Finite Group Databasebeta

← D61×#2502 #29849189

Subgroups — D61×#2502 #29849189

Order 15616.

3094 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C22×C4⋊C4×D61C211yesmaximal, characteristic
3#21450298C211yesmaximal, characteristic, complemented
4#21450297C211yesmaximal, characteristic, complemented
5#21450296C212yesmaximal, complemented
6#21450295C214yesmaximal, complemented
7#21450294C212yesmaximal, complemented
8#21450293C211yesmaximal, characteristic, complemented
9#21450292C214yesmaximal, complemented
10#398×D61C214yesmaximal, complemented
11#391×D61C212yesmaximal, complemented
12#385×D61C214yesmaximal
13#384×D61C212yesmaximal, complemented
14C4×C22⋊C4×D61C211yesmaximal, characteristic
15C61×#2502C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C23×C4×D61C2211yescharacteristic
18C23×C4×D61C2212yes—
19C2×#14519757C2211yescharacteristic
20C2×#14519756C2211yescharacteristic
21C2×#14519755C2212yes—
22C2×#14519754C2212yes—
23C2×#14519753C2211yescharacteristic
24C22⋊Q8×D61—28no—
25C22×C4⋊Dic61C2211yescharacteristic
26C22×#11758196C2212yes—
27C2×C4⋊C4×D61—28no—
28C2×C4⋊C4×D61C2211yescharacteristic
29C2×C4⋊C4×D61C2212yes—
30C2×C4⋊C4×D61C2212yes—
31C2×C4⋊C4×D61C2212yes—
32C2×C4⋊C4×D61C2212yes—
33C2×C4⋊C4×D61C2212yes—
34C2×C4⋊C4×D61C2218yes—
35C2×C22⋊C4×D61C2212yes—
36C2×C22⋊C4×D61C2212yes—
37C2×C42×D61C2212yes—
38#16839696C2218yescomplemented
39#16839695C2218yescomplemented
40#16839694C2214yescomplemented
41#16839693C2214yescomplemented
42#16839692C2214yescomplemented
43#16839669C2212yescomplemented
44#16839668C2212yescomplemented
45#16839667C2212yescomplemented
46#16839666C2218yescomplemented
47#16839637C2214yes—
48#16839636C2218yes—
49#16839635C2214yes—
50#16839634C2214yes—
51#16839633C2218yes—
52#16839632C2212yescomplemented
53#16839631C2218yescomplemented
54#16839631C2212yescomplemented
55#16839631C2212yescomplemented
56C22⋊C4×Dic61C2212yes—
57C4×#11758198C2212yes—
58C4×C22⋊Dic61C2211yescharacteristic
59#16839609C2212yes—
60#95×D61C2218yes—
61#95×D61C2212yes—
62C2×C22⋊Q8×C61C2211yescharacteristic
63C22×C4⋊C4×C61C2211yescharacteristic
64C61×#398C2214yescomplemented
65C61×#391C2212yescomplemented
66C61×#385C2214yes—
67C61×#384C2212yescomplemented
68C4×C22⋊C4×C61C2211yescharacteristic
69C24×D61C2311yescharacteristic
70C22×Dic122C2312yes—
71C22×Dic122C2311yescharacteristic
72C2×Q8×D61—24no—
73C2×Q8×D61—28no—
74C23×Dic61C2311yescharacteristic
75C23×Dic61C2312yes—
76C22×C4×D61—28no—
77C22×C4×D61—28no—
78C22×C4×D61—48no—
79C22×C4×D61D418yes—
80C22×C4×D61C2312yes—
81C22×C4×D61C2312yes—
82C22×C4×D61C2314yes—
83C22×C4×D61C2311yescharacteristic
84C22×C4×D61C2311yescharacteristic
85C22×C4×D61C2311yescharacteristic
86C22×C4×D61C2314yes—
87C22×C4×D61C2314yes—
88C22×C4×D61C2311yescharacteristic
89#14519757—24no—
90#14519756—24no—
91#14519755—28no—
92#14519754—28no—
93#14519753—24no—
94C2×C4⋊Dic61—24no—
95C2×C4⋊Dic61C2312yes—
96C2×C4⋊Dic61C2312yes—
97C2×C4⋊Dic61C2312yes—
98C2×C4⋊Dic61C2311yescharacteristic
99C2×C4⋊Dic61C2318yes—
100C2×#11758196—28no—
101C2×#11758196C2312yes—
102C2×#11758196C2314yes—
103C2×#11758196C2312yes—
104C2×#11758196C2314yes—
105C2×#11758196C2312yes—
106C2×#11758196C2318yes—
107C2×#11758196C2318yes—
108C4⋊C4×D61—28no—
109C4⋊C4×D61—28no—
110C4⋊C4×D61—28no—
111C4⋊C4×D61—28no—
112C4⋊C4×D61—28no—
113C4⋊C4×D61—24no—
114C4⋊C4×D61—416no—
115C2×C22⋊Dic61C2312yes—
116C2×C22⋊Dic61C2312yes—
117C2×#11758198C2312yes—
118C2×#11758198C2312yes—
119C2×#11758198C2314yes—
120C22⋊C4×D61—28no—
121C22⋊C4×D61Q8116yescomplemented
122C2×C4×Dic61C2314yes—
123C2×C4×Dic61C2312yes—
124C42×D61—28no—
125#14519684C2312yes—
126#14519684C2318yes—
127#14519684C2318yes—
128#14519684C2318yes—
129#14519684C2314yes—
130C22×Q8×C61C2311yescharacteristic
131C23×C4×C61C2311yescharacteristic
132C23×C4×C61C2312yes—
133C22⋊Q8×C61—24no—
134C2×C4⋊C4×C61—24no—
135C2×C4⋊C4×C61C2312yes—
136C2×C4⋊C4×C61C2312yes—
137C2×C4⋊C4×C61C2312yes—
138C2×C4⋊C4×C61C2311yescharacteristic
139C2×C4⋊C4×C61C2318yes—
140C2×C22⋊C4×C61C2312yes—
141C2×C22⋊C4×C61C2312yes—
142C2×C42×C61C2312yes—
143#95×C61C2312yes—
144#95×C61C2318yes—
145C24×C61C2411yescharacteristic
146C23×D61—24no—
147C23×D61—24no—
148C23×D61C2×D414yes—
149C23×D61C2×Q818yes—
150C23×D61C2411yescharacteristic
151C23×D61C2411yescharacteristic
152C2×Q8×C61—22no—
153C2×Q8×C61—24no—
154C22×C4×C61—24no—
155C22×C4×C61—24no—
156C22×C4×C61—44no—
157C22×C4×C61C2×D414yes—
158C22×C4×C61C2414yes—
159C22×C4×C61C2412yes—
160C22×C4×C61C2412yes—
161C22×C4×C61C2411yescharacteristic
162C22×C4×C61C2414yes—
163C22×C4×C61C2411yescharacteristic
164C22×Dic61—24no—
165C22×Dic61—24no—
166C22×Dic61—44no—
167C22×Dic61C2×D414yes—
168C22×Dic61C2414yes—
169C22×Dic61C2412yes—
170C22×Dic61C2412yes—
171C22×Dic61C2411yescharacteristic
172C22×Dic61C2414yes—
173C22×Dic61C2411yescharacteristic
174Q8×D61—416no—
175C2×C4×D61—24no—
176C2×C4×D61—28no—
177C2×C4×D61—28no—
178C2×C4×D61—24no—
179C2×C4×D61—24no—
180C2×C4×D61—232no—
181C2×C4×D61—28no—
182C2×C4×D61—28no—
183C2×C4×D61—24no—
184C2×C4×D61—24no—
185C2×C4×D61—28no—
186C2×C4×D61—216no—
187C2×C4×D61—28no—
188C2×C4×D61—48no—
189C2×C4×D61—432no—
190C2×C4×D61C2×D414yes—
191C2×C4×D61C2×D414yes—
192C2×C4×D61C2×D414yes—
193C2×C4×D61C2×D414yes—
194C2×C4×D61C2×Q8116yes—
195C2×C4×D61#4118yes—
196C2×Dic122—22no—
197C2×Dic122—24no—
198C2×Dic122—24no—
199C2×Dic122—28no—
200C4⋊C4×C61—24no—
201C4⋊C4×C61—24no—
202C4⋊C4×C61—24no—
203C4⋊C4×C61—22no—
204C4⋊C4×C61—48no—
205C22⋊C4×C61—24no—
206C22⋊C4×C61C2×Q818yescomplemented
207C42×C61—24no—
208C22⋊Dic61—24no—
209C22⋊Dic61C2×Q818yescomplemented
210#11758198—24no—
211#11758198—24no—
212#11758198C2×Q8116yescomplemented
213C4⋊Dic61—24no—
214C4⋊Dic61—24no—
215C4⋊Dic61—22no—
216C4⋊Dic61—24no—
217C4⋊Dic61—48no—
218#11758196—24no—
219#11758196—28no—
220#11758196—28no—
221#11758196—24no—
222#11758196—24no—
223#11758196—48no—
224#11758196—48no—
225C4×Dic61—24no—
226C4×Dic61—28no—
227C23×C61—22no—
228C23×C61—22no—
229C23×C61C22×D412yes—
230C23×C61C22×Q814yes—
231C23×C61C2511yescharacteristic
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—28no—
235C22×D61—28no—
236C22×D61—232no—
237C22×D61C4⋊Q8116yes—
238C22×D61C22×D412yes—
239C22×D61C22×D412yes—
240C22×D61C22×Q814yes—
241C22×D61C22×Q814yes—
242C22×D61C2×#4114yes—
243C22×D61H(5,𝔽2)14yes—
244Q8×C61—44no—
245C2×C4×C61—24no—
246C2×C4×C61—22no—
247C2×C4×C61—24no—
248C2×C4×C61—28no—
249C2×C4×C61—24no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—24no—
253C2×C4×C61—24no—
254C2×C4×C61—28no—
255C2×C4×C61—48no—
256C2×C4×C61C22×D412yes—
257C2×C4×C61C22×D412yes—
258C2×C4×C61C22×Q818yes—
259C2×C4×C61C2×#4114yes—
260C2×Dic61—24no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—28no—
264C2×Dic61—22no—
265C2×Dic61—24no—
266C2×Dic61—22no—
267C2×Dic61—24no—
268C2×Dic61—24no—
269C2×Dic61—28no—
270C2×Dic61—48no—
271C2×Dic61C22×D412yes—
272C2×Dic61C22×D412yes—
273C2×Dic61C22×Q818yes—
274C2×Dic61C2×#4114yes—
275C4×D61—28no—
276C4×D61—232no—
277C4×D61—28no—
278C4×D61—28no—
279C4×D61—416no—
280C4×D61—48no—
281C4×D61—48no—
282C4×D61—416no—
283C4×D61#123116yes—
284Dic122—44no—
285Dic122—48no—
286C2×#2502—611nomaximal, Sylow
287C22×C61—22no—
288C22×C61—22no—
289C22×C61—28no—
290C22×C61C2×C4⋊Q814yes—
291C22×C61C23×D411yescharacteristic
292C22×C61C23×Q812yes—
293C22×C61C22×#4112yes—
294C22×C61C2×H(5,𝔽2)12yes—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—28no—
298C2×D61—232no—
299C2×D61C2×#12314yes—
300C2×D61C2×C4⋊Q814yes—
301C2×D61C2×C4⋊Q814yes—
302C2×D61#53514yes—
303C2×D61#554116yes—
304C4×C61—22no—
305C4×C61—28no—
306C4×C61—42no—
307C4×C61—44no—
308C4×C61—42no—
309C4×C61C2×#12314yes—
310Dic61—22no—
311Dic61—28no—
312Dic61—42no—
313Dic61—44no—
314Dic61—42no—
315Dic61C2×#12314yes—
316C22×C22⋊Q8—611no—
317C23×C4⋊C4—611no—
318#2502—614no—
319#2502—612no—
320#2502—614no—
321#2502—612no—
322#2502—611no—
323#2502—611no—
324#2502—611no—
325#2502D6111yescharacteristic, direct product factor
326C2×#398—614no—
327C2×#391—612no—
328C2×#385—614no—
329C2×#384—612no—
330C2×C4×C22⋊C4—611no—
331C2×C61—24no—
332C2×C61C22×#12311yescharacteristic
333C2×C61C22×C4⋊Q811yescharacteristic
334C2×C61C2×#53511yescharacteristic
335C2×C61C2×#55414yes—
336D61—24no—
337D61#250218yesdirect product factor
338C23×Q8—611no—
339C24×C4—611no—
340C24×C4—612no—
341C2×C22⋊Q8—611no—
342C2×C22⋊Q8—611no—
343C2×C22⋊Q8—612no—
344C2×C22⋊Q8—611no—
345C2×C22⋊Q8—612no—
346C2×C22⋊Q8—1228no—
347C2×C22⋊Q8C2×D6111yescharacteristic
348C22×C4⋊C4—612no—
349C22×C4⋊C4—611no—
350C22×C4⋊C4—612no—
351C22×C4⋊C4—612no—
352C22×C4⋊C4—612no—
353C22×C4⋊C4—612no—
354C22×C4⋊C4—612no—
355C22×C4⋊C4—611no—
356C22×C4⋊C4—618no—
357C22×C4⋊C4—1228no—
358C22×C4⋊C4C2×D6111yescharacteristic
359C22×C22⋊C4—612no—
360C22×C22⋊C4—612no—
361C22×C42—612no—
362#398—614no—
363#398—614no—
364#398—618no—
365#398—614no—
366#398—618no—
367#398C2×D6114yescomplemented
368#391—612no—
369#391—612no—
370#391—612no—
371#391—618no—
372#391C2×D6112yescomplemented
373#385—614no—
374#385—618no—
375#385—614no—
376#385—614no—
377#385—618no—
378#385C2×D6114yes—
379#384—618no—
380#384—612no—
381#384—612no—
382#384—612no—
383#384C2×D6112yescomplemented
384C4×C22⋊C4—611no—
385C4×C22⋊C4—612no—
386C4×C22⋊C4—612no—
387C4×C22⋊C4—612no—
388C4×C22⋊C4C2×D6111yescharacteristic
389C2×#95—612no—
390C2×#95—618no—
391C61C2×#250211yescharacteristic, complemented, Sylow
392C25—611no—
393C22×Q8—612no—
394C22×Q8—611no—
395C22×Q8—1224no—
396C22×Q8—1228no—
397C22×Q8C22×D6111yescharacteristic
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—618no—
401C23×C4—614no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—611no—
405C23×C4—612no—
406C23×C4—612no—
407C23×C4—612no—
408C23×C4—614no—
409C23×C4—614no—
410C23×C4—1228no—
411C23×C4—1228no—
412C23×C4—2448no—
413C23×C4C22×D6111yescharacteristic
414C23×C4C22×D6112yes—
415C22⋊Q8—24no—
416C22⋊Q8—1228no—
417C22⋊Q8—1228no—
418C22⋊Q8—1224no—
419C22⋊Q8—1224no—
420C22⋊Q8—1224no—
421C2×C4⋊C4—24no—
422C2×C4⋊C4—614no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—614no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—612no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—612no—
429C2×C4⋊C4—612no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—618no—
432C2×C4⋊C4—618no—
433C2×C4⋊C4—618no—
434C2×C4⋊C4—1228no—
435C2×C4⋊C4—1228no—
436C2×C4⋊C4—1228no—
437C2×C4⋊C4—1228no—
438C2×C4⋊C4—1224no—
439C2×C4⋊C4—1228no—
440C2×C4⋊C4—1228no—
441C2×C4⋊C4—1224no—
442C2×C4⋊C4—24416no—
443C2×C4⋊C4C22×D6112yes—
444C2×C4⋊C4C22×D6112yes—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6112yes—
447C2×C4⋊C4C22×D6118yes—
448C2×C22⋊C4—612no—
449C2×C22⋊C4—614no—
450C2×C22⋊C4—6116no—
451C2×C22⋊C4—612no—
452C2×C22⋊C4—612no—
453C2×C22⋊C4—612no—
454C2×C22⋊C4—1228no—
455C2×C22⋊C4C22×D6112yes—
456C2×C22⋊C4C22×D6112yes—
457C2×C42—614no—
458C2×C42—612no—
459C2×C42—1228no—
460C2×C42C22×D6112yes—
461#95—612no—
462#95—614no—
463#95—618no—
464#95—618no—
465#95—618no—
466#95C22×D6112yes—
467#95C22×D6118yes—
468C24—611no—
469C24—614no—
470C24—618no—
471C24—611no—
472C24—1224no—
473C24—1224no—
474C24C23×D6111yescharacteristic
475C2×Q8—22no—
476C2×Q8—24no—
477C2×Q8—1224no—
478C2×Q8—1228no—
479C2×Q8—1222no—
480C2×Q8—1224no—
481C2×Q8—24416no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—44no—
485C22×C4—611no—
486C22×C4—614no—
487C22×C4—618no—
488C22×C4—614no—
489C22×C4—614no—
490C22×C4—611no—
491C22×C4—614no—
492C22×C4—614no—
493C22×C4—614no—
494C22×C4—6116no—
495C22×C4—612no—
496C22×C4—614no—
497C22×C4—612no—
498C22×C4—1224no—
499C22×C4—12232no—
500C22×C4—1228no—
501C22×C4—1228no—
502C22×C4—1224no—
503C22×C4—1224no—
504C22×C4—1224no—
505C22×C4—1228no—
506C22×C4—1228no—
507C22×C4—1228no—
508C22×C4—12216no—
509C22×C4—1228no—
510C22×C4—1224no—
511C22×C4—1224no—
512C22×C4—1224no—
513C22×C4—2448no—
514C22×C4—24432no—
515C22×C4—2444no—
516C22×C4D4×D6114yes—
517C22×C4C23×D6112yes—
518C22×C4C23×D6114yes—
519C22×C4C23×D6112yes—
520C22×C4C23×D6111yescharacteristic
521C22×C4C23×D6114yes—
522C22×C4C23×D6111yescharacteristic
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—24no—
527C4⋊C4—48no—
528C4⋊C4—1228no—
529C4⋊C4—1228no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1222no—
537C4⋊C4—2448no—
538C4⋊C4—2448no—
539C4⋊C4—2448no—
540C22⋊C4—24no—
541C22⋊C4—618no—
542C22⋊C4—6116no—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4Q8×D6118yescomplemented
547C42—24no—
548C42—1228no—
549C42—1224no—
550C23—22no—
551C23—22no—
552C23—612no—
553C23—614no—
554C23—614no—
555C23—614no—
556C23—6116no—
557C23—612no—
558C23—614no—
559C23—1228no—
560C23—1228no—
561C23—12232no—
562C23—1222no—
563C23—1222no—
564C23C2×D4×D6112yes—
565C23C2×Q8×D6114yes—
566C23C24×D6111yescharacteristic
567Q8—44no—
568Q8—2448no—
569Q8—2444no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—48no—
581C2×C4—612no—
582C2×C4—614no—
583C2×C4—6116no—
584C2×C4—612no—
585C2×C4—618no—
586C2×C4—1228no—
587C2×C4—12232no—
588C2×C4—1228no—
589C2×C4—1228no—
590C2×C4—1224no—
591C2×C4—1222no—
592C2×C4—1224no—
593C2×C4—1228no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1224no—
597C2×C4—1224no—
598C2×C4—1224no—
599C2×C4—1228no—
600C2×C4—2448no—
601C2×C4—24416no—
602C2×C4—2448no—
603C2×C4—24416no—
604C2×C4—2448no—
605C2×C4C2×D4×D6112yes—
606C2×C4C2×D4×D6112yes—
607C2×C4C2×Q8×D6118yes—
608C2×C4#41×D6114yes—
609C22—22no—
610C22—28no—
611C22—22no—
612C22—614no—
613C22—6116no—
614C22—614no—
615C22—614no—
616C22—614no—
617C22—12232no—
618C22—1222no—
619C22—1228no—
620C22—1222no—
621C22C4⋊Q8×D6114yes—
622C22C22×D4×D6111yescharacteristic
623C22C22×Q8×D6112yes—
624C22C2×#41×D6112yes—
625C22H(5,𝔽2)×D6112yes—
626C4—22no—
627C4—28no—
628C4—42no—
629C4—44no—
630C4—42no—
631C4—614no—
632C4—1222no—
633C4—1228no—
634C4—2442no—
635C4—2442no—
636C4—2444no—
637C4#123×D6114yes—
638C2—24no—
639C2—618no—
640C2—1224no—
641C2C2×#123×D6111yescharacteristic
642C2C2×C4⋊Q8×D6111yescharacteristic
643C2#535×D6111yescharacteristic
644C2#554×D6114yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2502C2
prime radicalC24×D61C23
hypercenter#2502D61
hypocenterC61C2×#2502
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2502

Complements

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

NH ≅ G/Ntype
3. #21450298C2semidirect
4. #21450297C2semidirect
5. #21450296C2semidirect
6. #21450295C2semidirect
7. #21450294C2semidirect
8. #21450293C2semidirect
9. #21450292C2semidirect
10. #398×D61C2semidirect
11. #391×D61C2semidirect
13. #384×D61C2semidirect
15. C61×#2502C2semidirect
38. #16839696C22semidirect
39. #16839695C22semidirect
40. #16839694C22semidirect
41. #16839693C22semidirect
42. #16839692C22semidirect
43. #16839669C22semidirect
44. #16839668C22semidirect
45. #16839667C22semidirect
46. #16839666C22semidirect
52. #16839632C22semidirect
53. #16839631C22semidirect
54. #16839631C22semidirect
55. #16839631C22semidirect
64. C61×#398C22semidirect
65. C61×#391C22semidirect
67. C61×#384C22semidirect
121. C22⋊C4×D61Q8semidirect
206. C22⋊C4×C61C2×Q8semidirect
209. C22⋊Dic61C2×Q8semidirect
212. #11758198C2×Q8semidirect
325. #2502D61direct (N × H)
337. D61#2502direct (N × H)
367. #398C2×D61semidirect
372. #391C2×D61semidirect
383. #384C2×D61semidirect
391. C61C2×#2502semidirect
546. C22⋊C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
C22×C4⋊C4×D612
#214502982
#214502972
#214502962
#214502952
#214502942
#214502932
#214502922
#398×D612
#391×D612
#385×D612
#384×D612
C4×C22⋊C4×D612
C61×#25022
C2×#250261