Finite Group Databasebeta

← #29658583

Subgroups — #29658583

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×#16840417C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21455279C211yesmaximal, characteristic, complemented
4#21445053C214yesmaximal, complemented
5#21444907C214yesmaximal, complemented
6#21444490C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444209C211yesmaximal, characteristic, complemented
9C61×#3100C211yesmaximal, characteristic, complemented
10C22×#11758222—21no—
11C2×#14519802—21no—
12#16840417—22no—
13#16840417—22no—
14#16840417—22no—
15#16840417C2214yes—
16C2×#14519777C2211yescharacteristic
17C2×#14519759—22no—
18C2×#14519756—21no—
19C2×#14519748—21no—
20C2×C4×D244C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16840165C2212yescomplemented
23#16840108C2212yescomplemented
24#16840073—22no—
25#16840042—22no—
26#16839974—22no—
27#16839964C2212yescomplemented
28#16839954C2212yescomplemented
29#16839815C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839763—21no—
32#16839507C2214yescomplemented
33D4⋊#92418C2214yescomplemented
34#16839489C2214yescomplemented
35C61×#532C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#480—21no—
38C61×#474C2212yescomplemented
39C61×#459C2212yescomplemented
40C61×#447—21no—
41C4×D8×C61—22no—
42C61×#431C2211yescharacteristic
43C2×#11758222—24no—
44C2×#11758222—24no—
45C2×#11758222—44no—
46C2×#11758222—44no—
47C22×Dic122—21no—
48C22×D244D411yescharacteristic
49C22×C22⋊D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52#14519802—44no—
53C2×#11758201—22no—
54C2×#11758199—22no—
55C2×#11758194—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777D412yes—
59#14519777C2312yes—
60#14519759—28no—
61#14519759—28no—
62#14519756—24no—
63#14519756—24no—
64#14519748—24no—
65#14519748—24no—
66C4×D244—21no—
67C4×D244—21no—
68C4×D244D412yes—
69C4×D244C2312yes—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D412yes—
73C2×#11758196—22no—
74C4⋊C4×D61—24no—
75C4⋊C4×D61—24no—
76C2×C22⋊Dic61—22no—
77C2×#11758198—21no—
78#14519710—24no—
79C4⋊#92418—21no—
80C4⋊#92418—21no—
81C4⋊#92418C2312yes—
82#14519701—22no—
83#14519698—22no—
84#14519698—24no—
85C22⋊#92418—21no—
86C4×#92418C2312yes—
87C2×#41×C61—21no—
88C22×D4×C61D411yescharacteristic
89C23×C4×C61—21no—
90C2×D8×C61—22no—
91C4⋊Q8×C61C2311yescharacteristic
92#127×C61C2311yescharacteristic
93#124×C61—21no—
94#121×C61—21no—
95#121×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61D412yes—
101C4×D4×C61C2312yes—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C42×C61C2311yescharacteristic
105#107×C61—22no—
106C4⋊C8×C61C2312yes—
107#102×C61—22no—
108#102×C61—22no—
109C22⋊C8×C61—21no—
110C4×C8×C61—21no—
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—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—42no—
121C2×D4×C61—41no—
122C2×D4×C61C2×D412yes—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
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—22no—
135C22×Dic61—21no—
136#11758222—48no—
137#11758222—416no—
138#11758222—48no—
139C2×D244—21no—
140C2×D244—22no—
141C2×D244D814yescomplemented
142C2×D244C2×D412yes—
143C2×C4×D61—28no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—44no—
147C2×C4×D61—48no—
148C2×Dic122—22no—
149C2×Dic122—22no—
150C2×Dic122—42no—
151C2×Dic122—42no—
152D8×C61—44no—
153C2×C8×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—41no—
161C22⋊C4×C61—41no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2411yescharacteristic
166C22⋊Dic61—28no—
167#11758201—44no—
168#11758199—44no—
169#11758198—22no—
170#11758198—22no—
171C4⋊Dic61—22no—
172C4⋊Dic61—21no—
173C4⋊Dic61—22no—
174C4⋊Dic61—22no—
175C4⋊Dic61C2×D412yes—
176C4⋊Dic61C2×D414yes—
177#11758196—28no—
178#11758194—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C23×C61—22no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—41no—
186C23×C61#12011yescharacteristic
187C22×D61—22no—
188C22×D61—24no—
189C22×D61—42no—
190C22×D61—42no—
191Q8×C61—41no—
192Q8×C61—41no—
193D4×C61—22no—
194D4×C61—22no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—41no—
198D4×C61—41no—
199D4×C61—42no—
200D4×C61—82no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—41no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61—42no—
207C2×C4×C61—44no—
208C2×C4×C61—41no—
209C2×C4×C61—82no—
210C2×C4×C61#12011yescharacteristic
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61C2×D812yes—
214C2×C4×C61AGL(1,ℤ/8)12yes—
215C2×C4×C61C22×D411yescharacteristic
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C22⋊D61—416no—
219C2×Dic61—22no—
220C2×Dic61—24no—
221C2×Dic61—42no—
222C2×Dic61—44no—
223D244—22no—
224D244—24no—
225D244C2×D814yes—
226C4×D61—44no—
227C4×D61—416no—
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—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—84no—
242C22×C61—81no—
243C22×C61—81no—
244C22×C61—81no—
245C22×C61#44712yes—
246C22×C61C2×#12011yescharacteristic
247C2×D61—24no—
248C2×D61—48no—
249C2×D61—42no—
250C2×D61—42no—
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#3100D6111yescharacteristic, complemented
265#3036—611no—
266#1647—614no—
267#1625—614no—
268#1546—611no—
269#1486—611no—
270#1478—611no—
271C2×C61—22no—
272C2×C61—82no—
273C2×C61—161no—
274C2×C61C2×#44711yescharacteristic
275C2×C61#300511yescharacteristic
276C2×C61#302011yescharacteristic
277D61—44no—
278C22×#41—1221no—
279C2×AGL(1,ℤ/8)—1221no—
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—21no—
295#480—1222no—
296#474—1222no—
297#474C2×D6112yescomplemented
298#463—1222no—
299#461—612no—
300#459—612no—
301#459C2×D6112yescomplemented
302#447—21no—
303C4×D8—22no—
304#431—611no—
305#431C2×D6111yescharacteristic
306#423—611no—
307#417—1221no—
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)—2444no—
325C2×SD16—1222no—
326C2×D8—22no—
327C2×D8—1222no—
328C2×M16—1221no—
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—1221no—
359C2×C4⋊C4—1224no—
360C2×C4⋊C4C22⋊D6111yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—1222no—
363C2×C22⋊C4—1221no—
364C2×C42C22×D6111yescharacteristic
365#107—22no—
366#107—1224no—
367C4⋊C8—612no—
368C4⋊C8—1221no—
369C4⋊C8—1221no—
370C4⋊C8C22×D6112yes—
371#103—1222no—
372#102—22no—
373#102—22no—
374#102—1222no—
375#102—1224no—
376C22⋊C8—21no—
377C22⋊C8—1221no—
378C4×C8—21no—
379C4×C8—612no—
380C24—21no—
381C24—1221no—
382#41—42no—
383#41—42no—
384#41—2448no—
385#41—2448no—
386#41—24416no—
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—42no—
397C2×D4—41no—
398C2×D4—612no—
399C2×D4—614no—
400C2×D4—1228no—
401C2×D4—1221no—
402C2×D4—1222no—
403C2×D4—2448no—
404C2×D4C2×C22⋊D6112yes—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—1228no—
412C22×C4—1221no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—2444no—
417C22×C4—2448no—
418C22×C4D4×D6111yescharacteristic
419C22×C4D4×D6111yescharacteristic
420C22×C4C2×C22⋊D6111yescharacteristic
421SD16—2444no—
422D8—44no—
423D8—2444no—
424M16—2442no—
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—42no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—1222no—
456C23—1224no—
457C23—2442no—
458C23—2442no—
459C23#1451974511yescharacteristic
460Q8—41no—
461Q8—41no—
462Q8—2442no—
463Q8—2442no—
464Q8—2444no—
465D4—22no—
466D4—22no—
467D4—42no—
468D4—42no—
469D4—41no—
470D4—41no—
471D4—42no—
472D4—82no—
473D4—614no—
474D4—1222no—
475D4—1224no—
476D4—24416no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—1222no—
487C2×C4—1224no—
488C2×C4—2444no—
489C2×C4—24416no—
490C2×C4—2442no—
491C2×C4—2444no—
492C2×C4—2444no—
493C2×C4#1451974511yescharacteristic
494C2×C4#1451974511yescharacteristic
495C2×C4#1451974511yescharacteristic
496C2×C4D8×D6112yes—
497C2×C4#1451980112yes—
498C2×C4C2×D4×D6111yescharacteristic
499C2×C4C2×D4×D6111yescharacteristic
500C2×C4C22×C22⋊D6111yescharacteristic
501C8—42no—
502C8—2442no—
503C8—2442no—
504C22—21no—
505C22—21no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—1224no—
514C22—2448no—
515C22—2442no—
516C22—2442no—
517C22#1683989412yes—
518C22C2×#1451974511yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—81no—
524C4—2444no—
525C4—2442no—
526C4#1683989412yes—
527C4C2×D8×D6111yescharacteristic
528C4C2×#1451980111yescharacteristic
529C4#1684056012yes—
530C2—22no—
531C2—82no—
532C2—161no—
533C2—2444no—
534C2C2×#1683989411yescharacteristic
535C2#2145505911yescharacteristic
536C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3100C2
socleC22×C61C2×#120
hypercenter#3100D61
hypocenterC61#18756
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. C2×#16840417C2semidirect
2. #21456937C2semidirect
3. #21455279C2semidirect
4. #21445053C2semidirect
5. #21444907C2semidirect
6. #21444490C2semidirect
7. #21444244C2semidirect
8. #21444209C2semidirect
9. C61×#3100C2semidirect
22. #16840165C22semidirect
23. #16840108C22semidirect
27. #16839964C22semidirect
28. #16839954C22semidirect
32. #16839507C22semidirect
33. D4⋊#92418C22semidirect
34. #16839489C22semidirect
38. C61×#474C22semidirect
39. C61×#459C22semidirect
141. C2×D244D8semidirect
264. #3100D61semidirect
297. #474C2×D61semidirect
301. #459C2×D61semidirect
311. C61#18756semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
#214569372
#214552792
#214450532
#214449072
#214444902
#214442442
#214442092
C61×#31002
#1875661