Finite Group Databasebeta

← #30616757

Subgroups — #30616757

Order 15616.

1162 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462296C211yesmaximal, characteristic, complemented
2#21462257C211yesmaximal, characteristic, complemented
3#21462010C212yesmaximal, complemented
4#21461959C212yesmaximal
5#21461843C212yesmaximal, complemented
6#21461843C212yesmaximal, complemented
7C4×#14519836C212yesmaximal, complemented
8#21461564C212yesmaximal, complemented
9C2×#16840360C211yesmaximal, characteristic
10C2×#14519856C2212yes—
11C2×#14519836C2214yes—
12C2×#14519836C2214yes—
13#16840711—24no—
14#16840705C2212yes—
15#16840704C2212yes—
16C4⋊C4×#6923—24no—
17C4×C22⋊#6923C2212yes—
18C4×C22⋊#6923C2212yes—
19C4×C22⋊#6923C2212yes—
20#16840640C2214yes—
21#16840640C2214yes—
22#16840360—24no—
23#16840360—24no—
24C2×#14519772C411yescharacteristic, complemented
25C2×#14519771C411yescharacteristic, complemented
26C2×#14519767C411yescharacteristic, complemented
27C2×#14519762C411yescharacteristic, complemented
28C2×#14519762C411yescharacteristic, complemented
29C2×#14519762C2211yescharacteristic
30C2×#14519762C2211yescharacteristic
31C2×#14519757C411yescharacteristic, complemented
32C2×#14519755C411yescharacteristic, complemented
33C2×#14519755C2211yescharacteristic
34C2×#14519752C2211yescharacteristic
35C2×C4×D244C2211yescharacteristic
36C2×#14519739C2211yescharacteristic
37C2×C4⋊C4×D61C2211yescharacteristic
38C2×#125×C61C411yescharacteristic, complemented
39#14519856—28no—
40C2×C4⋊#6923—22no—
41C2×C4⋊#6923—22no—
42C2×C22⋊#6923C2314yes—
43C2×C22⋊#6923C2314yes—
44C2×C4×#6923—22no—
45C2×C4×#6923—22no—
46C2×C4×#6923—22no—
47C2×C4×#6923—22no—
48#14519836—28no—
49#14519836—28no—
50(C2×#6923)⋊C4—22no—
51(C2×#6923)⋊C4—24no—
52(C2×#6923)⋊C4—22no—
53(C2×#6923)⋊C4—22no—
54(C2×#6923)⋊C4—22no—
55C22×Dic122C2×C411yescharacteristic
56C22×D244C2×C411yescharacteristic
57C22×D244C2311yescharacteristic
58C22×D244C2311yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64#14519772—24no—
65#14519771—24no—
66#14519767—24no—
67#14519762—22no—
68#14519762—22no—
69#14519762—22no—
70#14519762—22no—
71#14519762—24no—
72#14519762—24no—
73#14519757—24no—
74#14519755—22no—
75#14519755—22no—
76#14519755—24no—
77#14519752—22no—
78#14519752—22no—
79C4×D244—22no—
80C4×D244—22no—
81#14519739—22no—
82#14519739—22no—
83C2×C4⋊Dic61C2×C411yescharacteristic
84C2×C4⋊Dic61C2×C411yescharacteristic
85C2×#11758196C2×C411yescharacteristic
86C2×#11758196C2×C411yescharacteristic
87C2×#11758196C2×C411yescharacteristic
88C2×#11758196C2×C411yescharacteristic
89C2×#11758196C2×C411yescharacteristic
90C2×#11758196C2×C411yescharacteristic
91C4⋊C4×D61D414yes—
92C4⋊C4×D61D414yes—
93C2×#11758198C2×C411yescharacteristic
94C2×#11758198C2×C411yescharacteristic
95C2×#11758198C2×C411yescharacteristic
96C2×#11758198C2×C411yescharacteristic
97C2×#11758198C2×C411yescharacteristic
98C2×#11758198C2×C411yescharacteristic
99C2×#11758198C2×C411yescharacteristic
100C2×#11758198C2×C411yescharacteristic
101C2×#11758198C2311yescharacteristic
102C2×#11758198C2311yescharacteristic
103C2×C4×Dic61C2×C411yescharacteristic
104#125×C61—24no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C4⋊C4×C61C2×C411yescharacteristic
107C2×C4⋊C4×C61C2×C411yescharacteristic
108C2×C4⋊C4×C61C2×C411yescharacteristic
109C2×C4⋊C4×C61C2×C411yescharacteristic
110C2×C4⋊C4×C61C2×C411yescharacteristic
111C2×C42×C61C2×C411yescharacteristic
112C23×D61C22×C411yescharacteristic
113C23×D61C22×C411yescharacteristic
114C23×D61C2411yescharacteristic
115C22×#6923—24no—
116C22×#6923—24no—
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C22×C411yescharacteristic
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C22×Dic61C22×C411yescharacteristic
125C22×Dic61C22×C411yescharacteristic
126C22×Dic61C22×C411yescharacteristic
127C22×Dic61C22×C411yescharacteristic
128C22×Dic61C22×C411yescharacteristic
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—24no—
134C2×D244—24no—
135C2×D244—22no—
136C2×D244—81no—
137C2×C4×D61—24no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—81no—
141C2×C4×D61—81no—
142C2×C4×D61—81no—
143C2×C4×D61—81no—
144C2×C4×D61C2×D412yes—
145C2×C4×D61C2×D412yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×Dic122—22no—
151C2×Dic122—81no—
152C22⋊#6923—24no—
153C22⋊#6923—24no—
154C22⋊#6923—24no—
155C22⋊#6923#4118yes—
156C4⋊#6923—28no—
157C4⋊#6923—82no—
158C4×#6923—28no—
159C4×#6923—82no—
160C4×#6923—82no—
161C4×#6923—82no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61C22⋊C414yes—
168C42×C61—22no—
169#11758198—21no—
170#11758198—21no—
171#11758198—21no—
172#11758198—21no—
173#11758198—22no—
174#11758198—22no—
175#11758198—22no—
176#11758198—22no—
177#11758198—22no—
178#11758198—22no—
179#11758198—22no—
180#11758198—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61C22⋊C414yes—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—22no—
187#11758196C22⋊C414yes—
188#11758196C22⋊C414yes—
189C4×Dic61—22no—
190C23×C61C23×C411yescharacteristic
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—22no—
194C22×D61—22no—
195C22×D61—41no—
196C22×D61—41no—
197C22×D61—41no—
198C22×D61—41no—
199C22×D61—41no—
200C22×D61—41no—
201C22×D61C22×D411yescharacteristic
202C22×D61C22×D411yescharacteristic
203C22×D61C2×#4114yes—
204C2×#6923—28no—
205C2×#6923—44no—
206C2×#6923—44no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—44no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C412yes—
223C2×C4×C61C2×C22⋊C412yes—
224C2×Dic61—22no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C412yes—
234D244—42no—
235D244—42no—
236D244—42no—
237D244—42no—
238D244—82no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—24no—
242C4×D61—42no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—42no—
246C4×D61—82no—
247C4×D61—82no—
248Dic122—82no—
249#21423—611nomaximal, Sylow
250C22×C61—21no—
251C22×C61—21no—
252C22×C61C22×C22⋊C411yescharacteristic
253C22×C61C4×#4112yes—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—44no—
259C2×D61—41no—
260C2×D61—41no—
261C2×D61—41no—
262C2×D61—41no—
263C2×D61—41no—
264C2×D61—41no—
265C2×D61—82no—
266C2×D61—82no—
267C2×D61#52512yes—
268C2×D61#53212yes—
269#6923—48no—
270#6923—84no—
271C4×C61—22no—
272C4×C61—22no—
273C4×C61—22no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—41no—
278Dic61—22no—
279Dic61—22no—
280Dic61—22no—
281Dic61—41no—
282Dic61—41no—
283C2×#539—611no—
284#2313—611no—
285#2274—611no—
286#1996—612no—
287#1964—612no—
288#1899—612no—
289#1899—612no—
290C4×#99—612no—
291C4×#99—612no—
292C2×C61—22no—
293C2×C61#229711yescharacteristic
294C2×C61#288311yescharacteristic
295C2×C61#288411yescharacteristic
296D61—22no—
297D61—22no—
298D61—81no—
299D61—81no—
300#539—1224no—
301#539—1224no—
302C2×#126—611no—
303C2×#126—611no—
304C2×#125—611no—
305C2×#125#692311yescharacteristic, complemented
306C2×#123—611no—
307C2×#123—611no—
308C2×#123—611no—
309C2×#123—611no—
310C2×C22⋊Q8—611no—
311C2×C22⋊Q8—611no—
312C2×C22⋊Q8—611no—
313C2×#121—611no—
314C2×C4×D4—611no—
315C2×C4×D4—611no—
316C2×#117—612no—
317C22×C4⋊C4—611no—
318C2×#99—614no—
319C2×#99—614no—
320#387—1224no—
321#381—612no—
322#380—612no—
323C4×C4⋊C4—1224no—
324C4×C22⋊C4—612no—
325C4×C22⋊C4—612no—
326C4×C22⋊C4—612no—
327#342—614no—
328#342—614no—
329C61#2142311yescharacteristic, complemented, Sylow
330C22×Q8—611no—
331C22×D4—611no—
332C22×D4—611no—
333C22×D4—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—611no—
338C23×C4—611no—
339#126—1224no—
340#126—1224no—
341#125—24no—
342#125—1224no—
343#123—1224no—
344#123—1222no—
345#123—1222no—
346#123—1222no—
347#123—1222no—
348#123—1224no—
349C22⋊Q8—1224no—
350C22⋊Q8—1224no—
351C22⋊Q8—1222no—
352C22⋊Q8—1222no—
353#121—1222no—
354#121—1222no—
355C4×D4—1222no—
356C4×D4—1222no—
357C4×D4—1222no—
358C4×D4—1222no—
359#117—1228no—
360C2×C4⋊C4—611no—
361C2×C4⋊C4—611no—
362C2×C4⋊C4—611no—
363C2×C4⋊C4—614no—
364C2×C4⋊C4—614no—
365C2×C4⋊C4—611no—
366C2×C4⋊C4—611no—
367C2×C4⋊C4—611no—
368C2×C4⋊C4—611no—
369C2×C4⋊C4—611no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4C2×#692311yescharacteristic
373C2×C4⋊C4C2×#692311yescharacteristic
374C2×C4⋊C4C2×#692311yescharacteristic
375C2×C4⋊C4C2×#692311yescharacteristic
376C2×C4⋊C4C2×#692311yescharacteristic
377C2×C4⋊C4C2×#692311yescharacteristic
378C2×C22⋊C4—614no—
379C2×C22⋊C4—611no—
380C2×C22⋊C4—611no—
381C2×C22⋊C4—611no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—611no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—614no—
390C2×C42—611no—
391C2×C42—1222no—
392C2×C42—1222no—
393C2×C42—1222no—
394C2×C42—1222no—
395C2×C42C2×#692311yescharacteristic
396#99—1228no—
397#99—1228no—
398#95—1222no—
399#95—1222no—
400#95—1222no—
401#95—1222no—
402#95—1224no—
403C24—611no—
404C24—611no—
405C24—611no—
406C2×Q8—1222no—
407C2×Q8—4881no—
408C2×D4—1224no—
409C2×D4—1224no—
410C2×D4—1221no—
411C2×D4—1221no—
412C2×D4—1221no—
413C2×D4—1221no—
414C2×D4—1222no—
415C2×D4—4881no—
416C22×C4—611no—
417C22×C4—611no—
418C22×C4—611no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—612no—
422C22×C4—612no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—611no—
426C22×C4—611no—
427C22×C4—1222no—
428C22×C4—1224no—
429C22×C4—1224no—
430C22×C4—1222no—
431C22×C4—1224no—
432C22×C4—4881no—
433C22×C4—4881no—
434C22×C4—4881no—
435C22×C4—4881no—
436C22×C4C22×#692311yescharacteristic
437C22×C4C22×#692311yescharacteristic
438C22×C4C22×#692311yescharacteristic
439C22×C4C22×#692311yescharacteristic
440C22×C4C22×#692311yescharacteristic
441C22×C4C22×#692311yescharacteristic
442C22×C4C22×#692311yescharacteristic
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—614no—
449C4⋊C4—614no—
450C4⋊C4—614no—
451C4⋊C4—1222no—
452C4⋊C4—1222no—
453C4⋊C4—1222no—
454C4⋊C4—1222no—
455C4⋊C4—1228no—
456C4⋊C4—1222no—
457C4⋊C4—4882no—
458C4⋊C4C22⋊#692314yes—
459C22⋊C4—618no—
460C22⋊C4—1222no—
461C22⋊C4—1222no—
462C22⋊C4—1222no—
463C22⋊C4—1222no—
464C22⋊C4—1222no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—1221no—
468C22⋊C4—1221no—
469C22⋊C4—1224no—
470C22⋊C4—1222no—
471C22⋊C4—1222no—
472C22⋊C4—1222no—
473C22⋊C4—1224no—
474C22⋊C4—1224no—
475C42—22no—
476C42—1222no—
477C42—1228no—
478C42—4882no—
479C42—4882no—
480C42—4882no—
481C23—611no—
482C23—611no—
483C23—614no—
484C23—1222no—
485C23—1222no—
486C23—1221no—
487C23—1221no—
488C23—2441no—
489C23—2441no—
490C23—2441no—
491C23—2441no—
492C23—2441no—
493C23—2441no—
494C23C23×#692311yescharacteristic
495Q8—4882no—
496D4—4882no—
497D4—2442no—
498D4—2442no—
499D4—2442no—
500D4—2442no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1224no—
518C2×C4—1222no—
519C2×C4—1228no—
520C2×C4—1221no—
521C2×C4—1221no—
522C2×C4—4882no—
523C2×C4—4882no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4—2441no—
529C2×C4—2441no—
530C2×C4—2444no—
531C2×C4—2444no—
532C2×C4—2444no—
533C2×C4—2444no—
534C2×C4—2444no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4C2×C22⋊#692312yes—
538C2×C4C2×C22⋊#692312yes—
539C2×C4C2×C22⋊#692312yes—
540C22—21no—
541C22—21no—
542C22—612no—
543C22—612no—
544C22—1221no—
545C22—1221no—
546C22—1221no—
547C22—1221no—
548C22—4882no—
549C22—4882no—
550C22—2444no—
551C22—2441no—
552C22—2441no—
553C22—2441no—
554C22—2441no—
555C22—2441no—
556C22—2441no—
557C22C22×C22⋊#692311yescharacteristic
558C22#41×#692312yes—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—1222no—
567C4—1222no—
568C4—1222no—
569C4—4884no—
570C4—2448no—
571C4—2441no—
572C4—2441no—
573C2—22no—
574C2—1222no—
575C2—1222no—
576C2—4881no—
577C2—4881no—
578C2#2146228011yescharacteristic
579C2#2146241911yescharacteristic
580C2#2146242011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#125×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519752C22
cosocleC23×D61C24
hypercenterC2×#125#6923
hypocenterC61#21423
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. #21462296C2semidirect
2. #21462257C2semidirect
3. #21462010C2semidirect
5. #21461843C2semidirect
6. #21461843C2semidirect
7. C4×#14519836C2semidirect
8. #21461564C2semidirect
24. C2×#14519772C4semidirect
25. C2×#14519771C4semidirect
26. C2×#14519767C4semidirect
27. C2×#14519762C4semidirect
28. C2×#14519762C4semidirect
31. C2×#14519757C4semidirect
32. C2×#14519755C4semidirect
38. C2×#125×C61C4semidirect
305. C2×#125#6923semidirect
329. C61#21423semidirect

Maximal subgroups

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

subgroupindex
#214622962
#214622572
#214620102
#214619592
#214618432
C4×#145198362
#214615642
C2×#168403602
#2142361