Finite Group Databasebeta

← #30616766

Subgroups — #30616766

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462497C211yesmaximal, characteristic
2C4×D61⋊M16C211yesmaximal, characteristic
3#21462012C212yesmaximal, complemented
4#21461962C212yesmaximal, complemented
5#21461847C214yesmaximal
6#21461572C214yesmaximal, complemented
7C2×#16840318C211yesmaximal, characteristic
8C2×D61⋊M16C2212yes—
9#16840785—21no—
10#16840784—21no—
11#16840781—21no—
12#16840780—21no—
13#16840765C2212yes—
14C2×#14519838C2218yes—
15#16840734C2212yes—
16C4×#11758213—21no—
17C4×#11758213—21no—
18C4×#11758217C2212yes—
19C4×#11758217C2212yes—
20C4×#11758217C2212yes—
21#16840721—21no—
22C4×D61⋊C8—21no—
23#16840642C2218yes—
24#16840318—21no—
25#16840318—21no—
26#16840318—21no—
27#16840318—21no—
28#16840318—21no—
29#16840318—21no—
30#16840318—21no—
31#16840318—21no—
32C2×#14519776C2211yescharacteristic
33C2×#14519776C2211yescharacteristic
34C2×#14519769C411yescharacteristic
35C2×#14519755C412yes—
36C2×#14519755C2212yes—
37C2×#14519742C412yes—
38C2×#14519742C2212yes—
39C2×#14519735C411yescharacteristic
40C2×#14519735C411yescharacteristic
41C2×C42×D61C2211yescharacteristic
42C2×#125×C61C411yescharacteristic
43C2×#11758213—22no—
44D61⋊M16—28no—
45C2×#11758217C2314yes—
46C2×#11758217C2314yes—
47C2×D61⋊C8—22no—
48#14519844—22no—
49C4⋊#92420—22no—
50#14519838—216no—
51#14519834—22no—
52#14519834—22no—
53#14519832—22no—
54#14519832—22no—
55#14519832—22no—
56C4×#92420—22no—
57C4×#92420—22no—
58C4×#92420—22no—
59C22×Dic122C2×C412yes—
60C22×Dic122C2312yes—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2×C411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64#14519776—21no—
65#14519776—21no—
66#14519776—21no—
67#14519776—21no—
68#14519776—21no—
69#14519776—21no—
70#14519776—21no—
71#14519776—21no—
72#14519769—21no—
73#14519769—21no—
74#14519769—21no—
75#14519769—21no—
76#14519755—22no—
77#14519755—22no—
78#14519755—22no—
79#14519755—22no—
80#14519755—22no—
81#14519755—22no—
82#14519755—22no—
83#14519755—22no—
84#14519742—22no—
85#14519742—22no—
86#14519742—22no—
87#14519742—22no—
88#14519742—22no—
89#14519742—22no—
90#14519742—22no—
91#14519742—22no—
92#14519735—21no—
93#14519735—21no—
94#14519735—21no—
95#14519735—21no—
96#14519735—21no—
97#14519735—21no—
98#14519735—21no—
99#14519735—21no—
100C2×C4⋊Dic61C2×C411yescharacteristic
101C2×C4⋊Dic61C2×C411yescharacteristic
102C2×#11758196C2×C412yes—
103C2×#11758196C2×C412yes—
104C2×#11758196C2×C412yes—
105C2×#11758196C2312yes—
106C2×#11758198C2×C412yes—
107C2×#11758198C2×C412yes—
108C2×C4×Dic61C2×C412yes—
109C2×C4×Dic61C2×C412yes—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121#125×C61—21no—
122#125×C61—21no—
123#125×C61—21no—
124#125×C61—21no—
125C2×C4⋊C4×C61C2×C411yescharacteristic
126C2×C4⋊C4×C61C2×C411yescharacteristic
127C2×C4⋊C4×C61C2×C412yes—
128C2×C4⋊C4×C61C2×C412yes—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C22×C411yescharacteristic
131C22×C4×C61C22×C412yes—
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C412yes—
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61C22×C412yes—
137C22×Dic61C22×C411yescharacteristic
138C22×Dic61C22×C411yescharacteristic
139C22×Dic61C22×C412yes—
140C22×Dic61C2411yescharacteristic
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—42no—
144C2×C4×D61C22⋊C411yescharacteristic
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×Dic122—22no—
157C2×Dic122—22no—
158C2×Dic122—22no—
159C2×Dic122—22no—
160C2×Dic122—28no—
161C2×Dic122—44no—
162#11758217—24no—
163#11758217—28no—
164#11758217#4118yes—
165C2×#92420—24no—
166C2×#92420—24no—
167#11758213—28no—
168D61⋊C8—28no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—22no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181#11758198—22no—
182#11758198—22no—
183#11758198—22no—
184#11758198—22no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C23×C61C23×C411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—41no—
217C22×D61C2×C22⋊C411yescharacteristic
218C22×D61C2×C22⋊C411yescharacteristic
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—24no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—42no—
238C2×Dic61C2×C22⋊C411yescharacteristic
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C22×D411yescharacteristic
243C2×Dic61C22×D411yescharacteristic
244C2×Dic61C2×#4114yes—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61#10114yescomplemented
257Dic122—44no—
258Dic122—44no—
259Dic122—44no—
260Dic122—44no—
261#92420—28no—
262#92420—44no—
263#21425—611nomaximal, Sylow
264C22×C61—21no—
265C22×C61—21no—
266C22×C61C22×C22⋊C411yescharacteristic
267C22×C61C4×#4112yes—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—44no—
274C2×D61C2×#10112yes—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—42no—
281C4×C61—42no—
282C4×C61C2×#10112yes—
283Dic61—21no—
284Dic61—21no—
285Dic61—42no—
286Dic61—42no—
287Dic61C2×#10112yes—
288Dic61#41312yes—
289Dic61#52512yes—
290Dic61#53212yes—
291C2×#533—611no—
292#2920—611no—
293C2×C4×M16—611no—
294#1997—612no—
295#1966—612no—
296#1901—614no—
297C4×#101—614no—
298C2×C61—22no—
299C2×C61#229711yescharacteristic
300C2×C61C22×#10111yescharacteristic
301C2×C61C2×#41311yescharacteristic
302D61—22no—
303D61—41no—
304C22×M16—612no—
305#533—1221no—
306#533—1221no—
307#533—1221no—
308#533—1221no—
309#533—1221no—
310#533—1221no—
311#533—1221no—
312#533—1221no—
313C2×C4⋊Q8—611no—
314C2×C4⋊Q8—611no—
315C2×#125—611no—
316C2×#125#692311yescharacteristic
317C2×C22⋊Q8—612no—
318C2×C22⋊Q8—612no—
319C2×C4×Q8—612no—
320C2×C4×Q8—612no—
321C2×#117—611no—
322C2×#117—611no—
323C22×C42—611no—
324#432—1221no—
325#432—1221no—
326#431—1221no—
327#431—1221no—
328#423—612no—
329C2×#101—618no—
330#407—612no—
331C4×M16—612no—
332C4×M16—612no—
333C4×M16—612no—
334C4×M16—1221no—
335C4×M16—1221no—
336C2×#97—1221no—
337C2×C4×C8—1221no—
338#343—618no—
339C61#2142511yescharacteristic, complemented, Sylow
340C22×Q8—612no—
341C22×Q8—612no—
342C23×C4—611no—
343C23×C4—611no—
344C23×C4—611no—
345C2×M16—614no—
346C2×M16—614no—
347C2×M16—1228no—
348C2×M16—1222no—
349C22×C8—1222no—
350C4⋊Q8—1221no—
351C4⋊Q8—1221no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358#125—21no—
359#125—21no—
360#125—21no—
361#125—21no—
362#125—1221no—
363#125—1221no—
364#125—1221no—
365#125—1221no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C4×Q8—1222no—
375C4×Q8—1222no—
376C4×Q8—1222no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×Q8—1222no—
382#117—1221no—
383#117—1221no—
384#117—1221no—
385#117—1221no—
386#117—1221no—
387#117—1221no—
388#117—1221no—
389#117—1221no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—612no—
393C2×C4⋊C4—612no—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4C2×#692311yescharacteristic
397C2×C4⋊C4C2×#692311yescharacteristic
398C2×C4⋊C4C2×#692312yes—
399C2×C4⋊C4C2×#692312yes—
400C2×C22⋊C4—612no—
401C2×C22⋊C4—612no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—612no—
414C2×C42—612no—
415C2×C42C2×#692311yescharacteristic
416C4⋊C8—1222no—
417C4⋊C8—1222no—
418#101—12216no—
419C22⋊C8—1222no—
420C22⋊C8—1222no—
421#97—1222no—
422#97—1222no—
423#97—1222no—
424C4×C8—1222no—
425C4×C8—1222no—
426C4×C8—1222no—
427C24—611no—
428C2×Q8—1228no—
429C2×Q8—1222no—
430C2×Q8—1222no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—2444no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—1224no—
452C22×C4—1224no—
453C22×C4—2442no—
454C22×C4C22×#692312yes—
455C22×C4C22×#692311yescharacteristic
456C22×C4C22×#692311yescharacteristic
457C22×C4C22×#692312yes—
458C22×C4C22×#692311yescharacteristic
459M16—618no—
460M16—1224no—
461M16—1228no—
462M16—1228no—
463C2×C8—1224no—
464C2×C8—1228no—
465C2×C8—1224no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—1222no—
503C42—1222no—
504C42—1222no—
505C42—1222no—
506C42C22⋊#692311yescharacteristic
507C42C22⋊#692311yescharacteristic
508C42C22⋊#692311yescharacteristic
509C42C22⋊#692311yescharacteristic
510C23—611no—
511C23—611no—
512C23—1222no—
513C23—1222no—
514C23—2441no—
515C23C23×#692311yescharacteristic
516Q8—2444no—
517Q8—2444no—
518Q8—2444no—
519Q8—2444no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—611no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—614no—
532C2×C4—614no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1224no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2442no—
552C2×C4C2×C22⋊#692311yescharacteristic
553C2×C4C2×C22⋊#692311yescharacteristic
554C2×C4C2×C22⋊#692311yescharacteristic
555C2×C4C2×C22⋊#692311yescharacteristic
556C2×C4C2×C22⋊#692311yescharacteristic
557C2×C4C2×C22⋊#692311yescharacteristic
558C8—1228no—
559C8—2444no—
560C22—21no—
561C22—21no—
562C22—612no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—1221no—
567C22—1221no—
568C22—2444no—
569C22C22×C22⋊#692311yescharacteristic
570C22#41×#692312yes—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—42no—
578C4—612no—
579C4—612no—
580C4—612no—
581C4—612no—
582C4—1221no—
583C4—1221no—
584C4—2442no—
585C4—2442no—
586C4D61⋊#10112yes—
587C2—22no—
588C2—1222no—
589C2—2441no—
590C2#2146228011yescharacteristic
591C2C2×D61⋊#10111yescharacteristic
592C2D61⋊#41311yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×#125×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenterC2×#125#6923
hypocenterC61#21425
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
3. #21462012C2semidirect
4. #21461962C2semidirect
6. #21461572C2semidirect
256. C4×D61#101semidirect
339. C61#21425semidirect

Maximal subgroups

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

subgroupindex
#214624972
C4×D61⋊M162
#214620122
#214619622
#214618472
#214615722
C2×#168403182
#2142561