaximal, characteristic, complemented
11#14063C214yesmaximal, complemented
12#13991C214yesmaximal, complemented
13#13940C212yesmaximal, complemented
14C2×#551—28no—
15C2×#551—216no—
16C22×#126C2212yes—
17C22×#124C2211yescharacteristic
18C22×#123C2212yes—
19C22×#123C2212yes—
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×#120C2211yescharacteristic
23C22×C4×D4C2212yes—
24C22×#117C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C42C2211yescharacteristic, complemented
27#2853—28no—
28C4×#123—216no—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2214yescomplemented
32C2×#387C2214yescomplemented
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#385C2212yescomplemented
37C2×#385C2212yescomplemented
38C2×#381C2211yescharacteristic, complemented
39C2×#381C2212yescomplemented
40C2×#379C2214yes—
41C2×C4×C4⋊C4C2214yescomplemented
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2212yescomplemented
44#1446C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1442C2214yescomplemented
48#1441C2214yescomplemented
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2211yescharacteristic, complemented
53#1440C2212yescomplemented
54#1440C2211yescharacteristic, complemented
55#1440C2212yescomplemented
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C26C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79#551—2128no—
80C2×#126—28no—
81C2×#126—28no—
82C2×#126—28no—
83C2×#124—24no—
84C2×#124—28no—
85C2×#123—28no—
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—216no—
91C2×#123D418yes—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×#120—28no—
96C2×#120—24no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4D4116yes—
100C2×#117—28no—
101C2×#117—28no—
102C2×#117—28no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C42—24no—
131C22×C42—28no—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2314yes—
136#388—216no—
137#388—216no—
138#387—216no—
139#387—216no—
140#386—28no—
141#386—28no—
142#385—28no—
143#385—28no—
144#385—216no—
145#381—24no—
146#381—28no—
147#379—216no—
148C4×C4⋊C4—216no—
149C4×C22⋊C4—216no—
150C4×C22⋊C4—28no—
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—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25—22no—
169C25—24no—
170C25—28no—
171C25—44no—
172C25—44no—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C25C2412yes—
176C22×Q8—24no—
177C22×Q8—22no—
178C22×Q8—44no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4C2×D418yes—
186C22×D4C2×D414yes—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4C2×D418yes—
203C23×C4C2×D414yes—
204C23×C4C2×D418yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215#126—264no—
216#124—232no—
217#123—264no—
218#123—264no—
219#123—232no—
220C22⋊Q8—264no—
221#120—232no—
222C4×D4—264no—
223#117—264no—
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—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D418yes—
244C2×C4⋊C4C2×D418yes—
245C2×C22⋊C4—264no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×D414yes—
291C2×C22⋊C4#41116yes—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—44no—
301C2×C42—44no—
302C2×C42—416no—
303C2×C42C2×D418yes—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—24no—
331C24—28no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—416no—
337C24—416no—
338C24—416no—
339C24—48no—
340C24—48no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24—84no—
345C24—88no—
346C24C22×D412yes—
347C24C22×D414yes—
348C24C22×D412yes—
349C24C22×D414yes—
350C24C2×#4118yes—
351C24C2511yescharacteristic
352C2×Q8—28no—
353C2×Q8—88no—
354C2×Q8—84no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—28no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—416no—
361C2×D4—416no—
362C2×D4—88no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
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—216no—
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—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—88no—
426C22×C4#120116yes—
427C22×C4C22×D412yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C2×#4118yes—
436C22×C4C2×#4118yes—
437C4⋊C4—216no—
438C4⋊C4—216no—
439C4⋊C4—232no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—416no—
443C4⋊C4—416no—
444C22⋊C4—216no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—28no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—28no—
451C22⋊C4—232no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C42—28no—
462C42—216no—
463C42—416no—
464C42—416no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—22no—
482C23—28no—
483C23—24no—
484C23—432no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—832no—
508C23—88no—
509C23—832no—
510C23—832no—
511C23C2×#12014yes—
512C23#52518yes—
513C23C23×D412yes—
514C23C23×D411yescharacteristic
515C23C22×#4114yes—
516C23#58514yes—
517C23#58514yes—
518Q8—88no—
519D4—432no—
520D4—88no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—416no—
543C2×C4—432no—
544C2×C4—432no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
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—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—816no—
580C2×C4C2×#12018yes—
581C22—24no—
582C22—216no—
583C22—28no—
584C22—28no—
585C22—48no—
586C22—48no—
587C22—48no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—44no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—42no—
597C22—44no—
598C22—832no—
599C22—84no—
600C22—84no—
601C22C22×#12011yescharacteristic
602C22C2×#52512yes—
603C22#344912yes—
604C22D4×#4114yes—
605C22D4×#4114yes—
606C22#347514yes—
607C22#348512yes—
608C22#348514yes—
609C22#351014yes—
610C22#352114yes—
611C22#356614yes—
612C4—28no—
613C4—44no—
614C4—42no—
615C4—48no—
616C4—44no—
617C4—44no—
618C4—48no—
619C4—44no—
620C4—84no—
621C4—84no—
622C2—44no—
623C2—48no—
624C2—82no—
625C2#3473612yes—
626C2#3544611yescharacteristic
627C2#3622414yes—
628C2#3987014yes—
629C2#5918114yes—
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×#2853C2semidirect
3. C2×C4×#123C2semidirect
4. #16032C2semidirect
5. #15982C2semidirect
6. #15949C2semidirect
8. #15072C2semidirect
9. #15001C2semidirect
10. #14167C2semidirect
11. #14063C2semidirect
12. #13991C2semidirect
13. #13940C2semidirect
26. C23×C42C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
36. C2×#385C22semidirect
37. C2×#385C22semidirect
38. C2×#381C22semidirect
39. C2×#381C22semidirect
41. C2×C4×C4⋊C4C22semidirect
43. C2×C4×C22⋊C4C22semidirect
44. #1446C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
52. #1440C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5512
C2×#28532
C2×C4×#1232
#160322
#159822
#159492
#159482
#150722
#150012
#141672
#140632
#139912
#139402