Finite Group Databasebeta

← #29658582

Subgroups — #29658582

Order 15616.

1224 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840416C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21455815C211yesmaximal, characteristic, complemented
4#21445052C214yesmaximal, complemented
5#21444906C214yesmaximal, complemented
6#21444492C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C61×#3036C211yesmaximal, characteristic, complemented
10C22×#11758224—21no—
11#16840416—22no—
12#16840416—22no—
13#16840416—22no—
14#16840416C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519761—22no—
17C2×#14519757—21no—
18C2×#14519751—21no—
19C2×C4×D244C2211yescharacteristic
20C22×C4⋊Dic61—21no—
21#16840165C2212yescomplemented
22#16840108C2212yescomplemented
23#16840073—22no—
24#16840073—21no—
25#16840042—22no—
26#16840039C2212yescomplemented
27#16839954C2212yescomplemented
28#16839895—21no—
29C4×#11758199—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839509C2214yescomplemented
33D4⋊#92418C2214yescomplemented
34#16839489C2214yescomplemented
35C2×AGL(1,ℤ/8)×C61—21no—
36C61×#532C2211yescharacteristic
37C2×C4×D4×C61C2211yescharacteristic
38C61×#463—22no—
39C61×#461C2212yescomplemented
40C61×#459C2212yescomplemented
41C61×#423C2211yescharacteristic
42C61×#417—21no—
43C2×#11758224—24no—
44C2×#11758224—24no—
45C2×#11758224—44no—
46C2×#11758224—44no—
47C22×Dic122—21no—
48C22×D244D411yescharacteristic
49C22×C22⋊D61—22no—
50C23×Dic61—22no—
51C22×C4×D61—21no—
52C2×#11758199—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777D412yes—
56#14519777C2312yes—
57#14519761—28no—
58#14519761—28no—
59#14519757—24no—
60#14519757—24no—
61#14519751—24no—
62#14519751—24no—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D412yes—
66C4×D244C2312yes—
67C2×C4⋊Dic61—22no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D412yes—
73C2×#11758198—22no—
74C2×#11758198—21no—
75#14519710—22no—
76#14519710—24no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79#14519698—22no—
80#14519698—24no—
81#14519697—22no—
82C22⋊#92418—21no—
83C22⋊#92418—21no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86C2×#41×C61—21no—
87C22×D4×C61D411yescharacteristic
88C23×C4×C61—21no—
89AGL(1,ℤ/8)×C61—44no—
90C2×SD16×C61—22no—
91C2×D8×C61—22no—
92C2×M16×C61—21no—
93C4⋊Q8×C61C2311yescharacteristic
94#127×C61C2311yescharacteristic
95#124×C61—21no—
96#121×C61—21no—
97#121×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61—21no—
102C4×D4×C61D412yes—
103C4×D4×C61C2312yes—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—21no—
106C2×C42×C61C2311yescharacteristic
107C4⋊C8×C61—21no—
108C4⋊C8×C61C2312yes—
109#103×C61—22no—
110#102×C61—22no—
111C24×C61—21no—
112C23×D61—21no—
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C2×D4×C61D814yescomplemented
123C2×D4×C61C2×D412yes—
124C22×C4×C61—21no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—44no—
128C22×C4×C61—41no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C2×C22⋊D61—28no—
133C2×C22⋊D61—48no—
134C22×Dic61—28no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—48no—
138#11758224—48no—
139#11758224—416no—
140#11758224—48no—
141C2×D244—21no—
142C2×D244—42no—
143C2×D244C2×D412yes—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—42no—
147C2×C4×D61—42no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—42no—
151C2×Dic122—42no—
152SD16×C61—44no—
153D8×C61—44no—
154M16×C61—42no—
155C2×C8×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61C2×D412yes—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—41no—
163C22⋊C4×C61—41no—
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C42×C61C2×D411yescharacteristic
167C42×C61C2411yescharacteristic
168#11758199—44no—
169#11758198—28no—
170#11758198—22no—
171#11758198—22no—
172C4⋊Dic61—28no—
173C4⋊Dic61—21no—
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—22no—
177C4⋊Dic61C2×D412yes—
178C4⋊Dic61C2×D414yes—
179C2×#92418—22no—
180C2×#92418—22no—
181C23×C61—22no—
182C23×C61—24no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61#12011yescharacteristic
187C22×D61—22no—
188C22×D61—42no—
189C22×D61—42no—
190C22×D61—42no—
191Q8×C61—41no—
192Q8×C61—41no—
193D4×C61—22no—
194D4×C61—24no—
195D4×C61—42no—
196D4×C61—41no—
197D4×C61—41no—
198D4×C61—42no—
199D4×C61—82no—
200D4×C61C2×D814yes—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—44no—
205C2×C4×C61—42no—
206C2×C4×C61—44no—
207C2×C4×C61—41no—
208C2×C4×C61—82no—
209C2×C4×C61#12011yescharacteristic
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61C2×D812yes—
213C2×C4×C61AGL(1,ℤ/8)12yes—
214C2×C4×C61C22×D411yescharacteristic
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C22⋊D61—416no—
218C2×Dic61—24no—
219C2×Dic61—22no—
220C2×Dic61—416no—
221C2×Dic61—44no—
222C2×Dic61—42no—
223D244—22no—
224D244—22no—
225D244—42no—
226C4×D61—42no—
227C4×D61—42no—
228C4×D61—44no—
229Dic122—42no—
230Dic122—42no—
231Dic122—44no—
232C8×C61—42no—
233#92418—42no—
234#92418—42no—
235#18756—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—24no—
239C22×C61—48no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61#44712yes—
246C22×C61C2×#12011yescharacteristic
247C2×D61—42no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—84no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—42no—
255C4×C61—81no—
256C4×C61#44712yes—
257C4×C61C22×D811yescharacteristic
258C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
259C4×C61#57512yes—
260Dic61—44no—
261Dic61—42no—
262C2×#547—611no—
263#3226—611no—
264#3100—611no—
265#3036D6111yescharacteristic, complemented
266#1647—614no—
267#1625—614no—
268#1546—611no—
269#1486—611no—
270#1478—611no—
271C2×C61—22no—
272C2×C61—44no—
273C2×C61—161no—
274C2×C61C2×#44711yescharacteristic
275C2×C61#300511yescharacteristic
276C2×C61#302011yescharacteristic
277D61—82no—
278C22×#41—1221no—
279C2×AGL(1,ℤ/8)—21no—
280#547—614no—
281#547—1222no—
282#547—1222no—
283#547—1222no—
284#532C2×D6111yescharacteristic
285C2×C4⋊Q8—611no—
286C2×#123—1222no—
287C2×C22⋊Q8—1221no—
288C2×#121—1221no—
289C2×C4×D4—611no—
290C2×C4×D4C2×D6111yescharacteristic
291C22×C4⋊C4—1221no—
292#500—612no—
293#486—612no—
294#480—1221no—
295#480—1222no—
296#474—612no—
297#474—1222no—
298#463—22no—
299#461C2×D6112yescomplemented
300#459—612no—
301#459C2×D6112yescomplemented
302#447—1221no—
303C4×D8—1222no—
304#431—611no—
305#431—611no—
306#423C2×D6111yescharacteristic
307#417—21no—
308#332—614no—
309D4⋊C8—614no—
310D4⋊C8—614no—
311C61#1875611yescharacteristic, complemented, Sylow
312C2×#41—21no—
313C2×#41—1224no—
314C2×#41—1224no—
315C2×#41—2444no—
316C2×#41—2444no—
317C22×Q8—1221no—
318C22×D4—611no—
319C22×D4—1222no—
320C22×D4C22⋊D6111yescharacteristic
321C23×C4—21no—
322C23×C4—1222no—
323C23×C4—1221no—
324AGL(1,ℤ/8)—44no—
325C2×SD16—22no—
326C2×D8—22no—
327C2×D8—1222no—
328C2×M16—21no—
329C4⋊Q8—612no—
330C4⋊Q8—612no—
331C4⋊Q8—1221no—
332C4⋊Q8—1221no—
333C4⋊Q8C22×D6111yescharacteristic
334#127C22×D6111yescharacteristic
335#124—21no—
336#123—1228no—
337#123—1228no—
338C22⋊Q8—1224no—
339C22⋊Q8—1224no—
340#121—21no—
341#121—21no—
342#121—1224no—
343#121—1224no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—612no—
349C4×D4—612no—
350C4×D4—1221no—
351C4×D4—1221no—
352C4×D4C22⋊D6112yes—
353C4×D4C22×D6112yes—
354C2×C4⋊C4—611no—
355C2×C4⋊C4—612no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1224no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1221no—
364C2×C42C22×D6111yescharacteristic
365#107—1222no—
366#107—1224no—
367C4⋊C8—21no—
368C4⋊C8—612no—
369C4⋊C8—1221no—
370C4⋊C8C22×D6112yes—
371#103—22no—
372#102—22no—
373#102—1222no—
374#102—1222no—
375#102—1224no—
376C22⋊C8—1221no—
377C22⋊C8—1221no—
378C4×C8—612no—
379C4×C8—1221no—
380C24—21no—
381C24—1221no—
382#41—42no—
383#41—42no—
384#41—2448no—
385#41—24416no—
386#41—2448no—
387C2×Q8—21no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—2442no—
391C2×Q8—2442no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—41no—
398C2×D4—612no—
399C2×D4—1228no—
400C2×D4—1221no—
401C2×D4—2442no—
402C2×D4—2448no—
403C2×D4#1175819914yescomplemented
404C2×D4C2×C22⋊D6112yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—44no—
409C22×C4—41no—
410C22×C4—1228no—
411C22×C4—1221no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—2448no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421SD16—44no—
422D8—44no—
423D8—2444no—
424M16—42no—
425C2×C8—22no—
426C2×C8—1222no—
427C2×C8—1222no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—612no—
432C4⋊C4—614no—
433C4⋊C4—1228no—
434C4⋊C4—1222no—
435C4⋊C4—1221no—
436C4⋊C4—1222no—
437C4⋊C4—1222no—
438C4⋊C4C2×C22⋊D6112yes—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—1228no—
444C22⋊C4—1222no—
445C22⋊C4—1222no—
446C42D4×D6111yescharacteristic
447C42D4×D6111yescharacteristic
448C42C2×C22⋊D6111yescharacteristic
449C42C23×D6111yescharacteristic
450C23—22no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—1222no—
456C23—2442no—
457C23—2442no—
458C23—2442no—
459C23#1451974511yescharacteristic
460Q8—41no—
461Q8—41no—
462Q8—2442no—
463Q8—2442no—
464Q8—2444no—
465D4—22no—
466D4—24no—
467D4—42no—
468D4—41no—
469D4—41no—
470D4—42no—
471D4—82no—
472D4—1222no—
473D4—1222no—
474D4—2442no—
475D4—24416no—
476D4C2×#1175819914yes—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—41no—
484C2×C4—82no—
485C2×C4—1222no—
486C2×C4—1224no—
487C2×C4—24416no—
488C2×C4—2442no—
489C2×C4—2444no—
490C2×C4—2442no—
491C2×C4—2442no—
492C2×C4—2444no—
493C2×C4#1451974511yescharacteristic
494C2×C4#1451974511yescharacteristic
495C2×C4#1451974511yescharacteristic
496C2×C4C2×#1175819912yes—
497C2×C4#1451980212yes—
498C2×C4C2×D4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4C22×C22⋊D6111yescharacteristic
501C8—42no—
502C8—2442no—
503C8—2442no—
504C22—21no—
505C22—21no—
506C22—24no—
507C22—48no—
508C22—42no—
509C22—42no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—4884no—
514C22—2442no—
515C22—2442no—
516C22—2442no—
517C22#1683989512yes—
518C22C2×#1451974511yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—81no—
524C4—2442no—
525C4—2444no—
526C4#1683989512yes—
527C4C22×#1175819911yescharacteristic
528C4C2×#1451980211yescharacteristic
529C4#1684055812yes—
530C2—22no—
531C2—44no—
532C2—161no—
533C2—4882no—
534C2C2×#1683989511yescharacteristic
535C2#2145505811yescharacteristic
536C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3036C2
socleC22×C61C2×#120
hypercenter#3036D61
hypocenterC61#18756
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
1. C2×#16840416C2semidirect
2. #21456937C2semidirect
3. #21455815C2semidirect
4. #21445052C2semidirect
5. #21444906C2semidirect
6. #21444492C2semidirect
7. #21444243C2semidirect
8. #21444210C2semidirect
9. C61×#3036C2semidirect
21. #16840165C22semidirect
22. #16840108C22semidirect
26. #16840039C22semidirect
27. #16839954C22semidirect
32. #16839509C22semidirect
33. D4⋊#92418C22semidirect
34. #16839489C22semidirect
39. C61×#461C22semidirect
40. C61×#459C22semidirect
122. C2×D4×C61D8semidirect
265. #3036D61semidirect
299. #461C2×D61semidirect
301. #459C2×D61semidirect
311. C61#18756semidirect
403. C2×D4#11758199semidirect

Maximal subgroups

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

subgroupindex
C2×#168404162
#214569372
#214558152
#214450522
#214449062
#214444922
#214442432
#214442102
C61×#30362
#1875661