Finite Group Databasebeta

← D61×#1888 #29681579

Subgroups — D61×#1888 #29681579

Order 15616.

1878 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#124×D61C211yesmaximal, characteristic
2#519×D61C211yesmaximal, characteristic, complemented
3#418×D61C212yesmaximal
4C2×#101×D61C211yesmaximal, characteristic, complemented
5#21446223C211yesmaximal, characteristic, complemented
6#21446222C211yesmaximal, characteristic, complemented
7#21446221C212yesmaximal, complemented
8#21446220C211yesmaximal, characteristic, complemented
9#21446219C212yesmaximal, complemented
10#339×D61C212yesmaximal, complemented
11C61×#1888C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C22×D4×D61C2211yescharacteristic
14C2×M16×D61C2212yes—
15C2×#14519766C2211yescharacteristic
16C2×#14519765C2211yescharacteristic
17C2×#14519764C2211yescharacteristic
18C2×#14519763—22no—
19#124×D61—24no—
20#124×D61C418yescomplemented
21#16840240C2211yescharacteristic, complemented
22#16840239—22no—
23#16840239C2212yescomplemented
24#16840238—24no—
25C4×Q8×D61—24no—
26#117×D61—22no—
27#117×D61C2212yes—
28C2×C22⋊C4×D61—22no—
29C2×C42×D61C2211yescharacteristic
30#16839768C2212yes—
31#16839767C2212yes—
32#16839766C2212yes—
33#16839765—24no—
34#102×D61—24no—
35C2×#14519696C2211yescharacteristic, complemented
36C2×#14519695C2212yescomplemented
37#101×D61—24no—
38#16839523C2212yescomplemented
39#16839522C2212yescomplemented
40#16839521C2212yescomplemented
41C2×#124×C61C2211yescharacteristic
42C61×#519C2211yescharacteristic, complemented
43C61×#418C2212yes—
44C2×#101×C61C2211yescharacteristic, complemented
45C61×#339C2212yescomplemented
46C24×D61—21no—
47C22×Dic122D412yes—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—22no—
51C2×Q8×D61—24no—
52C2×Q8×D61C2×C414yes—
53C2×Q8×D61D414yes—
54C22×D244C2311yescharacteristic
55C22×C22⋊D61—21no—
56C2×D4×D61—22no—
57C2×D4×D61—22no—
58C2×D4×D61—22no—
59C2×D4×D61—24no—
60C2×D4×D61C2×C414yes—
61C22×C4×D61—21no—
62C22×C4×D61D412yes—
63C22×C4×D61C2311yescharacteristic
64C2×#11758194C2312yes—
65C2×#11758189—22no—
66M16×D61—24no—
67M16×D61—24no—
68C2×C8×D61—22no—
69#14519766—22no—
70#14519766C2×C414yescomplemented
71#14519765—22no—
72#14519765C2×C414yescomplemented
73#14519764—22no—
74#14519764C2×C414yescomplemented
75#14519763—216no—
76Q8×Dic61—24no—
77C4×Dic122—28no—
78C4×Dic122—24no—
79#14519742—28no—
80#14519742—24no—
81#14519736—22no—
82#14519736C2312yes—
83#14519735—24no—
84#14519735D414yes—
85#14519734—24no—
86#14519734—22no—
87#14519733—24no—
88#14519733—22no—
89C4⋊C4×D61—24no—
90C4⋊C4×D61D414yes—
91C2×C22⋊Dic61—22no—
92C2×#11758198—22no—
93C2×#11758198—22no—
94C22⋊C4×D61—22no—
95C22⋊C4×D61—216no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98C42×D61—24no—
99C42×D61—22no—
100C42×D61—22no—
101C42×D61C2×C414yes—
102#14519699—24no—
103#14519698—24no—
104#14519697—24no—
105#14519696—22no—
106#14519695—24no—
107C22×Q8×C61C2311yescharacteristic
108C22×D4×C61C2311yescharacteristic
109C2×M16×C61C2312yes—
110#124×C61—22no—
111#124×C61C2×C414yescomplemented
112C4×Q8×C61—24no—
113#117×C61—22no—
114#117×C61C2312yes—
115C2×C22⋊C4×C61—22no—
116C2×C42×C61C2311yescharacteristic
117#102×C61—24no—
118#101×C61—22no—
119C24×C61—21no—
120C23×D61—21no—
121C23×D61—24no—
122C23×D61—48no—
123C23×D61—41no—
124C23×D61—41no—
125C23×D61—42no—
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C22×C412yes—
130C2×Q8×C61C2×D412yes—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—22no—
134C2×D4×C61C22×C412yes—
135C22×C4×C61—21no—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2411yescharacteristic
138C2×C22⋊D61—24no—
139C2×C22⋊D61—42no—
140C2×C22⋊D61—42no—
141C22×Dic61—21no—
142C22×Dic61C2×D412yes—
143C22×Dic61C2×D411yescharacteristic
144Q8×D61—28no—
145Q8×D61—28no—
146Q8×D61—28no—
147Q8×D61—28no—
148D4×D61—28no—
149D4×D61—28no—
150D4×D61—44no—
151D4×D61—44no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244—21no—
155C2×D244C22×C412yes—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—44no—
162C2×C4×D61C22⋊C418yes—
163C2×C4×D61C22×C412yes—
164C2×C4×D61C2×D412yes—
165C2×C4×D61C2×D412yes—
166C2×C4×D61C2×D412yes—
167C2×C4×D61#4114yes—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—24no—
171C2×Dic122—24no—
172C2×Dic122C22⋊C418yes—
173C2×Dic122C22×C412yes—
174C2×Dic122C2×D412yes—
175M16×C61—22no—
176M16×C61—22no—
177C2×C8×C61—22no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61C2×D414yes—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—28no—
182C42×C61—24no—
183C42×C61—21no—
184C42×C61—22no—
185C42×C61C22×C412yes—
186C22⋊Dic61—22no—
187C22⋊Dic61—28no—
188#11758198—24no—
189#11758198—28no—
190#11758198—28no—
191C4⋊Dic61—24no—
192C4⋊Dic61C2×D414yes—
193#11758196—28no—
194#11758196—24no—
195C4×Dic61—28no—
196C4×Dic61—24no—
197C4×Dic61—21no—
198C4×Dic61—22no—
199C4×Dic61—24no—
200C4×Dic61—24no—
201C4×Dic61C22×C412yes—
202#11758194—22no—
203#11758194—22no—
204C2×#92418—22no—
205#11758189—44no—
206C8×D61—44no—
207C23×C61—22no—
208C23×C61—44no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61C22×D411yescharacteristic
212C22×D61—24no—
213C22×D61—22no—
214C22×D61—416no—
215C22×D61—44no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—44no—
219C22×D61—44no—
220C22×D61—84no—
221C22×D61—88no—
222C22×D61—84no—
223C22×D61C2×C22⋊C412yes—
224C22×D61#12011yescharacteristic
225C22×D61#12011yescharacteristic
226C22×D61#12312yes—
227Q8×C61—24no—
228Q8×C61—24no—
229D4×C61—24no—
230D4×C61—42no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C414yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D412yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4112yes—
241C22⋊D61—48no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61C2×C22⋊C414yes—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61#12012yes—
250C2×Dic61#12312yes—
251D244—24no—
252D244—42no—
253C4×D61—28no—
254C4×D61—28no—
255C4×D61—48no—
256C4×D61—48no—
257C4×D61C4×D414yes—
258C4×D61C4×D414yes—
259C4×D61#12114yes—
260C4×D61#12114yes—
261Dic122—24no—
262Dic122—24no—
263Dic122—216no—
264C8×C61—42no—
265#92418—42no—
266C2×#1888—611nomaximal, Sylow
267C22×C61—22no—
268C22×C61—44no—
269C22×C61—42no—
270C22×C61—82no—
271C22×C61—82no—
272C22×C61C22×C22⋊C411yescharacteristic
273C22×C61C2×#12011yescharacteristic
274C22×C61C2×#12311yescharacteristic
275C2×D61—22no—
276C2×D61—24no—
277C2×D61—24no—
278C2×D61—44no—
279C2×D61—42no—
280C2×D61—88no—
281C2×D61—82no—
282C2×D61—82no—
283C2×D61—88no—
284C2×D61#38612yes—
285C2×D61#38612yes—
286C2×D61#45414yes—
287C4×C61—24no—
288C4×C61—42no—
289C4×C61—44no—
290C4×C61C2×C4×D412yes—
291C4×C61C2×#12112yes—
292Dic61—24no—
293Dic61—42no—
294Dic61—44no—
295Dic61#38614yes—
296C22×#124—611no—
297C2×#519—611no—
298C2×#418—612no—
299C22×#101—611no—
300#1888—612no—
301#1888—611no—
302#1888—612no—
303#1888—611no—
304#1888—611no—
305#1888D6111yescharacteristic, direct product factor
306C2×#339—612no—
307C2×C61—21no—
308C2×C61—21no—
309C2×C61—82no—
310C2×C61C2×#38611yescharacteristic
311C2×C61C2×#45412yes—
312D61—22no—
313D61—82no—
314D61#188814yesdirect product factor
315C23×Q8—611no—
316C23×D4—611no—
317C22×M16—612no—
318C2×#124—611no—
319C2×#124—611no—
320C2×#124—618no—
321C2×#124—611no—
322C2×#124—1222no—
323C2×#124—1224no—
324C2×#124C2×D6111yescharacteristic
325#519—611no—
326#519—612no—
327#519—1222no—
328#519—1224no—
329#519C2×D6111yescharacteristic, complemented
330C2×C4×Q8—1224no—
331C2×#117—612no—
332C2×#117—1222no—
333C22×C22⋊C4—1222no—
334C22×C42—611no—
335#418—612no—
336#418—612no—
337#418—612no—
338#418—1224no—
339#418C2×D6112yes—
340C2×#102—1224no—
341C2×#101—611no—
342C2×#101—612no—
343C2×#101—1224no—
344C2×#101C2×D6111yescharacteristic, complemented
345#339—612no—
346#339—612no—
347#339—612no—
348#339C2×D6112yescomplemented
349C61C2×#188811yescharacteristic, complemented, Sylow
350C25—1221no—
351C22×Q8—614no—
352C22×Q8—614no—
353C22×Q8—612no—
354C22×Q8—611no—
355C22×Q8—1222no—
356C22×Q8—1222no—
357C22×Q8—1224no—
358C22×Q8C22×D6111yescharacteristic
359C22×D4—614no—
360C22×D4—611no—
361C22×D4—1224no—
362C22×D4—1222no—
363C22×D4—1222no—
364C22×D4—1222no—
365C22×D4—1221no—
366C22×D4C22×D6111yescharacteristic
367C23×C4—611no—
368C23×C4—612no—
369C23×C4—1221no—
370C2×M16—612no—
371C2×M16—1222no—
372C2×M16—1224no—
373C2×M16—1224no—
374C2×M16C22×D6112yes—
375C22×C8—1222no—
376#124—22no—
377#124—614no—
378#124—614no—
379#124—614no—
380#124—1222no—
381#124—12216no—
382#124—1222no—
383#124—1222no—
384#124C4×D6114yescomplemented
385C4×Q8—24no—
386C4×Q8—1224no—
387C4×Q8—1228no—
388C4×Q8—1228no—
389C4×Q8—1224no—
390C4×Q8—1224no—
391#117—22no—
392#117—612no—
393#117—614no—
394#117—1222no—
395#117—1224no—
396#117—1222no—
397#117—1224no—
398#117—1224no—
399#117—1222no—
400#117C22×D6112yes—
401C2×C4⋊C4—614no—
402C2×C4⋊C4—1224no—
403C2×C22⋊C4—22no—
404C2×C22⋊C4—12216no—
405C2×C22⋊C4—1222no—
406C2×C22⋊C4—1222no—
407C2×C22⋊C4—1222no—
408C2×C22⋊C4—1222no—
409C2×C42—611no—
410C2×C42—614no—
411C2×C42—1222no—
412C2×C42—1222no—
413C2×C42—1224no—
414C2×C42—1221no—
415C2×C42C22×D6111yescharacteristic
416#102—24no—
417#102—1224no—
418#102—1224no—
419#102—1224no—
420#101—22no—
421#101—1224no—
422#101—1222no—
423C24—21no—
424C24—611no—
425C24—1224no—
426C24—1221no—
427C24—2441no—
428C24—2441no—
429C24—2442no—
430C24—2448no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—612no—
434C2×Q8—612no—
435C2×Q8—618no—
436C2×Q8—1224no—
437C2×Q8—1224no—
438C2×Q8—1228no—
439C2×Q8—1228no—
440C2×Q8—1228no—
441C2×Q8—1228no—
442C2×Q8—1222no—
443C2×Q8—1222no—
444C2×Q8C2×C4×D6112yes—
445C2×Q8D4×D6112yes—
446C2×D4—22no—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—612no—
450C2×D4—1228no—
451C2×D4—1228no—
452C2×D4—1224no—
453C2×D4—1222no—
454C2×D4—1221no—
455C2×D4—1222no—
456C2×D4—2444no—
457C2×D4—2444no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4C2×C4×D6112yes—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—614no—
468C22×C4—612no—
469C22×C4—618no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—1224no—
475C22×C4—1221no—
476C22×C4—2444no—
477C22×C4D4×D6112yes—
478C22×C4C23×D6111yescharacteristic
479M16—22no—
480M16—22no—
481M16—1222no—
482M16—1222no—
483M16—2444no—
484C2×C8—22no—
485C2×C8—1222no—
486C2×C8—2444no—
487C4⋊C4—24no—
488C4⋊C4—614no—
489C4⋊C4—1224no—
490C4⋊C4—1228no—
491C4⋊C4—1224no—
492C4⋊C4D4×D6114yes—
493C22⋊C4—22no—
494C22⋊C4—28no—
495C22⋊C4—1228no—
496C22⋊C4—1224no—
497C22⋊C4—1228no—
498C22⋊C4—1222no—
499C22⋊C4—1228no—
500C42—24no—
501C42—21no—
502C42—22no—
503C42—612no—
504C42—1224no—
505C42—1224no—
506C42—1224no—
507C42—1228no—
508C42—1221no—
509C42—1222no—
510C42C2×C4×D6112yes—
511C23—22no—
512C23—44no—
513C23—41no—
514C23—41no—
515C23—611no—
516C23—612no—
517C23—611no—
518C23—612no—
519C23—1224no—
520C23—1222no—
521C23—4884no—
522C23—4888no—
523C23—4884no—
524C23—2444no—
525C23—2444no—
526C23—2444no—
527C23—24416no—
528C23—2441no—
529C23—2441no—
530C23C2×D4×D6111yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—12216no—
534Q8—1224no—
535Q8—1224no—
536D4—24no—
537D4—42no—
538D4—1224no—
539D4—2448no—
540D4—2442no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—42no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—614no—
552C2×C4—614no—
553C2×C4—614no—
554C2×C4—1228no—
555C2×C4—1228no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1224no—
560C2×C4—2448no—
561C2×C4—2448no—
562C2×C4—2442no—
563C2×C4C22⋊C4×D6114yes—
564C2×C4C22×C4×D6111yescharacteristic
565C2×C4C2×D4×D6112yes—
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4#41×D6112yes—
568C8—42no—
569C8—2442no—
570C22—22no—
571C22—44no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—612no—
576C22—614no—
577C22—612no—
578C22—1224no—
579C22—1224no—
580C22—1222no—
581C22—4882no—
582C22—4888no—
583C22—4888no—
584C22—4882no—
585C22—2444no—
586C22—2442no—
587C22C2×C22⋊C4×D6111yescharacteristic
588C22#120×D6111yescharacteristic
589C22#123×D6111yescharacteristic
590C4—24no—
591C4—42no—
592C4—44no—
593C4—614no—
594C4—1224no—
595C4—2442no—
596C4—2444no—
597C4C4×D4×D6112yes—
598C4#121×D6112yes—
599C2—21no—
600C2—21no—
601C2—82no—
602C2—614no—
603C2—1222no—
604C2—4882no—
605C2#386×D6111yescharacteristic
606C2#454×D6112yes—
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1888C2
socleC22×C61C2×#120
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1888D61
hypocenterC61C2×#1888
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1888

Complements

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

NH ≅ G/Ntype
2. #519×D61C2semidirect
4. C2×#101×D61C2semidirect
5. #21446223C2semidirect
6. #21446222C2semidirect
7. #21446221C2semidirect
8. #21446220C2semidirect
9. #21446219C2semidirect
10. #339×D61C2semidirect
11. C61×#1888C2semidirect
20. #124×D61C4semidirect
21. #16840240C22semidirect
23. #16840239C22semidirect
35. C2×#14519696C22semidirect
36. C2×#14519695C22semidirect
38. #16839523C22semidirect
39. #16839522C22semidirect
40. #16839521C22semidirect
42. C61×#519C22semidirect
44. C2×#101×C61C22semidirect
45. C61×#339C22semidirect
70. #14519766C2×C4semidirect
72. #14519765C2×C4semidirect
74. #14519764C2×C4semidirect
111. #124×C61C2×C4semidirect
305. #1888D61direct (N × H)
314. D61#1888direct (N × H)
329. #519C2×D61semidirect
344. C2×#101C2×D61semidirect
348. #339C2×D61semidirect
349. C61C2×#1888semidirect
384. #124C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#124×D612
#519×D612
#418×D612
C2×#101×D612
#214462232
#214462222
#214462212
#214462202
#214462192
#339×D612
C61×#18882
C2×#188861