Finite Group Databasebeta

← #30610972

Subgroups — #30610972

Order 15616.

1108 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#432C211yesmaximal, characteristic, complemented
2C4×#14519856C211yesmaximal, characteristic, complemented
3#21461222C214yesmaximal, complemented
4#21461211C212yesmaximal, complemented
5#21461210C212yesmaximal, complemented
6#21461196C214yesmaximal, complemented
7#21459116C211yesmaximal, characteristic
8C2×#14519856—21no—
9#16840785C2212yes—
10#16840784C2212yes—
11#16840783—21no—
12#16840782—21no—
13D61⋊(C4⋊C8)—21no—
14D61⋊(C22⋊C8)C2212yes—
15#16840721—21no—
16C4×C4⋊#6923—21no—
17C4×C4⋊#6923C2212yes—
18C4×C22⋊#6923—21no—
19C4×C22⋊#6923C2212yes—
20#16840698C2212yes—
21C42×#6923—21no—
22#16840615C2218yes—
23#16840614C2218yes—
24#16840317C411yescharacteristic, complemented
25#16840317C2211yescharacteristic
26#16840313C411yescharacteristic, complemented
27#16840313C2211yescharacteristic
28#16840312C411yescharacteristic, complemented
29#16840299—21no—
30#16840299—21no—
31#16840298—21no—
32#16840298—21no—
33#16840297C412yescomplemented
34#16840297C2212yes—
35#16840275—21no—
36#16840275—21no—
37#16840274C2212yes—
38C4×#11758226—21no—
39C4×#11758226—21no—
40#16840230C412yescomplemented
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C2×D61⋊C8—22no—
44#14519856—28no—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923—22no—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50#14519844—22no—
51C4⋊#92420—22no—
52#14519834C2314yes—
53C22⋊#92420—22no—
54#14519832—22no—
55#14519832—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4C2314yes—
58C2×#11758226—21no—
59C2×#11758226—21no—
60C2×#11758222C2×C412yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519774C2×C411yescharacteristic
66#14519772—22no—
67#14519772—22no—
68#14519768—21no—
69#14519768—21no—
70#14519766—21no—
71#14519766—21no—
72#14519765C2×C412yes—
73#14519762—22no—
74#14519762—22no—
75#14519761—22no—
76#14519757C2×C412yes—
77#14519757C2312yes—
78#14519753—21no—
79#14519753—21no—
80#14519752—21no—
81#14519752—21no—
82#14519752C2×C412yes—
83#14519752C2312yes—
84#14519751—21no—
85#14519751—21no—
86#14519747—22no—
87Q8×Dic61—21no—
88Q8×Dic61—21no—
89#14519742C2×C412yes—
90#14519742C2×C412yes—
91C4×D244—22no—
92C4×D244—22no—
93C4×D244—21no—
94C4×D244—21no—
95#14519739—22no—
96#14519739—22no—
97#14519739—21no—
98#14519739—21no—
99#14519739C2×C412yes—
100#14519739C2×C412yes—
101#14519737—22no—
102#14519737—22no—
103#14519735—21no—
104#14519735—21no—
105#14519735—22no—
106#14519735—22no—
107#14519734—22no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C42×D61—22no—
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61C2311yescharacteristic
119C42×D61C2311yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic
121C4⋊Q8×C61C2×C411yescharacteristic
122#125×C61C2×C412yes—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127#117×C61C2×C412yes—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C22×#6923—24no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C2×C22⋊D61—22no—
137C22×Dic61C22⋊C411yescharacteristic
138C22×Dic61C22⋊C411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140#11758226—44no—
141#11758226—44no—
142#11758222—44no—
143C2×D244—22no—
144C2×D244—22no—
145C2×D244—21no—
146C2×D244—21no—
147C2×D244C22×C412yes—
148C2×C4×D61—24no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C22⋊C411yescharacteristic
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22×C411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2411yescharacteristic
166C2×Dic122C22×C412yes—
167C22⋊#6923—28no—
168C2×#92420—24no—
169C4⋊#6923—28no—
170C4×#6923—28no—
171C4×#6923—44no—
172C4×#6923—44no—
173D61⋊C8—44no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C412yes—
179C4⋊C4×C61C22×C412yes—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22⋊C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198#11758196—22no—
199#11758196—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22×C411yescharacteristic
209C4×Dic61C22×C411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C23×C61C2×C22⋊C411yescharacteristic
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—41no—
219C22×D61C2×C22⋊C411yescharacteristic
220C22×D61C22×D411yescharacteristic
221C2×#6923—28no—
222C2×#6923—44no—
223C2×#6923—44no—
224Q8×C61—42no—
225Q8×C61—42no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—41no—
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C23×C411yescharacteristic
237C22⋊D61—44no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C22×D411yescharacteristic
248D244—42no—
249D244—42no—
250D244—44no—
251D244—44no—
252D244—42no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—22no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—44no—
265C4×D61—44no—
266C4×D61—44no—
267C4×D61#10414yes—
268Dic122—42no—
269#92420—44no—
270#17543—611nomaximal, Sylow
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61C22×C22⋊C411yescharacteristic
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—21no—
278C2×D61—21no—
279C2×D61—44no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—42no—
283C2×D61C2×#10412yes—
284C2×D61#57512yes—
285#6923—48no—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—42no—
291C4×C61—42no—
292C4×C61#41012yes—
293C4×C61C2×#10412yes—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—42no—
300Dic61C2×#10412yes—
301#3444—611no—
302C2×#432—611no—
303C4×#117—611no—
304#1509—614no—
305#1506—612no—
306#1505—612no—
307#1501—614no—
308C2×C61—41no—
309C2×C61C2×#41011yescharacteristic
310C2×C61#289611yescharacteristic
311C2×C61C22×#10411yescharacteristic
312D61—22no—
313D61—22no—
314D61—82no—
315#533—611no—
316#533—611no—
317#533#692311yescharacteristic, complemented
318#532—611no—
319#532—611no—
320#532—611no—
321#529—612no—
322#529—612no—
323#529—1221no—
324#529—1221no—
325#529—1221no—
326#529—1221no—
327#525—612no—
328#525—1221no—
329#525—1221no—
330C4×#41—612no—
331C4×#41—1221no—
332C4×#41—1221no—
333C2×#117—1221no—
334C22×C42—611no—
335#432—612no—
336#432—612no—
337#432—1221no—
338#432—1221no—
339C2×C4⋊C8—1221no—
340C2×C22⋊C8—612no—
341C2×#97—1221no—
342C4×C4⋊C4—612no—
343C4×C4⋊C4—1221no—
344C4×C22⋊C4—612no—
345C4×C22⋊C4—1221no—
346#376—612no—
347C43—1221no—
348#328—618no—
349#327—618no—
350C61#1754311yescharacteristic, complemented, Sylow
351C2×#41—612no—
352C2×#41—1221no—
353C2×#41—1221no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C22×C8—1222no—
358C4⋊Q8—611no—
359C4⋊Q8C2×#692311yescharacteristic
360C4⋊Q8C2×#692311yescharacteristic
361#127—611no—
362#126—1222no—
363#126—1222no—
364#125—1221no—
365#125—1221no—
366#125C2×#692312yes—
367#124—612no—
368#124—1221no—
369#124—1221no—
370#123—1222no—
371#123—1222no—
372#123—1222no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—612no—
376C22⋊Q8—612no—
377C22⋊Q8—1221no—
378C22⋊Q8—1221no—
379#121—612no—
380#121—612no—
381#121—1221no—
382#121—1221no—
383#121—1221no—
384#121—1221no—
385#120—1222no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—612no—
389C4×Q8—612no—
390C4×Q8—1221no—
391C4×Q8—1221no—
392C4×D4—612no—
393C4×D4—612no—
394C4×D4—1222no—
395C4×D4—1222no—
396C4×D4—1221no—
397C4×D4—1221no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402C4×D4—1221no—
403C4×D4—1221no—
404#117—1222no—
405#117—1222no—
406#117—1222no—
407#117—1221no—
408#117—1221no—
409#117—1228no—
410#117C2×#692312yes—
411C2×C4⋊C4—1222no—
412C2×C22⋊C4—1222no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—1222no—
425C2×C42—1222no—
426C2×C42—1222no—
427C2×C42—1222no—
428C2×C42C2×#692311yescharacteristic
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431C22⋊C8—614no—
432C22⋊C8—1222no—
433#97—1222no—
434#97—1222no—
435#95—614no—
436#95—1222no—
437C24—611no—
438#41—2444no—
439#41—2444no—
440#41—2444no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—612no—
444C2×D4—612no—
445C2×D4—1222no—
446C2×D4—1222no—
447C2×D4—1222no—
448C2×D4—1221no—
449C2×D4—1221no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—1224no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4—1222no—
470C22×C4—2442no—
471C22×C4—2442no—
472C22×C4C22⋊#692311yescharacteristic
473C22×C4C22⋊#692311yescharacteristic
474C22×C4C22×#692311yescharacteristic
475C2×C8—1224no—
476C2×C8—2444no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4—1228no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4C22×#692312yes—
489C4⋊C4C22×#692312yes—
490C22⋊C4—22no—
491C22⋊C4—1222no—
492C22⋊C4—1222no—
493C22⋊C4—1222no—
494C22⋊C4—1222no—
495C22⋊C4—1222no—
496C22⋊C4—1222no—
497C22⋊C4—1228no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C42—22no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—611no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—1222no—
514C42—1222no—
515C42—1228no—
516C42—2444no—
517C42—2444no—
518C42C22⋊#692311yescharacteristic
519C42C22⋊#692311yescharacteristic
520C42C22×#692311yescharacteristic
521C42C22×#692311yescharacteristic
522C23—611no—
523C23—611no—
524C23—1222no—
525C23—1222no—
526C23—1222no—
527C23—2441no—
528C23C2×C22⋊#692311yescharacteristic
529Q8—42no—
530Q8—42no—
531Q8—2442no—
532D4—2442no—
533D4—2442no—
534D4—2444no—
535D4—2444no—
536D4—2444no—
537D4—2442no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—611no—
544C2×C4—614no—
545C2×C4—611no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—611no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1228no—
559C2×C4—1222no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2444no—
565C2×C4—2444no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4C2×C22⋊#692311yescharacteristic
572C2×C4C2×C22⋊#692311yescharacteristic
573C2×C4C2×C22⋊#692311yescharacteristic
574C2×C4C2×C22⋊#692311yescharacteristic
575C2×C4C2×C22⋊#692311yescharacteristic
576C2×C4C23×#692311yescharacteristic
577C8—2444no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—612no—
582C22—612no—
583C22—1221no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—2444no—
588C22—2442no—
589C22—2442no—
590C22—2442no—
591C22C22×C22⋊#692311yescharacteristic
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—42no—
597C4—42no—
598C4—612no—
599C4—1221no—
600C4—1221no—
601C4—1221no—
602C4—1221no—
603C4—1221no—
604C4—2448no—
605C4—2442no—
606C4#1684074312yes—
607C4D61⋊#10412yes—
608C2—41no—
609C2—1222no—
610C2—1222no—
611C2—4882no—
612C2C2×#1684074311yescharacteristic
613C2#2146243911yescharacteristic
614C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#17543
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. D61⋊#432C2semidirect
2. C4×#14519856C2semidirect
3. #21461222C2semidirect
4. #21461211C2semidirect
5. #21461210C2semidirect
6. #21461196C2semidirect
24. #16840317C4semidirect
26. #16840313C4semidirect
28. #16840312C4semidirect
33. #16840297C4semidirect
40. #16840230C4semidirect
42. C61×#533C4semidirect
317. #533#6923semidirect
350. C61#17543semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4322
C4×#145198562
#214612222
#214612112
#214612102
#214611962
#214591162
#1754361