Finite Group Databasebeta

← #29648010

Subgroups — #29648010

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#21459797C211yesmaximal, characteristic
2#21454247C211yesmaximal, characteristic, complemented
3#21444440C212yesmaximal, complemented
4#21444435C211yesmaximal, characteristic, complemented
5#21444427C211yesmaximal, characteristic, complemented
6#21444422C212yesmaximal, complemented
7#21444413C211yesmaximal, characteristic, complemented
8#21444408C211yesmaximal, characteristic, complemented
9#21444405C212yesmaximal, complemented
10C2×#16839503C212yesmaximal, complemented
11C61×#2970C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13#16840490C411yescharacteristic
14#16840472C411yescharacteristic
15#16840469C411yescharacteristic
16#16840458C412yes—
17#16840452C412yes—
18#16840451—21no—
19#16840445—22no—
20#16840416C411yescharacteristic
21#16840411—21no—
22#16840370—21no—
23#16840354—22no—
24#16840319C2211yescharacteristic
25#16840296C2212yes—
26#16840296C2211yescharacteristic
27C2×#14519769C2212yes—
28#16840241—21no—
29#16839718—21no—
30#16839718—22no—
31C4×#11758194C2212yes—
32C4×#11758194C2211yescharacteristic
33C2×#14519688C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839503C2211yescharacteristic
36#16839503C2212yes—
37#16839503C2212yes—
38#16839503C2212yes—
39#16839503C2212yes—
40#16839503C2211yescharacteristic
41#16839503C2212yes—
42#16839501C2211yescharacteristic
43#16839501C2212yes—
44#16839501C2211yescharacteristic
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2212yes—
50C61×#432C2211yescharacteristic
51C4×M16×C61C2211yescharacteristic
52C2×#97×C61C2212yes—
53C2×#11758224—21no—
54#14519784—28no—
55C2×#11758194—22no—
56C2×#11758194—21no—
57C22×#92418—22no—
58C22×#92418—21no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519775—21no—
62#14519772—22no—
63#14519771—22no—
64#14519771—21no—
65#14519770—22no—
66#14519769—22no—
67#14519769—22no—
68#14519769C2×C412yes—
69#14519769C2×C412yes—
70#14519769C2311yescharacteristic
71#14519769C2312yes—
72#14519768—21no—
73#14519767—22no—
74#14519767—22no—
75#14519765C2311yescharacteristic
76#14519761—22no—
77#14519760—21no—
78#14519760—22no—
79#14519759—22no—
80#14519758—22no—
81#14519757—21no—
82#14519756—21no—
83#14519756—21no—
84#14519756C2×C411yescharacteristic
85#14519756C2×C412yes—
86#14519755—22no—
87#14519754—22no—
88#14519753—21no—
89#14519751C2×C411yescharacteristic
90#14519751C2×C412yes—
91#14519749—22no—
92C4×Dic122—21no—
93C4×Dic122—21no—
94C4×Dic122C2×C412yes—
95C4×Dic122C2×C411yescharacteristic
96#14519742—21no—
97C4×D244C2×C412yes—
98C4×D244C2×C411yescharacteristic
99D4×Dic61—21no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102#14519737—22no—
103#14519736C2×C412yes—
104#14519734—22no—
105#14519734—21no—
106#14519733—22no—
107C2×C4⋊Dic61C2×C412yes—
108C2×C4⋊Dic61C2×C412yes—
109C2×#11758196—22no—
110C2×#11758196—22no—
111C4⋊C4×D61—21no—
112#14519688—22no—
113#14519688—21no—
114#14519688C2311yescharacteristic
115#14519688C2312yes—
116#14519688C2311yescharacteristic
117C4×#92418—22no—
118C4×#92418—21no—
119C2×#41×C61D411yescharacteristic
120C2×M16×C61—21no—
121C22×C8×C61—22no—
122C4×Q8×C61C2×C411yescharacteristic
123C4×Q8×C61D411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61C2×C411yescharacteristic
126C4×D4×C61D411yescharacteristic
127C4×D4×C61D412yes—
128#117×C61D411yescharacteristic
129#117×C61D412yes—
130C2×C42×C61C2312yes—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—21no—
133C4⋊C8×C61—22no—
134C22⋊C8×C61—22no—
135C22⋊C8×C61—21no—
136#97×C61—22no—
137#97×C61C2311yescharacteristic
138#97×C61C2312yes—
139#97×C61C2311yescharacteristic
140C4×C8×C61—21no—
141#41×C61—24no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C2×D4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155#11758224—44no—
156C2×D244C22×C411yescharacteristic
157C2×C4×D61—21no—
158C2×C4×D61—22no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122C22×C411yescharacteristic
162M16×C61—42no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—22no—
166C2×C8×C61—44no—
167C4⋊C4×C61C22⋊C411yescharacteristic
168C4⋊C4×C61C22⋊C412yes—
169C4⋊C4×C61C22⋊C411yescharacteristic
170C4⋊C4×C61C22⋊C412yes—
171C22⋊C4×C61C22⋊C412yes—
172C22⋊C4×C61C22⋊C412yes—
173C22⋊C4×C61C22⋊C412yes—
174C42×C61C22×C412yes—
175C42×C61C22×C411yescharacteristic
176C42×C61C2×D412yes—
177C42×C61C2×D411yescharacteristic
178C42×C61C2×D411yescharacteristic
179C42×C61C2×D412yes—
180C42×C61C2411yescharacteristic
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183C22⋊Dic61—21no—
184#11758198—21no—
185#11758198—22no—
186#11758198—22no—
187#11758198—21no—
188#11758198—21no—
189#11758198—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22×C411yescharacteristic
195C4⋊Dic61C22×C411yescharacteristic
196C4⋊Dic61C22×C412yes—
197C4⋊Dic61C22×C412yes—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—21no—
203#11758196—22no—
204#11758196—21no—
205#11758196—22no—
206#11758196—21no—
207#11758196—21no—
208#11758196—22no—
209#11758196—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—21no—
213#11758194—28no—
214#11758194—24no—
215C2×#92418—28no—
216C2×#92418—24no—
217C2×#92418—21no—
218C2×#92418—22no—
219C2×#92418—21no—
220C23×C61C2×C22⋊C411yescharacteristic
221C23×C61C2×C22⋊C412yes—
222C22×D61—21no—
223Q8×C61—22no—
224D4×C61—24no—
225D4×C61#10414yes—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—42no—
231C2×C4×C61—41no—
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C412yes—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C2×C22⋊C412yes—
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C23×C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C22⋊D61—44no—
242C22⋊D61—81no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—44no—
249C2×Dic61—82no—
250D244—41no—
251C4×D61—42no—
252C4×D61—82no—
253Dic122—42no—
254Dic122—41no—
255Dic122—81no—
256C8×C61—22no—
257C8×C61—42no—
258C8×C61—41no—
259#92418—22no—
260#92418—24no—
261#92418—22no—
262#17653—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61C2×#10414yes—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—41no—
271C2×D61—41no—
272C2×D61—41no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—22no—
276C4×C61—21no—
277C4×C61—22no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—22no—
281C4×C61C2×#10412yes—
282Dic61—41no—
283Dic61—42no—
284Dic61—41no—
285Dic61—42no—
286Dic61—41no—
287#3515—611no—
288#2970D6111yescharacteristic, complemented
289#2875—611no—
290#1532—612no—
291#1531—611no—
292#1529—611no—
293#1528—612no—
294#1526—611no—
295#1524—611no—
296#1524—612no—
297C2×#330—612no—
298C2×C61—24no—
299C2×C61—41no—
300C2×C61#288911yescharacteristic
301C2×C61C22×#10411yescharacteristic
302C2×C61#290111yescharacteristic
303D61—81no—
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—611no—
318#529—612no—
319C2×#125—612no—
320#520—1221no—
321C4×#41C2×D6111yescharacteristic
322#446—21no—
323#435—22no—
324C8×D4—21no—
325#432C2×D6112yes—
326#432C2×D6111yescharacteristic
327#405—1221no—
328#405—1222no—
329C4×M16—612no—
330C4×M16—611no—
331C4×M16C2×D6111yescharacteristic
332C2×#97—612no—
333C2×#97—611no—
334C2×#97C2×D6112yes—
335#330—611no—
336#330—612no—
337#330—612no—
338#330—612no—
339#330—611no—
340#330—612no—
341#330—612no—
342#329—611no—
343#329—612no—
344#329—611no—
345C61#1765311yescharacteristic, complemented, Sylow
346C2×#41—1221no—
347C2×#41D24411yescharacteristic
348#131—1228no—
349C2×M16—21no—
350C2×M16—1222no—
351C2×M16—1221no—
352C22×C8—22no—
353C22×C8—1222no—
354C22×C8—1221no—
355C4⋊Q8—611no—
356C4⋊Q8—611no—
357C4⋊Q8—1221no—
358#126—1222no—
359#126—1222no—
360#126—1221no—
361#126—1222no—
362#125—612no—
363#125—612no—
364#125—611no—
365#125—612no—
366#125—1222no—
367#125—1222no—
368#125—1221no—
369#125—1222no—
370#125—1222no—
371#124—611no—
372#123—1222no—
373#123—1222no—
374#123—1221no—
375#123—1222no—
376#123—1222no—
377C22⋊Q8—611no—
378C22⋊Q8—612no—
379C22⋊Q8—1221no—
380C22⋊Q8—1221no—
381C22⋊Q8—1221no—
382C22⋊Q8—1222no—
383C22⋊Q8—1221no—
384C22⋊Q8—1222no—
385#121—612no—
386#121—611no—
387#121—1222no—
388C4×Q8—612no—
389C4×Q8—611no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392C4×Q8—1221no—
393C4×Q8C4×D6111yescharacteristic
394C4×Q8C22⋊D6111yescharacteristic
395C4×D4—611no—
396C4×D4—612no—
397C4×D4—1221no—
398C4×D4—1221no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4C4×D6111yescharacteristic
402C4×D4C4×D6112yes—
403C4×D4C22⋊D6111yescharacteristic
404C4×D4C22⋊D6112yes—
405#117—612no—
406#117—1222no—
407#117—1222no—
408#117—1221no—
409#117D24411yescharacteristic
410#117D24412yes—
411C2×C4⋊C4—612no—
412C2×C4⋊C4—612no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1221no—
416C2×C42C22×D6112yes—
417C2×C42C22×D6111yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—22no—
420C22⋊C8—22no—
421C22⋊C8—21no—
422#97—22no—
423#97—611no—
424#97—611no—
425#97—612no—
426#97—1222no—
427#97—1221no—
428#97C22×D6111yescharacteristic
429#97C22×D6112yes—
430#97C22×D6111yescharacteristic
431C4×C8—21no—
432C4×C8—1222no—
433C4×C8—1221no—
434#41—24no—
435#41—2444no—
436C2×Q8—611no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8#1175819811yescharacteristic
440C2×D4—611no—
441C2×D4—1221no—
442C2×D4—1222no—
443C2×D4#1175819812yes—
444C2×D4#1175819811yescharacteristic
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1221no—
449C22×C4C2×C4×D6112yes—
450C22×C4C2×C4×D6111yescharacteristic
451C22×C4C2×D24412yes—
452C22×C4C2×D24411yescharacteristic
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455M16—42no—
456M16—1228no—
457M16—1224no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—22no—
461C2×C8—44no—
462C2×C8—1228no—
463C2×C8—1224no—
464C2×C8—1221no—
465C2×C8—1222no—
466C2×C8—1221no—
467C4⋊C4—611no—
468C4⋊C4—611no—
469C4⋊C4—612no—
470C4⋊C4—612no—
471C4⋊C4—1221no—
472C4⋊C4—1222no—
473C4⋊C4—1221no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4#1175819811yescharacteristic
488C4⋊C4#1175819811yescharacteristic
489C4⋊C4#1175819812yes—
490C4⋊C4#1175819812yes—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1221no—
494C22⋊C4—1221no—
495C22⋊C4—1221no—
496C22⋊C4—1221no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1221no—
500C22⋊C4#1175819812yes—
501C22⋊C4#1175819812yes—
502C22⋊C4#1175819812yes—
503C42—1222no—
504C42—1222no—
505C42—1221no—
506C42C2×C4×D6112yes—
507C42C2×C4×D6111yescharacteristic
508C42C2×D24412yes—
509C42C2×D24411yescharacteristic
510C42C2×C22⋊D6111yescharacteristic
511C42C2×C22⋊D6112yes—
512C42C23×D6111yescharacteristic
513C23—1221no—
514C23C2×#1175819811yescharacteristic
515C23C2×#1175819812yes—
516Q8—22no—
517Q8—4881no—
518Q8—2441no—
519Q8—2442no—
520D4—24no—
521D4—4881no—
522D4—2441no—
523D4—2444no—
524D4#1451970314yes—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—1221no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—4882no—
537C2×C4—4882no—
538C2×C4—2444no—
539C2×C4—2442no—
540C2×C4C2×#1175819811yescharacteristic
541C2×C4C2×#1175819812yes—
542C2×C4C2×#1175819811yescharacteristic
543C2×C4C2×#1175819812yes—
544C2×C4C2×#1175819812yes—
545C2×C4C2×#1175819811yescharacteristic
546C2×C4C22×C4×D6111yescharacteristic
547C2×C4C22×C22⋊D6111yescharacteristic
548C2×C4C22×D24411yescharacteristic
549C8—22no—
550C8—42no—
551C8—41no—
552C8—1222no—
553C8—1224no—
554C8—1222no—
555C22—22no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—2441no—
561C22—2441no—
562C22—2441no—
563C22C2×#1451970314yes—
564C22C22×#1175819811yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—22no—
573C4—2441no—
574C4—2442no—
575C4—2441no—
576C4—2442no—
577C4—2441no—
578C4C2×#1451970312yes—
579C2—24no—
580C2—41no—
581C2—4881no—
582C2#2145430111yescharacteristic
583C2C22×#1451970311yescharacteristic
584C2#2145436711yescharacteristic
subgroupG/H
centerC22C22×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2970C2
socleC22×C61C22×C22⋊C4
prime radical#21459797C2
cosocleC42×C61C24
hypercenter#2970D61
hypocenterC61#17653
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #21454247C2semidirect
3. #21444440C2semidirect
4. #21444435C2semidirect
5. #21444427C2semidirect
6. #21444422C2semidirect
7. #21444413C2semidirect
8. #21444408C2semidirect
9. #21444405C2semidirect
10. C2×#16839503C2semidirect
11. C61×#2970C2semidirect
288. #2970D61semidirect
345. C61#17653semidirect

Maximal subgroups

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

subgroupindex
#214597972
#214542472
#214444402
#214444352
#214444272
#214444222
#214444132
#214444082
#214444052
C2×#168395032
C61×#29702
#1765361