Finite Group Databasebeta

← #29648009

Subgroups — #29648009

Order 15616.

964 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459799C211yesmaximal, characteristic
2#21454831C211yesmaximal, characteristic, complemented
3#21454248C211yesmaximal, characteristic, complemented
4#21444436C212yesmaximal, complemented
5#21444435C211yesmaximal, characteristic, complemented
6#21444419C212yesmaximal, complemented
7#21444413C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic, complemented
9#21444404C212yesmaximal, complemented
10#330×D61C212yesmaximal, complemented
11C61×#1529C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840491C411yescharacteristic
14#16840472C411yescharacteristic
15#16840469C411yescharacteristic
16#16840459C412yes—
17#16840452—21no—
18#16840448—22no—
19#16840446C412yes—
20#16840417—21no—
21#16840417C411yescharacteristic
22#16840374—21no—
23#16840355—22no—
24#16840319C2211yescharacteristic
25#16840291C2212yes—
26#125×D61C2212yes—
27#16840242—21no—
28C4×#11758222C2211yescharacteristic
29#16839889—21no—
30#16839836—22no—
31#16839827—21no—
32#16839817C2212yes—
33#16839720—21no—
34#16839716—22no—
35C4×#11758194C2211yescharacteristic
36#16839714C2212yes—
37C2×#14519688C2211yescharacteristic
38#97×D61C2212yes—
39#97×D61C2212yes—
40#16839504C2211yescharacteristic
41#16839504C2211yescharacteristic
42#16839503C2212yes—
43#16839503C2212yes—
44#16839503C2212yes—
45#16839503C2212yes—
46#16839502C2211yescharacteristic
47#16839502C2211yescharacteristic
48C61×#529C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×#330C2212yes—
52C61×#329C2212yes—
53C2×#11758222—21no—
54C2×#11758222D411yescharacteristic
55#14519785—28no—
56C2×#11758194—21no—
57C2×#11758189—22no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60C2×C8×D61—22no—
61#14519777—21no—
62#14519777C2×C411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519770—22no—
65#14519770—22no—
66#14519770—22no—
67#14519769—21no—
68#14519769C2312yes—
69#14519768—22no—
70#14519768C2×C412yes—
71#14519768C2×C412yes—
72#14519767—22no—
73#14519767—22no—
74#14519767—22no—
75#14519759—22no—
76#14519759—22no—
77#14519759—22no—
78#14519758—22no—
79#14519756—21no—
80#14519756—21no—
81#14519756—21no—
82#14519756—21no—
83#14519755—22no—
84#14519754—22no—
85#14519753C2×C412yes—
86#14519748—22no—
87#14519748C2×C412yes—
88Q8×Dic61C2×C412yes—
89C4×Dic122—21no—
90C4×Dic122—21no—
91C4×Dic122—21no—
92C4×Dic122C2×C411yescharacteristic
93C4×Dic122D411yescharacteristic
94C4×D244—21no—
95C4×D244C2×C411yescharacteristic
96C4×D244D411yescharacteristic
97D4×Dic61C2×C412yes—
98C4×C22⋊D61C2×C412yes—
99C4×C22⋊D61D412yes—
100#14519737—22no—
101#14519737—22no—
102#14519736C2×C412yes—
103#14519736D411yescharacteristic
104#14519735—22no—
105#14519734D412yes—
106#14519733—22no—
107C2×C4⋊Dic61—21no—
108C4⋊C4×D61—22no—
109C4⋊C4×D61—22no—
110C4⋊C4×D61C2×C412yes—
111C4⋊C4×D61C2×C412yes—
112C42×D61C2312yes—
113#14519706—22no—
114#14519689—22no—
115#14519688—21no—
116#14519688C2311yescharacteristic
117#14519688C2311yescharacteristic
118#14519688C2311yescharacteristic
119#14519688C2311yescharacteristic
120#14519687—22no—
121#14519687—22no—
122C4×#92418—21no—
123C8×Dic61—22no—
124C2×M16×C61—21no—
125#126×C61—21no—
126#125×C61C2311yescharacteristic
127#124×C61C2311yescharacteristic
128#123×C61—21no—
129C22⋊Q8×C61C2×C411yescharacteristic
130#121×C61C2×C411yescharacteristic
131C4×Q8×C61C2×C411yescharacteristic
132C4×D4×C61—21no—
133C4×D4×C61C2×C411yescharacteristic
134#117×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61C2312yes—
139#97×C61C2312yes—
140C4×C8×C61—21no—
141C2×Q8×C61C22×C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61C22×C411yescharacteristic
144C22×C4×C61—21no—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×C22⋊D61—22no—
149C2×C22⋊D61C22⋊C412yes—
150C22×Dic61—22no—
151#11758222—24no—
152#11758222—44no—
153C2×D244C22⋊C411yescharacteristic
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61C22×C412yes—
157C2×C4×D61C2×D412yes—
158C2×C4×D61C2×D412yes—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C22⋊C411yescharacteristic
162M16×C61—24no—
163C2×C8×C61—22no—
164C2×C8×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C22×C411yescharacteristic
170C4⋊C4×C61C22×C411yescharacteristic
171C22⋊C4×C61—21no—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C42×C61—21no—
177C42×C61C22×C411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D411yescharacteristic
180C42×C61C2411yescharacteristic
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—22no—
184C22⋊Dic61C22⋊C412yes—
185#11758198—22no—
186#11758198C22⋊C412yes—
187#11758198C22⋊C412yes—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C22⋊C411yescharacteristic
194C4⋊Dic61C22⋊C411yescharacteristic
195C4⋊Dic61C22×C412yes—
196C4⋊Dic61C22×C412yes—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196C22⋊C412yes—
205#11758196C22⋊C412yes—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61C22×C412yes—
209C4×Dic61C2×D412yes—
210C4×Dic61C2×D412yes—
211#11758194—42no—
212C2×#92418—24no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217#11758189—28no—
218C8×D61—28no—
219C8×D61—44no—
220C23×C61—21no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C412yes—
223Q8×C61—41no—
224D4×C61—41no—
225D4×C61—81no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—42no—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C23×C411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C22⋊D61—24no—
240C22⋊D61—44no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—82no—
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247D244#10414yes—
248C4×D61—24no—
249C4×D61—42no—
250C4×D61—42no—
251C4×D61—44no—
252C4×D61—82no—
253Dic122—22no—
254Dic122—42no—
255Dic122—81no—
256C8×C61—24no—
257C8×C61—42no—
258#92418—22no—
259#92418—22no—
260#92418—22no—
261#92418—41no—
262#17653—611nomaximal, Sylow
263C22×C61—21no—
264C22×C61—21no—
265C22×C61—21no—
266C22×C61—41no—
267C22×C61—41no—
268C22×C61—41no—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—22no—
271C2×D61—22no—
272C2×D61C2×#10414yes—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—41no—
279C4×C61—41no—
280C4×C61—41no—
281C4×C61C2×#10412yes—
282Dic61—22no—
283Dic61—22no—
284Dic61—22no—
285Dic61—42no—
286Dic61—42no—
287#3515—611no—
288#2970—611no—
289#2875—611no—
290#1532—612no—
291#1531—611no—
292#1529D6111yescharacteristic, complemented
293#1528—612no—
294#1526—611no—
295#1524—612no—
296#1524—611no—
297C2×#330—612no—
298C2×C61—41no—
299C2×C61—81no—
300C2×C61#288911yescharacteristic
301C2×C61C22×#10411yescharacteristic
302C2×C61#290111yescharacteristic
303D61—24no—
304C2×#131—1221no—
305#557—611no—
306#554—611no—
307#554—611no—
308#553—612no—
309#552—612no—
310#552—1221no—
311#552—1222no—
312#547—611no—
313#547—1221no—
314#541—1221no—
315#539—1222no—
316#533—611no—
317#529—612no—
318#529C2×D6111yescharacteristic
319C2×#125—612no—
320#520—1221no—
321C4×#41—611no—
322#446—1221no—
323#435—1222no—
324C8×D4—1221no—
325#432—612no—
326#432C2×D6111yescharacteristic
327#405—1222no—
328#405—1221no—
329C4×M16—611no—
330C4×M16—612no—
331C4×M16C2×D6111yescharacteristic
332C2×#97—611no—
333C2×#97—612no—
334C2×#97—612no—
335#330—611no—
336#330—611no—
337#330—612no—
338#330—612no—
339#330—612no—
340#330—612no—
341#330C2×D6112yes—
342#329—611no—
343#329—611no—
344#329C2×D6112yes—
345C61#1765311yescharacteristic, complemented, Sylow
346C2×#41—611no—
347C2×#41—1221no—
348#131—1228no—
349C2×M16—21no—
350C2×M16—1222no—
351C2×M16—1221no—
352C22×C8—1222no—
353C22×C8—1222no—
354C22×C8—1221no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358#126—21no—
359#126—1222no—
360#126—1222no—
361#126—1222no—
362#125—612no—
363#125—612no—
364#125—612no—
365#125—1221no—
366#125—1222no—
367#125—1222no—
368#125—1222no—
369#125—1222no—
370#125C22×D6111yescharacteristic
371#124C22×D6111yescharacteristic
372#123—21no—
373#123—1222no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377C22⋊Q8—612no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383C22⋊Q8—1221no—
384C22⋊Q8C4×D6111yescharacteristic
385#121—612no—
386#121—1222no—
387#121C4×D6111yescharacteristic
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8—1221no—
394C4×Q8C4×D6111yescharacteristic
395C4×D4—21no—
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—612no—
399C4×D4—611no—
400C4×D4—611no—
401C4×D4—1221no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4C4×D6111yescharacteristic
405#117—21no—
406#117—611no—
407#117—612no—
408#117—612no—
409#117—1222no—
410#117—1222no—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C42—612no—
417C2×C42C22×D6111yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—1222no—
420C22⋊C8—21no—
421C22⋊C8—1222no—
422#97—611no—
423#97—611no—
424#97—611no—
425#97—611no—
426#97—1222no—
427#97—1222no—
428#97—1221no—
429#97C22×D6112yes—
430#97C22×D6112yes—
431C4×C8—21no—
432C4×C8—1222no—
433C4×C8—1221no—
434#41—1224no—
435#41—2444no—
436C2×Q8—611no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8C2×C4×D6111yescharacteristic
440C2×D4—21no—
441C2×D4—612no—
442C2×D4—611no—
443C2×D4—1222no—
444C2×D4C2×C4×D6111yescharacteristic
445C22×C4—21no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4C2×C4×D6111yescharacteristic
453C22×C4D4×D6111yescharacteristic
454C22×C4D4×D6111yescharacteristic
455M16—24no—
456M16—1228no—
457M16—2442no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—1228no—
461C2×C8—1224no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—2444no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—611no—
472C4⋊C4—612no—
473C4⋊C4—612no—
474C4⋊C4—611no—
475C4⋊C4—612no—
476C4⋊C4—612no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1222no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4C2×C4×D6111yescharacteristic
490C4⋊C4C2×C4×D6111yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—612no—
497C22⋊C4—612no—
498C22⋊C4—612no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C42—21no—
504C42—612no—
505C42—612no—
506C42—612no—
507C42—1222no—
508C42—1222no—
509C42C2×C4×D6111yescharacteristic
510C42D4×D6111yescharacteristic
511C42D4×D6111yescharacteristic
512C42C23×D6111yescharacteristic
513C23—21no—
514C23—612no—
515C23C22⋊C4×D6111yescharacteristic
516Q8—41no—
517Q8—1222no—
518Q8—4881no—
519Q8—2442no—
520D4—41no—
521D4—81no—
522D4—614no—
523D4—1224no—
524D4—2444no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—612no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—4882no—
539C2×C4—4882no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2444no—
543C2×C4C22⋊C4×D6111yescharacteristic
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C22⋊C4×D6111yescharacteristic
546C2×C4C22×C4×D6111yescharacteristic
547C2×C4C2×D4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C8—24no—
550C8—42no—
551C8—1222no—
552C8—1222no—
553C8—1222no—
554C8—2441no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—614no—
562C22—1222no—
563C22—1222no—
564C22C2×C22⋊C4×D6111yescharacteristic
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—1222no—
574C4—1222no—
575C4—1222no—
576C4—2442no—
577C4—2442no—
578C4#104×D6112yes—
579C2—41no—
580C2—81no—
581C2—1224no—
582C2#2145430011yescharacteristic
583C2C2×#104×D6111yescharacteristic
584C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1529C2
socleC22×C61C22×C22⋊C4
prime radical#21459799C2
cosocleC42×C61C24
hypercenter#1529D61
hypocenterC61#17653
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1529

Complements

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

NH ≅ G/Ntype
2. #21454831C2semidirect
3. #21454248C2semidirect
4. #21444436C2semidirect
5. #21444435C2semidirect
6. #21444419C2semidirect
7. #21444413C2semidirect
8. #21444407C2semidirect
9. #21444404C2semidirect
10. #330×D61C2semidirect
11. C61×#1529C2semidirect
292. #1529D61semidirect
345. C61#17653semidirect

Maximal subgroups

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

subgroupindex
#214597992
#214548312
#214542482
#214444362
#214444352
#214444192
#214444132
#214444072
#214444042
#330×D612
C61×#15292
#1765361