Finite Group Databasebeta

← #29648282

Subgroups — #29648282

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459717C211yesmaximal, characteristic
2#21454836C211yesmaximal, characteristic, complemented
3#21454831C211yesmaximal, characteristic, complemented
4#21444449C212yesmaximal, complemented
5#21444438C212yesmaximal, complemented
6#21444425C212yesmaximal, complemented
7#21444418C211yesmaximal, characteristic, complemented
8#21444418C211yesmaximal, characteristic, complemented
9#21444414C211yesmaximal, characteristic, complemented
10#21444406C212yesmaximal, complemented
11C61×#1537C211yesmaximal, characteristic, complemented
12#16840497—21no—
13#16840497—21no—
14#16840485—21no—
15#16840485—21no—
16#16840439—22no—
17#16840439—22no—
18#16840386C412yes—
19#16840386C412yes—
20#16840338C411yescharacteristic
21#16840331C411yescharacteristic
22#16840297C2212yes—
23#16840297C2212yes—
24C2×#14519765C2211yescharacteristic
25#16840239C411yescharacteristic
26#16840236C411yescharacteristic
27C4×#11758222C2211yescharacteristic
28#16839889—21no—
29#16839886—21no—
30#16839839—21no—
31#16839837—22no—
32#16839836—22no—
33#16839827—21no—
34#16839817C2212yescomplemented
35#16839817C2212yescomplemented
36C4×#11758194C2211yescharacteristic
37C4×#11758194C2211yescharacteristic
38#16839714C2212yescomplemented
39#97×D61C2212yescomplemented
40#16839504C2211yescharacteristic, complemented
41#16839504C2211yescharacteristic, complemented
42#16839504C2211yescharacteristic, complemented
43#16839504C2211yescharacteristic, complemented
44#16839501C2212yes—
45#16839501C2212yes—
46#16839501C2212yes—
47#16839501C2212yes—
48C61×#533C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C61×#432C2211yescharacteristic
51C61×#330C2212yescomplemented
52C61×#330C2212yescomplemented
53C2×#11758222C2×C411yescharacteristic
54C22×Dic122C2×C411yescharacteristic
55C22×D244C2×C411yescharacteristic
56C2×#11758194—21no—
57C2×#11758194—21no—
58C2×#11758189—22no—
59C2×C8×D61—22no—
60#14519777—21no—
61#14519774—21no—
62#14519772—22no—
63#14519772—22no—
64#14519772—22no—
65#14519772—22no—
66#14519766—22no—
67#14519766—22no—
68#14519765—21no—
69#14519765—21no—
70#14519765—21no—
71#14519765—21no—
72#14519765C2×C412yes—
73#14519765C2312yes—
74#14519762—22no—
75#14519762—22no—
76#14519762—22no—
77#14519762—22no—
78#14519761—22no—
79#14519757C2×C412yes—
80#14519757C2×C412yes—
81#14519753—22no—
82#14519753—22no—
83#14519752C2×C412yes—
84#14519752C2×C412yes—
85#14519747—22no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122D411yescharacteristic
89C4×Dic122D411yescharacteristic
90#14519742C2×C412yes—
91#14519742C2×C412yes—
92C4×D244—21no—
93C4×D244—21no—
94C4×D244D411yescharacteristic
95C4×D244D411yescharacteristic
96#14519739—22no—
97#14519739—22no—
98#14519739C2×C412yes—
99#14519739C2×C412yes—
100C4×C22⋊D61D412yes—
101C4×C22⋊D61D412yes—
102#14519736C2×C411yescharacteristic
103#14519735—22no—
104#14519735—22no—
105#14519734C2×C412yes—
106#14519733—22no—
107C2×#11758198—21no—
108C2×#11758198—21no—
109C22⋊C4×D61—22no—
110C42×D61C2312yes—
111#14519706—22no—
112#14519706—22no—
113#14519689—22no—
114#14519689—22no—
115#14519688C2311yescharacteristic
116#14519688C2311yescharacteristic
117#14519688C2311yescharacteristic
118#14519688C2311yescharacteristic
119#14519687—22no—
120C4×#92418—21no—
121C4×#92418—21no—
122C8×Dic61—22no—
123C4⋊Q8×C61C2×C411yescharacteristic
124C4⋊Q8×C61C2×C411yescharacteristic
125#125×C61C2311yescharacteristic
126#125×C61C2311yescharacteristic
127C22⋊Q8×C61—21no—
128C22⋊Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61—21no—
131#117×C61C2×C411yescharacteristic
132#117×C61C2×C411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61C2312yes—
139#97×C61C2312yes—
140C23×D61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C2×C22⋊D61C22⋊C412yes—
147C22×Dic61—21no—
148#11758222—42no—
149C2×D244—21no—
150C2×D244—22no—
151C2×D244—22no—
152C2×D244—42no—
153C2×D244C22⋊C411yescharacteristic
154C2×D244C22×C412yes—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61C22×C412yes—
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×Dic122—21no—
161C2×Dic122—42no—
162C2×Dic122C22⋊C411yescharacteristic
163C2×Dic122C22×C412yes—
164C2×C8×C61—22no—
165C2×C8×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61C22×C411yescharacteristic
173C4⋊C4×C61C22×C411yescharacteristic
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61C22⋊C412yes—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198C22⋊C412yes—
196#11758198C22⋊C412yes—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C411yescharacteristic
201#11758196—22no—
202#11758196—22no—
203#11758196C22⋊C412yes—
204#11758196C22⋊C412yes—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61C22×C412yes—
208C4×Dic61C2×D412yes—
209C4×Dic61C2×D412yes—
210#11758194—42no—
211#11758194—42no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216#11758189—44no—
217C8×D61—44no—
218C23×C61C2×C22⋊C411yescharacteristic
219C22×D61—22no—
220C22×D61—42no—
221C22×D61—81no—
222C22×D61—81no—
223C22×D61C2×C22⋊C412yes—
224Q8×C61—81no—
225Q8×C61—81no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
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—42no—
240C2×Dic61—22no—
241C2×Dic61—42no—
242C2×Dic61C2×C22⋊C412yes—
243C2×Dic61C2×C22⋊C412yes—
244C2×Dic61C2×C22⋊C412yes—
245D244—22no—
246D244—42no—
247D244—42no—
248D244—82no—
249D244—82no—
250C4×D61—42no—
251C4×D61—42no—
252C4×D61—42no—
253C4×D61—82no—
254C4×D61—82no—
255Dic122—22no—
256Dic122—42no—
257Dic122—42no—
258C8×C61—42no—
259C8×C61—42no—
260#92418—22no—
261#92418—22no—
262#92418—41no—
263#92418—41no—
264#17691—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61C22×C22⋊C411yescharacteristic
269C2×D61—22no—
270C2×D61—22no—
271C2×D61—22no—
272C2×D61—42no—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—162no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61#41312yes—
286Dic61—22no—
287Dic61—22no—
288Dic61—22no—
289Dic61—42no—
290#3510—611no—
291#2971—611no—
292#2970—611no—
293#1537D6111yescharacteristic, complemented
294#1535—612no—
295#1532—612no—
296#1529—612no—
297#1527—611no—
298#1527—611no—
299#1526—611no—
300#1524—612no—
301C2×C61—41no—
302C2×C61C2×#41311yescharacteristic
303C2×C61#290111yescharacteristic
304C2×C61#290111yescharacteristic
305D61—42no—
306D61—82no—
307#559—1221no—
308#559—1221no—
309#556—1221no—
310#556—1221no—
311#551—1222no—
312#551—1222no—
313#542—612no—
314#542—612no—
315#536—611no—
316#535—611no—
317#533C2×D6111yescharacteristic
318#529—612no—
319#529—612no—
320C2×#124—611no—
321#519—611no—
322#518—611no—
323C4×#41—611no—
324#446—1221no—
325C8×Q8—1221no—
326#436—1222no—
327#436—1221no—
328#435—1222no—
329C8×D4—1221no—
330#432—612no—
331#432—612no—
332#432C2×D6111yescharacteristic
333#432C2×D6111yescharacteristic
334C4×M16—611no—
335C4×M16—611no—
336C4×M16—612no—
337C2×#97—612no—
338#330—611no—
339#330—611no—
340#330—611no—
341#330—611no—
342#330C2×D6112yescomplemented
343#330C2×D6112yescomplemented
344#329—612no—
345#329—612no—
346#329—612no—
347#329—612no—
348C61#1769111yescharacteristic, complemented, Sylow
349C2×#41—611no—
350C22×Q8—611no—
351C22×D4—611no—
352C2×M16—1222no—
353C2×M16—1221no—
354C2×M16—1221no—
355C22×C8—1222no—
356C4⋊Q8—1221no—
357C4⋊Q8C4×D6111yescharacteristic
358C4⋊Q8C4×D6111yescharacteristic
359#127—1221no—
360#126—1222no—
361#126—1222no—
362#126—1222no—
363#126—1222no—
364#125C22×D6111yescharacteristic
365#125C22×D6111yescharacteristic
366#124—612no—
367#124—612no—
368#124—1221no—
369#124—1221no—
370#124—1222no—
371#124—1222no—
372#124—1221no—
373#124—1221no—
374#123—1222no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379C22⋊Q8—21no—
380C22⋊Q8—21no—
381C22⋊Q8—612no—
382C22⋊Q8—612no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385#121—612no—
386#121—612no—
387#120—1222no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—611no—
391C4×Q8—611no—
392C4×Q8—612no—
393C4×Q8—612no—
394C4×Q8—1221no—
395C4×Q8—1221no—
396C4×D4—612no—
397C4×D4—612no—
398C4×D4—612no—
399C4×D4—612no—
400C4×D4—611no—
401C4×D4—611no—
402C4×D4—1221no—
403C4×D4—1221no—
404C4×D4—1222no—
405C4×D4—1222no—
406#117—611no—
407#117—612no—
408#117—1222no—
409#117—1222no—
410#117—1222no—
411#117C4×D6111yescharacteristic
412#117C4×D6111yescharacteristic
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C42—612no—
417C2×C42C22×D6111yescharacteristic
418C4⋊C8—21no—
419C4⋊C8—21no—
420C4⋊C8—1222no—
421C4⋊C8—1222no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—1222no—
425C22⋊C8—1222no—
426#97—611no—
427#97—611no—
428#97—611no—
429#97—611no—
430#97—1222no—
431#97C22×D6112yes—
432#97C22×D6112yes—
433C4×C8—1222no—
434C4×C8—1221no—
435C4×C8—1221no—
436C24—1221no—
437#41—2442no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—611no—
441C2×Q8—612no—
442C2×Q8—1221no—
443C2×Q8—2442no—
444C2×D4—612no—
445C2×D4—612no—
446C2×D4—611no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1221no—
450C2×D4—2442no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—612no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1221no—
457C22×C4C2×C4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460M16—2444no—
461M16—2442no—
462M16—2442no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—2444no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—612no—
477C4⋊C4—612no—
478C4⋊C4—611no—
479C4⋊C4—611no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4C2×C4×D6111yescharacteristic
485C4⋊C4C2×C4×D6111yescharacteristic
486C4⋊C4C2×C4×D6111yescharacteristic
487C4⋊C4C2×C4×D6111yescharacteristic
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—612no—
491C22⋊C4—612no—
492C22⋊C4—612no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C42—21no—
504C42—21no—
505C42—612no—
506C42—612no—
507C42—612no—
508C42—1222no—
509C42—1222no—
510C42C2×C4×D6111yescharacteristic
511C42D4×D6111yescharacteristic
512C42D4×D6111yescharacteristic
513C42C23×D6111yescharacteristic
514C23—612no—
515C23—1222no—
516C23—4881no—
517C23—4881no—
518C23—2442no—
519C23C22⋊C4×D6111yescharacteristic
520Q8—81no—
521Q8—81no—
522Q8—1222no—
523Q8—2442no—
524Q8—2442no—
525D4—1222no—
526D4—4882no—
527D4—4882no—
528D4—2442no—
529D4—2442no—
530D4—2442no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1222no—
542C2×C4—4882no—
543C2×C4—4882no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4C22⋊C4×D6111yescharacteristic
549C2×C4C22⋊C4×D6111yescharacteristic
550C2×C4C22⋊C4×D6111yescharacteristic
551C2×C4C22×C4×D6111yescharacteristic
552C2×C4C2×D4×D6111yescharacteristic
553C2×C4C2×D4×D6111yescharacteristic
554C8—42no—
555C8—42no—
556C8—1222no—
557C8—1222no—
558C8—2441no—
559C8—2441no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—1222no—
564C22—1222no—
565C22—1222no—
566C22—9762no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22C2×C22⋊C4×D6111yescharacteristic
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—1222no—
581C4—1222no—
582C4—1222no—
583C4—2442no—
584C4#1683974512yes—
585C2—41no—
586C2—4882no—
587C2—2442no—
588C2C2×#1683974511yescharacteristic
589C2#2145436511yescharacteristic
590C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1537C2
socleC22×C61C22×C22⋊C4
prime radical#16840331C4
cosocleC42×C61C24
hypercenter#1537D61
hypocenterC61#17691
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1537

Complements

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

NH ≅ G/Ntype
2. #21454836C2semidirect
3. #21454831C2semidirect
4. #21444449C2semidirect
5. #21444438C2semidirect
6. #21444425C2semidirect
7. #21444418C2semidirect
8. #21444418C2semidirect
9. #21444414C2semidirect
10. #21444406C2semidirect
11. C61×#1537C2semidirect
34. #16839817C22semidirect
35. #16839817C22semidirect
38. #16839714C22semidirect
39. #97×D61C22semidirect
40. #16839504C22semidirect
41. #16839504C22semidirect
42. #16839504C22semidirect
43. #16839504C22semidirect
51. C61×#330C22semidirect
52. C61×#330C22semidirect
293. #1537D61semidirect
342. #330C2×D61semidirect
343. #330C2×D61semidirect
348. C61#17691semidirect

Maximal subgroups

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

subgroupindex
#214597172
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#214548312
#214444492
#214444382
#214444252
#214444182
#214444142
#214444062
C61×#15372
#1769161