Finite Group Databasebeta

← D61×#1515 #29647275

Subgroups — D61×#1515 #29647275

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#432×D61C211yesmaximal, characteristic
3#383×D61C211yesmaximal, characteristic
4#21444371C211yesmaximal, characteristic, complemented
5#21444370C211yesmaximal, characteristic, complemented
6#21444369C211yesmaximal, characteristic, complemented
7#21444368C214yesmaximal, complemented
8#328×D61C214yesmaximal
9C61×#1515C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840318—21no—
12#16840317C411yescharacteristic, complemented
13#16840317C411yescharacteristic, complemented
14#16840316—21no—
15#16840316C412yescomplemented
16#16840315—21no—
17#16840314—21no—
18#16840314C2212yes—
19C4⋊Q8×D61C411yescharacteristic, complemented
20C4⋊Q8×D61C411yescharacteristic, complemented
21#125×D61—21no—
22C22⋊Q8×D61—21no—
23C22⋊Q8×D61C2212yes—
24C4×Q8×D61—21no—
25C4×Q8×D61C412yescomplemented
26#117×D61—21no—
27C2×C42×D61C2211yescharacteristic
28#16839819C2211yescharacteristic
29#16839818—21no—
30#16839817—21no—
31#16839816C2212yes—
32C4⋊C8×D61—21no—
33C22⋊C8×D61C2212yes—
34#97×D61—21no—
35#16839630C2211yescharacteristic
36#16839629—21no—
37#16839629—21no—
38#16839629C2212yes—
39#95×D61—21no—
40#95×D61—21no—
41#95×D61C2212yes—
42#16839500C2214yes—
43#16839499C2214yes—
44#16839498C2214yes—
45C61×#533C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#383C2211yescharacteristic
48C61×#328C2214yes—
49C2×Q8×D61—21no—
50C2×Q8×D61C2×C412yes—
51C22×C4×D61—22no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×C8×D61—22no—
56#14519777C2×C411yescharacteristic, complemented
57#14519777C2×C411yescharacteristic, complemented
58#14519776—21no—
59#14519776—21no—
60#14519776—21no—
61#14519776C2×C412yescomplemented
62#14519775—21no—
63#14519775C2×C412yescomplemented
64#14519769—21no—
65#14519768—21no—
66#14519768—21no—
67#14519767—22no—
68#14519767—22no—
69#14519757—21no—
70#14519757C2×C412yes—
71#14519756—21no—
72#14519756C2312yes—
73#14519755—22no—
74#14519755—22no—
75#14519754—22no—
76#14519754—22no—
77#14519753—21no—
78#14519753C2×C412yes—
79Q8×Dic61—21no—
80Q8×Dic61C2×C412yescomplemented
81C4×Dic122—21no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122C2×C412yescomplemented
85#14519742—21no—
86#14519742—22no—
87#14519742—22no—
88#14519742C2×C412yescomplemented
89#14519736—21no—
90#14519735—21no—
91#14519735—21no—
92#14519734—22no—
93#14519733—22no—
94C4⋊C4×D61—21no—
95C4⋊C4×D61—21no—
96C4⋊C4×D61—21no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61C2×C412yes—
100C22⋊C4×D61—22no—
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—22no—
105C42×D61C2×C411yescharacteristic
106C42×D61C2×C411yescharacteristic
107C42×D61C2×C411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C4⋊#92418—21no—
114#14519706—22no—
115C22⋊#92418C2312yes—
116#14519689—22no—
117#14519688—21no—
118#14519687—22no—
119#14519684—21no—
120#14519684—21no—
121#14519684—22no—
122#14519684—22no—
123#14519684—22no—
124#14519684C2312yes—
125C4⋊Q8×C61C2×C411yescharacteristic, complemented
126C4⋊Q8×C61C2×C411yescharacteristic, complemented
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2312yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yescomplemented
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C22⋊C8×C61C2312yes—
136#97×C61—21no—
137#95×C61—21no—
138#95×C61—21no—
139#95×C61C2312yes—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—22no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151Q8×D61—44no—
152Q8×D61—44no—
153C2×C4×D61—24no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—44no—
159C2×C4×D61—44no—
160C2×C4×D61—44no—
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22⋊C411yescharacteristic
163C2×C4×D61C22⋊C411yescharacteristic
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22×C411yescharacteristic
170C2×C4×D61C22×C411yescharacteristic
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×Dic122—21no—
174C2×Dic122—22no—
175C2×Dic122—22no—
176C2×Dic122C22×C412yes—
177C2×C8×C61—22no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—22no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C42×C61—22no—
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—22no—
191#11758198—22no—
192#11758198—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61C22×C412yes—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C2×#92418—22no—
223C8×D61—44no—
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—22no—
227C22×D61C2×C22⋊C411yescharacteristic
228C22×D61C2×C22⋊C411yescharacteristic
229C22×D61#13714yes—
230Q8×C61—42no—
231Q8×C61—42no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—42no—
238C2×C4×C61—42no—
239C2×C4×C61—42no—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—44no—
273C4×D61—44no—
274C4×D61—88no—
275Dic122—44no—
276Dic122—44no—
277Dic122—42no—
278Dic122—42no—
279C8×C61—42no—
280#92418—42no—
281C2×#1515—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61C2×#13712yes—
286C2×D61—21no—
287C2×D61—21no—
288C2×D61—28no—
289C2×D61#41012yes—
290C2×D61#41912yes—
291C2×D61#41912yes—
292C2×D61#42112yes—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—82no—
302Dic61—21no—
303Dic61—21no—
304Dic61—21no—
305Dic61—21no—
306Dic61—21no—
307Dic61—21no—
308Dic61—42no—
309Dic61—42no—
310Dic61—82no—
311C2×#533—611no—
312C2×#432—611no—
313C2×#383—611no—
314#1515—611no—
315#1515—611no—
316#1515—614no—
317#1515—611no—
318#1515D6111yescharacteristic, direct product factor
319C2×#328—614no—
320C2×C61—22no—
321C2×C61C2×#41011yescharacteristic
322C2×C61C2×#41911yescharacteristic
323C2×C61C2×#42111yescharacteristic
324D61—22no—
325D61#151514yesdirect product factor
326#533—611no—
327#533—611no—
328#533—612no—
329#533—612no—
330#533—611no—
331#533—1221no—
332#533—1221no—
333#533—1221no—
334#533—1221no—
335#533C2×D6111yescharacteristic
336C2×C4⋊Q8—611no—
337C2×C4⋊Q8—611no—
338C2×#125—1221no—
339C2×C22⋊Q8—612no—
340C2×C22⋊Q8—1221no—
341C2×C4×Q8—612no—
342C2×C4×Q8—1221no—
343C2×#117—1221no—
344C22×C42—611no—
345#432—611no—
346#432—612no—
347#432—1221no—
348#432—1221no—
349#432C2×D6111yescharacteristic
350C2×C4⋊C8—1221no—
351C2×C22⋊C8—612no—
352C2×#97—1221no—
353#383—611no—
354#383—612no—
355#383—1221no—
356#383—1221no—
357#383C2×D6111yescharacteristic
358C2×#95—612no—
359C2×#95—1221no—
360C2×#95—1221no—
361#328—614no—
362#328—614no—
363#328—614no—
364#328C2×D6114yes—
365C61C2×#151511yescharacteristic, complemented, Sylow
366C22×Q8—612no—
367C22×Q8—1221no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—1222no—
372C22×C8—1222no—
373C4⋊Q8—612no—
374C4⋊Q8—612no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8—1221no—
378C4⋊Q8—1221no—
379C4⋊Q8—1221no—
380C4⋊Q8—1221no—
381C4⋊Q8C4×D6111yescharacteristic, complemented
382C4⋊Q8C4×D6111yescharacteristic, complemented
383#125—21no—
384#125—1222no—
385#125—1222no—
386#125—1221no—
387#125—1221no—
388#125—1221no—
389C22⋊Q8—21no—
390C22⋊Q8—612no—
391C22⋊Q8—612no—
392C22⋊Q8—612no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8C22×D6112yes—
401C4×Q8—21no—
402C4×Q8—612no—
403C4×Q8—612no—
404C4×Q8—612no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1221no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×Q8C4×D6112yescomplemented
413#117—21no—
414#117—1222no—
415#117—1222no—
416#117—1221no—
417#117—1221no—
418#117—1221no—
419C2×C4⋊C4—612no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4—1221no—
425C2×C22⋊C4—1222no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—1222no—
438C2×C42C22×D6111yescharacteristic
439C4⋊C8—21no—
440C4⋊C8—1222no—
441C4⋊C8—1221no—
442C22⋊C8—612no—
443C22⋊C8—1222no—
444C22⋊C8C22×D6112yes—
445#97—21no—
446#97—1222no—
447#97—1221no—
448#95—21no—
449#95—21no—
450#95—612no—
451#95—1222no—
452#95—1222no—
453#95—1222no—
454#95—1221no—
455#95—1221no—
456#95C22×D6112yes—
457C24—611no—
458C2×Q8—21no—
459C2×Q8—612no—
460C2×Q8—1222no—
461C2×Q8—1222no—
462C2×Q8—1221no—
463C2×Q8—2444no—
464C2×Q8—2444no—
465C2×Q8C2×C4×D6112yes—
466C22×C4—22no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1224no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2444no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494C2×C8—22no—
495C2×C8—1222no—
496C2×C8—2444no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—612no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4C2×C4×D6112yes—
517C22⋊C4—22no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C42—22no—
522C42—611no—
523C42—611no—
524C42—611no—
525C42—611no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—1222no—
535C42—1222no—
536C42—1222no—
537C42C2×C4×D6111yescharacteristic
538C42C2×C4×D6111yescharacteristic
539C42D4×D6111yescharacteristic
540C42D4×D6111yescharacteristic
541C23—611no—
542C23—611no—
543C23—614no—
544C23—1222no—
545C23—1222no—
546C23C2×D4×D6111yescharacteristic
547Q8—42no—
548Q8—42no—
549Q8—2442no—
550Q8—2442no—
551Q8—2444no—
552Q8—2444no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—611no—
562C2×C4—611no—
563C2×C4—611no—
564C2×C4—611no—
565C2×C4—611no—
566C2×C4—611no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—4888no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2442no—
588C2×C4—2442no—
589C2×C4—2442no—
590C2×C4C22⋊C4×D6111yescharacteristic
591C2×C4C22⋊C4×D6111yescharacteristic
592C2×C4C22⋊C4×D6111yescharacteristic
593C2×C4C22⋊C4×D6111yescharacteristic
594C2×C4C22×C4×D6111yescharacteristic
595C2×C4C2×D4×D6111yescharacteristic
596C8—42no—
597C8—2442no—
598C22—21no—
599C22—21no—
600C22—612no—
601C22—612no—
602C22—612no—
603C22—612no—
604C22—1228no—
605C22—1221no—
606C22—1221no—
607C22C2×C22⋊C4×D6111yescharacteristic
608C22#137×D6112yes—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—42no—
616C4—42no—
617C4—82no—
618C4—1221no—
619C4—1221no—
620C4—1221no—
621C4—1221no—
622C4—1221no—
623C4—1221no—
624C4—4882no—
625C4—2442no—
626C4—2442no—
627C2—22no—
628C2—614no—
629C2—1222no—
630C2#410×D6111yescharacteristic
631C2#419×D6111yescharacteristic
632C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1515C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1515D61
hypocenterC61C2×#1515
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
4. #21444371C2semidirect
5. #21444370C2semidirect
6. #21444369C2semidirect
7. #21444368C2semidirect
9. C61×#1515C2semidirect
12. #16840317C4semidirect
13. #16840317C4semidirect
15. #16840316C4semidirect
19. C4⋊Q8×D61C4semidirect
20. C4⋊Q8×D61C4semidirect
25. C4×Q8×D61C4semidirect
56. #14519777C2×C4semidirect
57. #14519777C2×C4semidirect
61. #14519776C2×C4semidirect
63. #14519775C2×C4semidirect
80. Q8×Dic61C2×C4semidirect
84. C4×Dic122C2×C4semidirect
88. #14519742C2×C4semidirect
125. C4⋊Q8×C61C2×C4semidirect
126. C4⋊Q8×C61C2×C4semidirect
131. C4×Q8×C61C2×C4semidirect
318. #1515D61direct (N × H)
325. D61#1515direct (N × H)
365. C61C2×#1515semidirect
381. C4⋊Q8C4×D61semidirect
382. C4⋊Q8C4×D61semidirect
412. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#432×D612
#383×D612
#214443712
#214443702
#214443692
#214443682
#328×D612
C61×#15152
C2×#151561