Finite Group Databasebeta

← #12115754

Subgroups — #12115754

Order 1280.

1246 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65897C211yesmaximal, characteristic
2C5×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#65301C211yesmaximal, characteristic, complemented
4#10602353C212yesmaximal, complemented
5#10602351C212yesmaximal, complemented
6#10600377C211yesmaximal, characteristic, complemented
7#10600375C211yesmaximal, characteristic, complemented
8#10600310C212yesmaximal, complemented
9#10600296C212yesmaximal, complemented
10#10600282C211yesmaximal, characteristic, complemented
11C4×#3941C211yesmaximal, characteristic, complemented
12C2×C4×C5×D4—21no—
13C2×C4×C5×D4C2211yescharacteristic
14C22×C5×C4⋊C4C2212yes—
15C2×#4084—21no—
16C2×#4076C2212yes—
17C2×#4072C2212yes—
18C22×#681C2212yes—
19#65897—24no—
20#65897C418yescomplemented
21C2×#4038—21no—
22C2×#4031—22no—
23C2×C4×D20C2211yescharacteristic
24C2×#4015C2211yescharacteristic
25C5×#389—21no—
26C5×#385—22no—
27C5×#384C2211yescharacteristic
28C5×#380C2212yes—
29C4×C5×C4⋊C4C2211yescharacteristic
30C2×C22⋊#168C2212yes—
31#65301—24no—
32C2×#3941C2211yescharacteristic
33C2×#3941C2211yescharacteristic
34C2×#3941C2212yes—
35C2×C4⋊#168C2211yescharacteristic
36C2×C4×#168C2211yescharacteristic
37#64749C2211yescharacteristic
38#64749C2211yescharacteristic
39#64749C2212yes—
40#64749C2212yes—
41#64749C2212yes—
42#64749C2212yes—
43#64749C2212yes—
44#22503—51nomaximal, Sylow
45C22×C5×D4D411yescharacteristic
46C23×C4×C5—22no—
47C23×C4×C5C2312yes—
48C22×C22⋊D5—22no—
49C22×D20C2311yescharacteristic
50C22×C4×D5—21no—
51C22×Dic10—21no—
52C4×C5×D4—44no—
53C4×C5×D4C2×C414yes—
54C4×C5×D4D414yes—
55C2×C5×C4⋊C4—24no—
56C2×C5×C4⋊C4—21no—
57C2×C5×C4⋊C4—22no—
58C2×C5×C4⋊C4—21no—
59C2×C5×C4⋊C4—42no—
60C2×C5×C4⋊C4D411yescharacteristic
61C2×C5×C4⋊C4C2311yescharacteristic
62C2×C5×C4⋊C4C2312yes—
63C2×C5×C4⋊C4C2311yescharacteristic
64C2×C5×C22⋊C4—22no—
65C2×C5×C22⋊C4D412yes—
66C2×C42×C5—21no—
67C2×C42×C5C2311yescharacteristic
68C2×C22⋊Dic5—22no—
69#4084—28no—
70#4076—24no—
71#4076C2×C418yescomplemented
72C2×#685—21no—
73C2×#685—22no—
74#4072—24no—
75#4072C2×C418yescomplemented
76C2×C4⋊Dic5—21no—
77C2×C4⋊Dic5C2312yes—
78C2×C4⋊Dic5C2311yescharacteristic
79C2×C4×Dic5—22no—
80C2×#681—24no—
81C2×#681—24no—
82C2×#681—42no—
83C22×#168C2312yes—
84C22×#168C2311yescharacteristic
85C22×#168C2311yescharacteristic
86#4038—28no—
87#4031—216no—
88C4×D20—22no—
89C4×D20C2×C414yescomplemented
90#4015—22no—
91#4015C2×C414yescomplemented
92C5×#95—21no—
93C5×#95—22no—
94C22⋊#168—24no—
95#3941—22no—
96#3941—22no—
97#3941—42no—
98C4⋊#168—22no—
99C4×#168—22no—
100C2×#555—51no—
101C2×#436—51no—
102D4×C4⋊C4D511yescharacteristic, complemented
103#1990—51no—
104#1987—51no—
105#1895—52no—
106#1874—52no—
107#1825—52no—
108#1802—51no—
109C4×#102—51no—
110#1736—52no—
111C24×C5C2×D412yes—
112C23×D5—21no—
113C2×C5×D4—22no—
114C2×C5×D4—22no—
115C2×C5×D4C22⋊C414yes—
116C22×C4×C5—22no—
117C22×C4×C5—22no—
118C22×C4×C5—22no—
119C22×C4×C5—22no—
120C22×C4×C5—21no—
121C22×C4×C5—21no—
122C22×C4×C5—22no—
123C22×C4×C5—44no—
124C22×C4×C5—44no—
125C22×C4×C5—82no—
126C22×C4×C5C22×C414yes—
127C22×C4×C5C2×D414yes—
128C22×C4×C5C2×D411yescharacteristic
129C22×C4×C5C2×D412yes—
130C22×C4×C5C2×D411yescharacteristic
131C22×C4×C5C2411yescharacteristic
132C2×C22⋊D5—28no—
133C2×C22⋊D5—82no—
134C2×C22⋊D5—82no—
135C22×Dic5—21no—
136C22×Dic5—22no—
137C2×D20—21no—
138C2×D20—21no—
139C2×D20—21no—
140C2×D20—21no—
141C2×D20—21no—
142C2×D20C22×C412yes—
143C2×C4×D5—24no—
144C2×C4×D5—42no—
145C2×C4×D5—42no—
146C2×Dic10—24no—
147C2×Dic10—81no—
148C2×Dic10—81no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4—22no—
151C5×C4⋊C4—21no—
152C5×C4⋊C4—22no—
153C5×C4⋊C4—21no—
154C5×C4⋊C4—44no—
155C5×C4⋊C4—42no—
156C5×C4⋊C4—81no—
157C5×C4⋊C4C22⋊C414yes—
158C5×C4⋊C4#4112yes—
159C5×C22⋊C4—44no—
160C5×C22⋊C4C22⋊C418yes—
161C42×C5—42no—
162C42×C5C22×C412yes—
163C42×C5C2×D412yes—
164C22⋊Dic5—28no—
165#685—24no—
166#685—28no—
167C4⋊Dic5—24no—
168C4⋊Dic5—22no—
169C4⋊Dic5—21no—
170C4⋊Dic5C22×C414yes—
171C4⋊Dic5C22×C412yes—
172C4×Dic5—28no—
173#681—48no—
174C2×#168—21no—
175C2×#168—21no—
176C2×#168—21no—
177C2×#168—22no—
178C2×#168—21no—
179C2×#168—22no—
180C2×#168—22no—
181C2×#168—21no—
182C2×#168#4112yes—
183C22×M16—52no—
184#555—58no—
185#555—104no—
186C2×C4⋊Q8—51no—
187C2×#124—102no—
188C2×#121—52no—
189C2×#121—101no—
190C2×C4×D4—21no—
191C2×C4×D4—51no—
192C2×C4×D4C2×D511yescharacteristic
193C2×#117—52no—
194C2×#117—101no—
195C22×C4⋊C4C2×D512yes—
196#436—104no—
197C2×C4⋊C8—51no—
198C2×#102—51no—
199C2×#102—51no—
200C2×#102—52no—
201C2×C22⋊C8—52no—
202C2×C4×C8—51no—
203#389—21no—
204#385—22no—
205#384C2×D511yescharacteristic
206#380C2×D512yes—
207C4×C4⋊C4C2×D511yescharacteristic
208#340—52no—
209#340—51no—
210#340—52no—
211#340—52no—
212#340—52no—
213#340—52no—
214#340—51no—
215C23×C5—22no—
216C23×C5—22no—
217C23×C5—22no—
218C23×C5—22no—
219C23×C5—22no—
220C23×C5—22no—
221C23×C5C2×C22⋊C414yes—
222C23×C5C22×D411yescharacteristic
223C22×D5—22no—
224C22×D5—41no—
225C22×D5—41no—
226C22×D5—41no—
227C22×D5—41no—
228C22×D5—41no—
229C22×D5—41no—
230C5×D4—28no—
231C2×C4×C5—22no—
232C2×C4×C5—22no—
233C2×C4×C5—21no—
234C2×C4×C5—21no—
235C2×C4×C5—24no—
236C2×C4×C5—24no—
237C2×C4×C5—24no—
238C2×C4×C5—24no—
239C2×C4×C5—22no—
240C2×C4×C5—41no—
241C2×C4×C5—41no—
242C2×C4×C5—42no—
243C2×C4×C5—41no—
244C2×C4×C5—41no—
245C2×C4×C5—42no—
246C2×C4×C5—42no—
247C2×C4×C5—41no—
248C2×C4×C5—88no—
249C2×C4×C5C2×C22⋊C414yes—
250C2×C4×C5C2×C22⋊C412yes—
251C2×C4×C5C2×C22⋊C412yes—
252C2×C4×C5AGL(1,ℤ/8)12yes—
253C2×C4×C5#13712yes—
254C2×C4×C5C23×C411yescharacteristic
255C2×C4×C5C22×D411yescharacteristic
256C2×C4×C5C2×#4111yescharacteristic
257C2×C4×C5C2×#4111yescharacteristic
258C2×C4×C5H(5,𝔽2)11yescharacteristic
259C2×C4×C52−1+411yescharacteristic
260C22⋊D5—88no—
261C2×Dic5—22no—
262C2×Dic5—24no—
263C2×Dic5—41no—
264C2×Dic5—41no—
265C2×Dic5—42no—
266C2×Dic5—42no—
267D20—22no—
268D20—22no—
269D20—41no—
270D20—41no—
271C4×D5—48no—
272Dic10—84no—
273#168—22no—
274#168—42no—
275#168—41no—
276C22×Q8—101no—
277C22×D4—51no—
278C22×D4—102no—
279C22×D4C22⋊D511yescharacteristic
280C23×C4—22no—
281C23×C4—101no—
282C23×C4C22×D512yes—
283C2×M16—104no—
284C2×M16—104no—
285C2×M16—202no—
286C22×C8—51no—
287C22×C8—52no—
288C22×C8—51no—
289C4⋊Q8—54no—
290C4⋊Q8—102no—
291#124—1016no—
292#121—58no—
293#121—108no—
294#121—104no—
295C4×D4—44no—
296C4×D4—54no—
297C4×D4—102no—
298C4×D4C4×D514yes—
299C4×D4D2014yes—
300#117—58no—
301#117—104no—
302#117—108no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—51no—
309C2×C4⋊C4—52no—
310C2×C4⋊C4—101no—
311C2×C4⋊C4C22⋊D511yescharacteristic
312C2×C4⋊C4C22×D512yes—
313C2×C4⋊C4C22×D511yescharacteristic
314C2×C4⋊C4C22×D511yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—102no—
317C2×C22⋊C4—102no—
318C2×C22⋊C4—101no—
319C2×C22⋊C4C22⋊D512yes—
320C2×C42—21no—
321C2×C42—102no—
322C2×C42C22×D511yescharacteristic
323C4⋊C8—102no—
324#102—102no—
325#102—102no—
326#102—202no—
327C22⋊C8—104no—
328C4×C8—102no—
329#95—21no—
330#95—22no—
331C22×C5—24no—
332C22×C5—24no—
333C22×C5—24no—
334C22×C5—42no—
335C22×C5—42no—
336C22×C5—42no—
337C22×C5—42no—
338C22×C5C22×C22⋊C411yescharacteristic
339C22×C5#54611yescharacteristic
340C22×C5#54711yescharacteristic
341C22×C5C2×AGL(1,ℤ/8)11yescharacteristic
342C22×C5C2×#13711yescharacteristic
343C22×C5#57611yescharacteristic
344C22×C5#57711yescharacteristic
345C2×D5—42no—
346C2×D5—42no—
347C2×D5—42no—
348C2×D5—81no—
349C2×D5—81no—
350C2×D5—81no—
351C2×D5—81no—
352C4×C5—24no—
353C4×C5—22no—
354C4×C5—42no—
355C4×C5—81no—
356C4×C5—82no—
357C4×C5#41714yes—
358C4×C5C4×#4111yescharacteristic
359C4×C5#52011yescharacteristic
360C4×C5#54611yescharacteristic
361C4×C5#54711yescharacteristic
362Dic5—42no—
363Dic5—44no—
364C24—101no—
365C24C2×C22⋊D512yes—
366C2×Q8—104no—
367C2×Q8—401no—
368C2×Q8—401no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—52no—
372C2×D4—108no—
373C2×D4—101no—
374C2×D4—101no—
375C2×D4—101no—
376C2×D4—101no—
377C2×D4—101no—
378C2×D4—402no—
379C2×D4—402no—
380C2×D4#68514yes—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—82no—
391C22×C4—101no—
392C22×C4—102no—
393C22×C4—104no—
394C22×C4—202no—
395C22×C4—202no—
396C22×C4C2×C4×D514yes—
397C22×C4C2×D2014yes—
398C22×C4C2×C22⋊D512yes—
399C22×C4C2×C22⋊D511yescharacteristic
400C22×C4C2×C22⋊D511yescharacteristic
401C22×C4C23×D511yescharacteristic
402M16—208no—
403C2×C8—52no—
404C2×C8—101no—
405C2×C8—101no—
406C2×C8—101no—
407C2×C8—102no—
408C2×C8—101no—
409C2×C8—102no—
410C2×C8—102no—
411C2×C8—101no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—52no—
420C4⋊C4—54no—
421C4⋊C4—81no—
422C4⋊C4—101no—
423C4⋊C4—102no—
424C4⋊C4—104no—
425C4⋊C4#68514yes—
426C4⋊C4#71112yes—
427C22⋊C4—44no—
428C22⋊C4—108no—
429C22⋊C4—108no—
430C22⋊C4—104no—
431C22⋊C4#68518yes—
432C42—42no—
433C42—108no—
434C42C2×C4×D512yes—
435C42C2×D2012yes—
436C2×C5—44no—
437C2×C5#233911yescharacteristic
438C2×C5C2×#41711yescharacteristic
439C2×C5#289711yescharacteristic
440C2×C5#332511yescharacteristic
441C2×C5#332911yescharacteristic
442C2×C5#335411yescharacteristic
443C2×C5#335711yescharacteristic
444D5—82no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—102no—
452C23—201no—
453C23—201no—
454C23—201no—
455C23—201no—
456C23—201no—
457C23—201no—
458C23C2×#68514yes—
459C23C22×C22⋊D511yescharacteristic
460Q8—404no—
461D4—28no—
462D4—102no—
463D4—102no—
464D4—201no—
465D4—201no—
466D4—408no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—88no—
485C2×C4—102no—
486C2×C4—104no—
487C2×C4—201no—
488C2×C4—201no—
489C2×C4—202no—
490C2×C4—202no—
491C2×C4—208no—
492C2×C4C2×#68514yes—
493C2×C4C2×#68512yes—
494C2×C4C2×#68512yes—
495C2×C4#407812yes—
496C2×C4#408812yes—
497C2×C4C22×C4×D511yescharacteristic
498C2×C4C22×D2011yescharacteristic
499C2×C4C2×#71111yescharacteristic
500C2×C4#414411yescharacteristic
501C2×C4D5×#4111yescharacteristic
502C2×C4#415011yescharacteristic
503C8—102no—
504C8—202no—
505C8—201no—
506C5#2250311yescharacteristic, complemented, Sylow
507C22—24no—
508C22—24no—
509C22—24no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—202no—
515C22—202no—
516C22—202no—
517C22—401no—
518C22—401no—
519C22—401no—
520C22—401no—
521C22C22×#68511yescharacteristic
522C22C2×#407811yescharacteristic
523C22#6614011yescharacteristic
524C22C2×#408811yescharacteristic
525C22#6615811yescharacteristic
526C22#6616911yescharacteristic
527C22#6617111yescharacteristic
528C4—22no—
529C4—24no—
530C4—42no—
531C4—82no—
532C4—81no—
533C4—204no—
534C4—202no—
535C4#6525514yes—
536C4C4×#71111yescharacteristic
537C4#6588011yescharacteristic
538C4#6588811yescharacteristic
539C4#6588911yescharacteristic
540C2—44no—
541C2—402no—
542C2C2×#6525511yescharacteristic
543C2#1061406111yescharacteristic
544C2#1061429411yescharacteristic
545C2#1061521811yescharacteristic
546C2#1061525211yescharacteristic
547C2#1061528611yescharacteristic
548C2#1061535811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC22×C4C23×D5
FittingC5×D4×C4⋊C4C2
socleC23×C5C22×D4
prime radicalC2×#65897C2
cosocleC22×C4×C5C24
hypercenterD4×C4⋊C4D5
hypocenterC5#22503
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×D4×C4⋊C4C2semidirect
3. C2×#65301C2semidirect
4. #10602353C2semidirect
5. #10602351C2semidirect
6. #10600377C2semidirect
7. #10600375C2semidirect
8. #10600310C2semidirect
9. #10600296C2semidirect
10. #10600282C2semidirect
11. C4×#3941C2semidirect
20. #65897C4semidirect
71. #4076C2×C4semidirect
75. #4072C2×C4semidirect
89. C4×D20C2×C4semidirect
91. #4015C2×C4semidirect
102. D4×C4⋊C4D5semidirect
506. C5#22503semidirect

Maximal subgroups

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

subgroupindex
C2×#658972
C5×D4×C4⋊C42
C2×#653012
#106023532
#106023512
#106003772
#106003752
#106003102
#106002962
#106002822
C4×#39412
#225035