Finite Group Databasebeta

← #761615

Subgroups — #761615

Order 512.

4018 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#17005C214yesmaximal, complemented
4#15073C212yesmaximal, complemented
5#15015C212yesmaximal, complemented
6#14843C212yesmaximal, complemented
7#14841C214yesmaximal, complemented
8#14822C212yesmaximal, complemented
9#13972C214yesmaximal, complemented
10#13966C212yesmaximal, complemented
11#13923C212yesmaximal, complemented
12#13869C212yesmaximal
13#13869C212yesmaximal
14C4×#379C211yesmaximal, characteristic
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C2×#400C2214yescomplemented
24C2×#385—28no—
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#381C2214yes—
30C2×#381C2214yes—
31C2×#380—28no—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#379C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43#1445C2214yescomplemented
44#1443C2218yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2218yescomplemented
48#1442C2214yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1440C2218yes—
53#1440C2218yes—
54#1436C2218yescomplemented
55#1434C2218yescomplemented
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C24×C4—28no—
67C24×C4—44no—
68C24×C4—44no—
69C24×C4D418yes—
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—216no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2314yes—
114#400—48no—
115#385—28no—
116#385—28no—
117#385—28no—
118#385—28no—
119#381—216no—
120#381—48no—
121#380—28no—
122#380—28no—
123#380—28no—
124#380—28no—
125#379—28no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—44no—
128C4×C22⋊C4—44no—
129C2×#95—28no—
130C2×#95—24no—
131C2×#95—24no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—42no—
154C25—42no—
155C25C2×D414yes—
156C25#4118yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—232no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—28no—
180C23×C4—24no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—416no—
188C23×C4—42no—
189C23×C4—416no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4—464no—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4#4118yes—
212C23×C4C2418yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
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—28no—
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—416no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—416no—
243C2×C22⋊C4—232no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—216no—
254C2×C22⋊C4—216no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—416no—
267C2×C22⋊C4—432no—
268C2×C22⋊C4#4118yes—
269C2×C42—28no—
270C2×C42—44no—
271C2×C42—44no—
272#95—28no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—216no—
280#95—216no—
281#95—28no—
282#95—48no—
283#95—48no—
284#95—44no—
285#95—48no—
286#95—48no—
287#95—48no—
288#95—48no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—28no—
298C24—28no—
299C24—28no—
300C24—28no—
301C24—232no—
302C24—232no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—44no—
310C24—44no—
311C24—416no—
312C24—416no—
313C24—416no—
314C24—416no—
315C24—416no—
316C24#124116yes—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C22×D412yes—
320C24C2×#4114yes—
321C24C2×#4114yes—
322C24C2×#4114yes—
323C24C2×#4114yes—
324C24C2×#4114yes—
325C24H(5,𝔽2)14yes—
326C242−1+414yes—
327C24C2511yescharacteristic
328C22×C4—232no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—416no—
389C22×C4—416no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—416no—
393C22×C4—416no—
394C22×C4—4128no—
395C22×C4—416no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—432no—
413C22×C4—88no—
414C22×C4—816no—
415C22×C4—88no—
416C22×C4#123116yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)14yes—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C22⋊C4—216no—
435C22⋊C4—216no—
436C22⋊C4—416no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C42—88no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—232no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—232no—
475C23—416no—
476C23—416no—
477C23—416no—
478C23—416no—
479C23—4128no—
480C23—416no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C2×#12314yes—
486C23#52514yes—
487C23C2×#12414yes—
488C23C2×#12414yes—
489C23C2×#12414yes—
490C23#53414yes—
491C23#53614yes—
492C23#552116yes—
493C23#553116yes—
494C23C23×D411yescharacteristic
495C23C22×#4112yes—
496C23C22×#4112yes—
497C23C22×#4112yes—
498C23C2×H(5,𝔽2)12yes—
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+412yes—
501C23#58512yes—
502C2×C4—232no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—416no—
520C2×C4—416no—
521C2×C4—416no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—48no—
546C2×C4—48no—
547C2×C4—416no—
548C2×C4—48no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—416no—
553C2×C4—48no—
554C2×C4—48no—
555C2×C4—48no—
556C2×C4—88no—
557C2×C4—88no—
558C2×C4—816no—
559C2×C4—816no—
560C2×C4C2×#12314yes—
561C2×C4C2×#12314yes—
562C2×C4#53414yes—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—232no—
571C22—232no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—464no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—416no—
585C22—42no—
586C22#250318yes—
587C22#259118yes—
588C22C22×#12311yescharacteristic
589C22C2×#52511yescharacteristic
590C22C22×#12411yescharacteristic
591C22C2×#53411yescharacteristic
592C22C2×#53411yescharacteristic
593C22C2×#53611yescharacteristic
594C22#344911yescharacteristic
595C22C2×#55214yes—
596C22C2×#55314yes—
597C22#347514yes—
598C22#349014yes—
599C22#350514yes—
600C22#352014yes—
601C22#353514yes—
602C4—24no—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—42no—
607C4—44no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—88no—
612C4—84no—
613C2—28no—
614C2—28no—
615C2—44no—
616C2C2×#250311yescharacteristic
617C2#3480911yescharacteristic
618C2C2×#259111yescharacteristic
619C2#3499211yescharacteristic
620C2#3501911yescharacteristic
621C2#3508411yescharacteristic
622C2#3744111yescharacteristic
623C2#5932818yes—
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. #17005C2semidirect
4. #15073C2semidirect
5. #15015C2semidirect
6. #14843C2semidirect
7. #14841C2semidirect
8. #14822C2semidirect
9. #13972C2semidirect
10. #13966C2semidirect
11. #13923C2semidirect
23. C2×#400C22semidirect
43. #1445C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
54. #1436C22semidirect
55. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C22×#3802
#170052
#150732
#150152
#148432
#148412
#148222
#139722
#139662
#139232
#138692
C4×#3792