Finite Group Databasebeta

← #29662795

Subgroups — #29662795

Order 15616.

760 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456279C211yesmaximal, characteristic, complemented
2#21455815C211yesmaximal, characteristic
3C2×#16840043C211yesmaximal, characteristic, complemented
4#21454554C211yesmaximal, characteristic, complemented
5#21445307C212yesmaximal, complemented
6#21445283C212yesmaximal, complemented
7#21445183C212yesmaximal, complemented
8#21445159C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444492C211yesmaximal, characteristic, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12C2×#14519807—21no—
13C2×#14519807—21no—
14C2×M16×D61—21no—
15C2×#14519777C2211yescharacteristic
16C2×C4×D244C2211yescharacteristic
17#16840157—22no—
18#16840141C2212yescomplemented
19#16840134C2212yescomplemented
20#16840073—21no—
21#16840043—21no—
22#16840043—21no—
23#16840043C2212yes—
24#16840043C2212yes—
25#16840039C2211yescharacteristic
26#16840039C2211yescharacteristic
27#16839954C2211yescharacteristic
28#16839954C2211yescharacteristic
29#16839895—21no—
30C4×#11758199—21no—
31C4×#11758199—21no—
32#16839839C2212yes—
33#16839832—22no—
34#16839827C2212yes—
35#16839815C2211yescharacteristic
36C2×#14519711—21no—
37C2×#14519699—21no—
38#16839725—21no—
39C4×#11758194C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839509C2212yescomplemented
43#16839509C2212yescomplemented
44C61×#532C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#331C2212yescomplemented
49C22×Dic122—21no—
50C22×D244D411yescharacteristic
51C22×C4×D61—21no—
52#14519807—44no—
53#14519807—44no—
54C2×#11758202—22no—
55C2×#11758202—22no—
56C2×#11758200—22no—
57C2×#11758200—22no—
58C2×#11758199—21no—
59C2×#11758199—21no—
60C2×#11758194D411yescharacteristic
61C2×#11758194D411yescharacteristic
62C2×#11758189—22no—
63M16×D61—44no—
64C2×C8×D61—22no—
65#14519777—21no—
66#14519777—21no—
67#14519777—21no—
68#14519777C2312yes—
69C4×D244—21no—
70C4×D244—21no—
71C4×D244D411yescharacteristic
72C4×D244D411yescharacteristic
73C4×D244C2311yescharacteristic
74C4×D244C2311yescharacteristic
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C2×#11758198—21no—
79#14519711—22no—
80#14519711—22no—
81#14519710—21no—
82#14519710—21no—
83C4⋊#92418C2311yescharacteristic
84C4⋊#92418C2311yescharacteristic
85#14519699—22no—
86#14519699—22no—
87#14519698—21no—
88#14519698—21no—
89#14519697—21no—
90#14519697—21no—
91C22⋊#92418—21no—
92#14519689—22no—
93#14519689—22no—
94#14519688D412yes—
95C4×#92418—21no—
96C4×#92418C2312yes—
97C2×#41×C61—21no—
98C2×M16×C61—21no—
99C22×C8×C61—21no—
100C4⋊Q8×C61C2311yescharacteristic
101#127×C61C2311yescharacteristic
102#124×C61—21no—
103#121×C61—21no—
104#121×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C4⋊C8×C61—21no—
109C4⋊C8×C61—21no—
110C4⋊C8×C61C2312yes—
111C4⋊C8×C61C2312yes—
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—41no—
120C22×C4×C61—41no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—41no—
126C2×D244—22no—
127C2×D244—21no—
128C2×D244—41no—
129C2×D244C2×D411yescharacteristic
130C2×D244C2×D411yescharacteristic
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—42no—
136C2×C4×D61—42no—
137C2×Dic122—22no—
138C2×Dic122—42no—
139C2×Dic122—42no—
140C2×Dic122—41no—
141M16×C61—42no—
142C2×C8×C61—22no—
143C2×C8×C61—22no—
144C2×C8×C61—42no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C22⋊C4×C61—41no—
148C22⋊C4×C61—41no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61#4111yescharacteristic
152C42×C61C2411yescharacteristic
153#11758202—44no—
154#11758202—44no—
155#11758200—44no—
156#11758200—44no—
157#11758199—42no—
158#11758199—42no—
159#11758198—21no—
160#11758198—21no—
161#11758198—21no—
162#11758198—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—41no—
167C4⋊Dic61—41no—
168C4⋊Dic61C2×D411yescharacteristic
169C4⋊Dic61C2×D411yescharacteristic
170#11758194—22no—
171#11758194D814yescomplemented
172C2×#92418—21no—
173C2×#92418—21no—
174C2×#92418C2×D412yes—
175C2×#92418C2×D412yes—
176#11758189—44no—
177C8×D61—44no—
178C23×C61—41no—
179C23×C61#12111yescharacteristic
180C22×D61—21no—
181C22×D61—21no—
182C22×D61—42no—
183C22×D61—42no—
184C22×D61—81no—
185Q8×C61—41no—
186Q8×C61—41no—
187D4×C61—41no—
188D4×C61—41no—
189D4×C61—41no—
190D4×C61—41no—
191D4×C61—81no—
192D4×C61—82no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—82no—
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61C2×D812yes—
202C2×C4×C61AGL(1,ℤ/8)12yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—82no—
211D244—21no—
212D244—21no—
213D244—21no—
214D244—42no—
215D244#13712yescomplemented
216C4×D61—41no—
217C4×D61—41no—
218C4×D61—41no—
219C4×D61—41no—
220C4×D61—42no—
221C4×D61—42no—
222Dic122—42no—
223Dic122—42no—
224Dic122—82no—
225C8×C61—42no—
226C8×C61—42no—
227#92418—22no—
228#92418—41no—
229#92418—41no—
230#92418C2×D814yes—
231#19163—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61—81no—
237C22×C61#45912yes—
238C22×C61C2×#12111yescharacteristic
239C2×D61—22no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C2×D61—84no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—21no—
249C4×C61—81no—
250C4×C61#45912yes—
251C4×C61C22×D811yescharacteristic
252C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
253C4×C61C2×#13711yescharacteristic
254C4×C61#57511yescharacteristic
255Dic61—41no—
256Dic61—41no—
257Dic61—82no—
258#3152—611no—
259#3100—611no—
260C2×#474—611no—
261D4×M16—611no—
262#1689—612no—
263#1685—612no—
264#1669—612no—
265#1665—612no—
266#1565—611no—
267#1547D6111yescharacteristic, complemented
268#1546—611no—
269C2×C61—22no—
270C2×C61—161no—
271C2×C61C2×#45911yescharacteristic
272C2×C61#305611yescharacteristic
273C2×C61#306911yescharacteristic
274D61—42no—
275D61—81no—
276C2×#137—1221no—
277C2×#137—1221no—
278C22×M16—1221no—
279#532C2×D6111yescharacteristic
280C2×C4⋊Q8—611no—
281C2×C4×D4—611no—
282#497—1222no—
283#494—612no—
284#492—612no—
285#480—1221no—
286#474—611no—
287#474—611no—
288#474—612no—
289#474—612no—
290#474—1221no—
291#474—1221no—
292#459—611no—
293#459—611no—
294#447—1221no—
295C4×D8—1221no—
296C4×D8—1221no—
297#436—612no—
298#435—1222no—
299C8×D4—612no—
300#431—611no—
301C2×#107—1221no—
302#423C2×D6111yescharacteristic
303C2×C4⋊C8C2×D6111yescharacteristic
304C2×#102—1221no—
305#407—1221no—
306C4×M16—611no—
307#335—612no—
308#334—612no—
309#332—612no—
310#332—612no—
311#332C2×D6112yescomplemented
312#331C2×D6112yescomplemented
313C61#1916311yescharacteristic, complemented, Sylow
314C2×#41—21no—
315C22×Q8—1221no—
316C22×D4—611no—
317C23×C4—1221no—
318#137—2444no—
319#137—2444no—
320C2×Q16—1222no—
321C2×Q16—1222no—
322C2×SD16—1222no—
323C2×SD16—1222no—
324C2×D8—1221no—
325C2×D8—1221no—
326C2×M16—21no—
327C2×M16—611no—
328C2×M16—611no—
329C2×M16—1222no—
330C2×M16—2444no—
331C22×C8—21no—
332C22×C8—1222no—
333C4⋊Q8—612no—
334C4⋊Q8—1221no—
335C4⋊Q8—1221no—
336C4⋊Q8—1221no—
337C4⋊Q8C22×D6111yescharacteristic
338#127C22×D6111yescharacteristic
339#124—21no—
340#121—21no—
341#121—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4—611no—
345C4×D4—611no—
346C4×D4—611no—
347C4×D4—611no—
348C4×D4—1221no—
349C4×D4—1221no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1221no—
352C2×C4⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C42C22×D6111yescharacteristic
355#107—1221no—
356#107—1221no—
357#107—1222no—
358#107—1222no—
359C4⋊C8—21no—
360C4⋊C8—21no—
361C4⋊C8—611no—
362C4⋊C8—611no—
363C4⋊C8C22×D6112yes—
364C4⋊C8C22×D6112yes—
365#102—1221no—
366#102—1221no—
367#102—1221no—
368#102—1221no—
369#102—1222no—
370#102—1222no—
371C22⋊C8—1222no—
372C22⋊C8—1222no—
373C22⋊C8—1221no—
374#97—612no—
375C4×C8—612no—
376C4×C8—1221no—
377C24—1221no—
378#41—42no—
379#41—42no—
380C2×Q8—21no—
381C2×Q8—1222no—
382C2×Q8—2442no—
383C2×Q8—2442no—
384C2×Q8—2441no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—611no—
390C2×D4—611no—
391C2×D4—1222no—
392C2×D4—1221no—
393C2×D4—2441no—
394C22×C4—41no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—1221no—
399C22×C4—1221no—
400C22×C4—2441no—
401C22×C4—2442no—
402C22×C4—2442no—
403C22×C4#1175822211yescharacteristic
404C22×C4D4×D6111yescharacteristic
405C22×C4D4×D6111yescharacteristic
406Q16—2444no—
407Q16—2444no—
408SD16—2444no—
409SD16—2444no—
410D8—2442no—
411D8—2442no—
412M16—42no—
413M16—614no—
414M16—1222no—
415M16—2444no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—42no—
419C2×C8—612no—
420C2×C8—612no—
421C2×C8—1221no—
422C2×C8—1221no—
423C2×C8—2444no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—611no—
427C4⋊C4—611no—
428C4⋊C4—1221no—
429C4⋊C4—1222no—
430C4⋊C4—1222no—
431C4⋊C4—2441no—
432C4⋊C4—2441no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—1221no—
436C22⋊C4—1221no—
437C22⋊C4—1221no—
438C22⋊C4—1221no—
439C42#1175822211yescharacteristic
440C42D4×D6111yescharacteristic
441C42D4×D6111yescharacteristic
442C42C23×D6111yescharacteristic
443C23—41no—
444C23—1221no—
445C23—1221no—
446C23—4881no—
447C23—2442no—
448C23—2442no—
449C23#1451975011yescharacteristic
450Q8—41no—
451Q8—41no—
452Q8—4882no—
453Q8—2442no—
454Q8—2442no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—41no—
459D4—81no—
460D4—82no—
461D4—612no—
462D4—1221no—
463D4—1221no—
464D4—1221no—
465D4—2442no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—1221no—
472C2×C4—1221no—
473C2×C4—4882no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4#1451975011yescharacteristic
483C2×C4#1451975011yescharacteristic
484C2×C4#1451975011yescharacteristic
485C2×C4D8×D6112yes—
486C2×C4#1451980412yes—
487C2×C4C2×D4×D6111yescharacteristic
488C2×C4C2×D4×D6111yescharacteristic
489C2×C4C2×#1175822211yescharacteristic
490C8—42no—
491C8—42no—
492C8—614no—
493C8—1222no—
494C8—2441no—
495C8—2441no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—1222no—
502C22—4884no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22—2441no—
507C22—2441no—
508C22#1683995612yes—
509C22C2×#1451975011yescharacteristic
510C4—21no—
511C4—21no—
512C4—21no—
513C4—81no—
514C4—4882no—
515C4—2441no—
516C4—2441no—
517C4#1683995612yes—
518C4C2×D8×D6111yescharacteristic
519C4C2×#1451980411yescharacteristic
520C4#137×D6111yescharacteristic
521C4#1684055811yescharacteristic
522C2—22no—
523C2—161no—
524C2—4881no—
525C2—2442no—
526C2C2×#1683995611yescharacteristic
527C2#2145545211yescharacteristic
528C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C2×#121
prime radical#21455815C2
hypercenter#1547D61
hypocenterC61#19163
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
1. #21456279C2semidirect
3. C2×#16840043C2semidirect
4. #21454554C2semidirect
5. #21445307C2semidirect
6. #21445283C2semidirect
7. #21445183C2semidirect
8. #21445159C2semidirect
9. #21444569C2semidirect
10. #21444492C2semidirect
11. C61×#1547C2semidirect
18. #16840141C22semidirect
19. #16840134C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839509C22semidirect
43. #16839509C22semidirect
47. C61×#332C22semidirect
48. C61×#331C22semidirect
171. #11758194D8semidirect
215. D244#137semidirect
267. #1547D61semidirect
311. #332C2×D61semidirect
312. #331C2×D61semidirect
313. C61#19163semidirect

Maximal subgroups

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

subgroupindex
#214562792
#214558152
C2×#168400432
#214545542
#214453072
#214452832
#214451832
#214451592
#214445692
#214444922
C61×#15472
#1916361