Finite Group Databasebeta

← #4381114

Subgroups — #4381114

Order 512.

2638 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2482C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic, complemented
4C2×#2312C211yesmaximal, characteristic
5#16391C214yesmaximal
6#15840C214yesmaximal, complemented
7#15832C214yesmaximal, complemented
8#15758C214yesmaximal, complemented
9#15751C214yesmaximal, complemented
10#14837C214yesmaximal, complemented
11#14660C211yesmaximal, characteristic, complemented
12#14297C211yesmaximal, characteristic, complemented
13#13939C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C22⋊C4C2211yescharacteristic
17#2625—28no—
18#2561—28no—
19#2482—28no—
20C2×#400C2212yescomplemented
21C2×#399C2212yes—
22C2×#398C2212yescomplemented
23C2×#397C2214yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#393C2212yes—
27#2312—28no—
28C2×#391C2212yescomplemented
29C2×#389C2212yescomplemented
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#385C2212yes—
35C2×#381C2212yes—
36C2×#381C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×#379C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1446C2212yescomplemented
45#1445C2212yescomplemented
46#1445C2214yescomplemented
47#1443C2214yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66C4⋊#95C2212yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2214yescomplemented
69#1434C2214yescomplemented
70#1434C2214yescomplemented
71C26C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—24no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4D418yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2312yes—
110#400—28no—
111#399—28no—
112#398—28no—
113#397—28no—
114#397—216no—
115#397—28no—
116#393—28no—
117#391—28no—
118#389—28no—
119#388—216no—
120#388—216no—
121#388—216no—
122#388—216no—
123#385—28no—
124#381—28no—
125#381—24no—
126#380—24no—
127#380D418yes—
128#379—28no—
129#379—28no—
130#379—28no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—28no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C25—22no—
157C25—22no—
158C25—42no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25—44no—
163C25C2×D414yes—
164C25C2412yes—
165C25C2411yescharacteristic
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—42no—
170C22×Q8—42no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4C2×D414yes—
177C23×C4C2×D414yes—
178C23×C4C2×D414yes—
179C23×C4#4114yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C22⋊Q8—416no—
192#117—416no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4C2×D418yes—
220C2×C22⋊C4—232no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C2×D418yes—
260C2×C22⋊C4C2×D418yes—
261C2×C22⋊C4C2×D418yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—44no—
270C2×C42—44no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95C2×D418yes—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—28no—
310C24—48no—
311C24—48no—
312C24—48no—
313C24—44no—
314C24—44no—
315C24—48no—
316C24—84no—
317C24—82no—
318C24—82no—
319C24—84no—
320C24—84no—
321C24—88no—
322C24—88no—
323C24—88no—
324C24—88no—
325C24C22×D414yes—
326C24C22×D412yes—
327C24C22×D412yes—
328C24C2×#4112yes—
329C24C2511yescharacteristic
330C2×Q8—48no—
331C2×Q8—44no—
332C2×Q8—84no—
333C2×Q8—82no—
334C2×Q8—82no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
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—24no—
366C22×C4—24no—
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—24no—
379C22×C4—24no—
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—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4#121116yes—
399C22×C4C22×D414yes—
400C22×C4C22×D418yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—84no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C22⋊C4—216no—
417C22⋊C4—216no—
418C22⋊C4—48no—
419C22⋊C4—416no—
420C22⋊C4—416no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C42—48no—
431C42—84no—
432C42—88no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—416no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—88no—
473C23—88no—
474C23—832no—
475C23—816no—
476C23—816no—
477C23—816no—
478C23—816no—
479C23C2×#12114yes—
480C23C23×D411yescharacteristic
481C23C23×D411yescharacteristic
482C23C22×#4111yescharacteristic
483C23#58512yes—
484C23#58512yes—
485C23#58512yes—
486C23#58512yes—
487C23#58514yes—
488Q8—88no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—416no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
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—88no—
552C2×C4C2×#12114yes—
553C2×C4C2×#12114yes—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—48no—
566C22—48no—
567C22—832no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—84no—
574C22C22×#12111yescharacteristic
575C22#344912yes—
576C22#348512yes—
577C22#348511yescharacteristic
578C22#348512yes—
579C22#348512yes—
580C22#348711yescharacteristic
581C22#349111yescharacteristic
582C22#349111yescharacteristic
583C22#349312yes—
584C22#350012yes—
585C22#351012yes—
586C22#356614yes—
587C22#356614yes—
588C22#356614yes—
589C22#356814yes—
590C4—24no—
591C4—24no—
592C4—44no—
593C4—48no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—84no—
598C4—82no—
599C4—84no—
600C4—84no—
601C2—44no—
602C2—84no—
603C2#3515011yescharacteristic
604C2#3544711yescharacteristic
605C2#3561711yescharacteristic
606C2#4141414yes—
607C2#4721512yes—
608C2#4721812yes—
609C2#5018612yes—
610C2#5018912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2625C2semidirect
6. #15840C2semidirect
7. #15832C2semidirect
8. #15758C2semidirect
9. #15751C2semidirect
10. #14837C2semidirect
11. #14660C2semidirect
12. #14297C2semidirect
13. #13939C2semidirect
20. C2×#400C22semidirect
22. C2×#398C22semidirect
28. C2×#391C22semidirect
29. C2×#389C22semidirect
44. #1446C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect
70. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#24822
C2×#26252
C2×#23122
#163912
#158402
#158322
#157582
#157512
#148372
#146602
#142972
#139392