Finite Group Databasebeta

← D61×#2503 #29849300

Subgroups — D61×#2503 #29849300

Order 15616.

2918 conjugacy classes of subgroups.

The subgroup lattice has 636 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#21450305C211yesmaximal, characteristic, complemented
4#21450304C211yesmaximal, characteristic, complemented
5#21450303C212yesmaximal, complemented
6#21450302C214yesmaximal, complemented
7#21450301C212yesmaximal, complemented
8#21450300C211yesmaximal, characteristic, complemented
9#21450299C214yesmaximal, complemented
10#400×D61C214yesmaximal, complemented
11#391×D61C212yesmaximal, complemented
12#382×D61C212yesmaximal, complemented
13#381×D61C214yesmaximal
14C4×C22⋊C4×D61C211yesmaximal, characteristic
15C61×#2503C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C23×C4×D61C2212yes—
18C23×C4×D61C2211yescharacteristic
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×C22⋊C4×D61C2212yes—
35C2×C22⋊C4×D61C2212yes—
36C2×C42×D61C2212yes—
37#16839704C2218yescomplemented
38#16839703C2218yescomplemented
39#16839702C2214yescomplemented
40#16839701C2214yescomplemented
41#16839700C2214yescomplemented
42#16839669C2212yescomplemented
43#16839668C2212yescomplemented
44#16839667C2212yescomplemented
45#16839666C2218yescomplemented
46#16839628C2212yescomplemented
47#16839627C2212yescomplemented
48#16839627C2212yescomplemented
49#16839626C2218yescomplemented
50#16839625C2214yes—
51#16839624C2218yes—
52#16839624C2218yes—
53#16839623C2214yes—
54#16839623C2214yes—
55C22⋊C4×Dic61C2212yes—
56C4×#11758198C2212yes—
57C4×C22⋊Dic61C2211yescharacteristic
58#16839609C2212yes—
59#95×D61C2218yes—
60#95×D61C2218yes—
61#95×D61C2212yes—
62C2×C22⋊Q8×C61C2211yescharacteristic
63C22×C4⋊C4×C61C2211yescharacteristic
64C61×#400C2214yescomplemented
65C61×#391C2212yescomplemented
66C61×#382C2212yescomplemented
67C61×#381C2214yes—
68C4×C22⋊C4×C61C2211yescharacteristic
69C24×D61C2311yescharacteristic
70C22×Dic122C2312yes—
71C22×Dic122C2311yescharacteristic
72C2×Q8×D61—24no—
73C2×Q8×D61—28no—
74C23×Dic61C2312yes—
75C23×Dic61C2311yescharacteristic
76C22×C4×D61—28no—
77C22×C4×D61—28no—
78C22×C4×D61—216no—
79C22×C4×D61D418yes—
80C22×C4×D61C2314yes—
81C22×C4×D61C2312yes—
82C22×C4×D61C2312yes—
83C22×C4×D61C2314yes—
84C22×C4×D61C2314yes—
85C22×C4×D61C2311yescharacteristic
86C22×C4×D61C2311yescharacteristic
87C22×C4×D61C2311yescharacteristic
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×#11758196—28no—
100C2×#11758196C2314yes—
101C2×#11758196C2314yes—
102C2×#11758196C2312yes—
103C2×#11758196C2312yes—
104C2×#11758196C2312yes—
105C4⋊C4×D61—28no—
106C4⋊C4×D61—28no—
107C4⋊C4×D61—28no—
108C4⋊C4×D61—28no—
109C4⋊C4×D61—28no—
110C4⋊C4×D61—24no—
111C2×C22⋊Dic61C2312yes—
112C2×C22⋊Dic61C2312yes—
113C2×#11758198C2312yes—
114C2×#11758198C2312yes—
115C2×#11758198C2314yes—
116C22⋊C4×D61—28no—
117C22⋊C4×D61—28no—
118C2×C4×Dic61C2314yes—
119C2×C4×Dic61C2312yes—
120C42×D61—28no—
121#14519684C2312yes—
122#14519684C2318yes—
123#14519684C2318yes—
124#14519684C2318yes—
125#14519684C2318yes—
126#14519684C2318yes—
127#14519684C2318yes—
128#14519684C2314yes—
129C22×Q8×C61C2311yescharacteristic
130C23×C4×C61C2312yes—
131C23×C4×C61C2311yescharacteristic
132C22⋊Q8×C61—24no—
133C2×C4⋊C4×C61—24no—
134C2×C4⋊C4×C61C2312yes—
135C2×C4⋊C4×C61C2312yes—
136C2×C4⋊C4×C61C2312yes—
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C22⋊C4×C61C2312yes—
139C2×C22⋊C4×C61C2312yes—
140C2×C42×C61C2312yes—
141#95×C61C2312yes—
142#95×C61C2318yes—
143#95×C61C2318yes—
144C24×C61C2411yescharacteristic
145C23×D61—24no—
146C23×D61—24no—
147C23×D61C2×D414yes—
148C23×D61#4118yes—
149C23×D61C2411yescharacteristic
150C23×D61C2411yescharacteristic
151C2×Q8×C61—22no—
152C2×Q8×C61—24no—
153C22×C4×C61—24no—
154C22×C4×C61—24no—
155C22×C4×C61—28no—
156C22×C4×C61C2×D414yes—
157C22×C4×C61C2414yes—
158C22×C4×C61C2412yes—
159C22×C4×C61C2414yes—
160C22×C4×C61C2412yes—
161C22×C4×C61C2411yescharacteristic
162C22×C4×C61C2411yescharacteristic
163C22×Dic61—24no—
164C22×Dic61—24no—
165C22×Dic61—28no—
166C22×Dic61C2×D414yes—
167C22×Dic61C2414yes—
168C22×Dic61C2412yes—
169C22×Dic61C2414yes—
170C22×Dic61C2412yes—
171C22×Dic61C2411yescharacteristic
172C22×Dic61C2411yescharacteristic
173Q8×D61—416no—
174C2×C4×D61—24no—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—232no—
178C2×C4×D61—28no—
179C2×C4×D61—28no—
180C2×C4×D61—24no—
181C2×C4×D61—24no—
182C2×C4×D61—216no—
183C2×C4×D61—216no—
184C2×C4×D61—28no—
185C2×C4×D61—28no—
186C2×C4×D61—28no—
187C2×C4×D61—28no—
188C2×C4×D61—28no—
189C2×C4×D61—432no—
190C2×C4×D61C2×D414yes—
191C2×C4×D61C2×D414yes—
192C2×C4×D61C2×D414yes—
193C2×C4×D61C2×D414yes—
194C2×C4×D61#4118yes—
195C2×Dic122—22no—
196C2×Dic122—24no—
197C2×Dic122—24no—
198C2×Dic122—28no—
199C4⋊C4×C61—24no—
200C4⋊C4×C61—24no—
201C4⋊C4×C61—24no—
202C4⋊C4×C61—22no—
203C22⋊C4×C61—24no—
204C22⋊C4×C61—24no—
205C42×C61—24no—
206C22⋊Dic61—24no—
207C22⋊Dic61—24no—
208#11758198—24no—
209#11758198—24no—
210#11758198—28no—
211C4⋊Dic61—24no—
212C4⋊Dic61—24no—
213C4⋊Dic61—22no—
214C4⋊Dic61—24no—
215#11758196—24no—
216#11758196—28no—
217#11758196—28no—
218#11758196—24no—
219#11758196—24no—
220C4×Dic61—24no—
221C4×Dic61—28no—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61C22×D412yes—
225C23×C61C2×#4114yes—
226C23×C61C2511yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—28no—
230C22×D61—28no—
231C22×D61—232no—
232C22×D61#124116yes—
233C22×D61C22×D412yes—
234C22×D61C22×D412yes—
235C22×D61C2×#4114yes—
236C22×D61C2×#4114yes—
237C22×D61C2×#4114yes—
238C22×D612−1+414yes—
239Q8×C61—44no—
240C2×C4×C61—22no—
241C2×C4×C61—28no—
242C2×C4×C61—24no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—28no—
249C2×C4×C61—24no—
250C2×C4×C61—24no—
251C2×C4×C61—48no—
252C2×C4×C61C22×D412yes—
253C2×C4×C61C22×D412yes—
254C2×C4×C61C2×#4114yes—
255C2×Dic61—22no—
256C2×Dic61—28no—
257C2×Dic61—22no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—24no—
261C2×Dic61—24no—
262C2×Dic61—24no—
263C2×Dic61—28no—
264C2×Dic61—24no—
265C2×Dic61—24no—
266C2×Dic61—48no—
267C2×Dic61C22×D412yes—
268C2×Dic61C22×D412yes—
269C2×Dic61C2×#4114yes—
270C4×D61—28no—
271C4×D61—28no—
272C4×D61—28no—
273C4×D61—48no—
274C4×D61—416no—
275C4×D61—416no—
276C4×D61—48no—
277C4×D61—416no—
278C4×D61#123116yes—
279Dic122—44no—
280Dic122—48no—
281C2×#2503—611nomaximal, Sylow
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—28no—
285C22×C61C2×#12414yes—
286C22×C61C23×D411yescharacteristic
287C22×C61C22×#4112yes—
288C22×C61C22×#4112yes—
289C22×C61C2×2−1+412yes—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—28no—
293C2×D61—232no—
294C2×D61C2×#12314yes—
295C2×D61C2×#12414yes—
296C2×D61C2×#12414yes—
297C2×D61#53614yes—
298C2×D61#552116yes—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—44no—
302C4×C61—42no—
303C4×C61—44no—
304C4×C61C2×#12314yes—
305Dic61—22no—
306Dic61—42no—
307Dic61—44no—
308Dic61—42no—
309Dic61—44no—
310Dic61C2×#12314yes—
311C22×C22⋊Q8—611no—
312C23×C4⋊C4—611no—
313#2503—614no—
314#2503—612no—
315#2503—612no—
316#2503—614no—
317#2503—611no—
318#2503—611no—
319#2503—611no—
320#2503D6111yescharacteristic, direct product factor
321C2×#400—614no—
322C2×#391—612no—
323C2×#382—612no—
324C2×#381—614no—
325C2×C4×C22⋊C4—611no—
326C2×C61—24no—
327C2×C61C22×#12311yescharacteristic
328C2×C61C22×#12411yescharacteristic
329C2×C61C2×#53611yescharacteristic
330C2×C61C2×#55214yes—
331D61—24no—
332D61#250318yesdirect product factor
333C23×Q8—611no—
334C24×C4—611no—
335C24×C4—612no—
336C2×C22⋊Q8—611no—
337C2×C22⋊Q8—611no—
338C2×C22⋊Q8—612no—
339C2×C22⋊Q8—611no—
340C2×C22⋊Q8—612no—
341C2×C22⋊Q8—1228no—
342C2×C22⋊Q8C2×D6111yescharacteristic
343C22×C4⋊C4—612no—
344C22×C4⋊C4—611no—
345C22×C4⋊C4—612no—
346C22×C4⋊C4—612no—
347C22×C4⋊C4—612no—
348C22×C4⋊C4—612no—
349C22×C4⋊C4—612no—
350C22×C4⋊C4—611no—
351C22×C4⋊C4—1228no—
352C22×C4⋊C4C2×D6111yescharacteristic
353C22×C22⋊C4—612no—
354C22×C22⋊C4—612no—
355C22×C42—612no—
356#400—614no—
357#400—614no—
358#400—614no—
359#400—618no—
360#400—618no—
361#400C2×D6114yescomplemented
362#391—612no—
363#391—612no—
364#391—612no—
365#391—618no—
366#391C2×D6112yescomplemented
367#382—612no—
368#382—612no—
369#382—612no—
370#382—618no—
371#382C2×D6112yescomplemented
372#381—614no—
373#381—614no—
374#381—614no—
375#381—618no—
376#381—618no—
377#381C2×D6114yes—
378C4×C22⋊C4—611no—
379C4×C22⋊C4—612no—
380C4×C22⋊C4—612no—
381C4×C22⋊C4—612no—
382C4×C22⋊C4C2×D6111yescharacteristic
383C2×#95—612no—
384C2×#95—618no—
385C2×#95—618no—
386C61C2×#250311yescharacteristic, complemented, Sylow
387C25—611no—
388C22×Q8—612no—
389C22×Q8—611no—
390C22×Q8—1224no—
391C22×Q8—1228no—
392C22×Q8C22×D6111yescharacteristic
393C23×C4—611no—
394C23×C4—611no—
395C23×C4—618no—
396C23×C4—611no—
397C23×C4—611no—
398C23×C4—611no—
399C23×C4—612no—
400C23×C4—612no—
401C23×C4—612no—
402C23×C4—614no—
403C23×C4—614no—
404C23×C4—614no—
405C23×C4—1228no—
406C23×C4—1228no—
407C23×C4—12216no—
408C23×C4C22×D6112yes—
409C23×C4C22×D6111yescharacteristic
410C22⋊Q8—24no—
411C22⋊Q8—1228no—
412C22⋊Q8—1228no—
413C22⋊Q8—1224no—
414C22⋊Q8—1224no—
415C22⋊Q8—1224no—
416C2×C4⋊C4—24no—
417C2×C4⋊C4—614no—
418C2×C4⋊C4—612no—
419C2×C4⋊C4—614no—
420C2×C4⋊C4—612no—
421C2×C4⋊C4—612no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—612no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—1228no—
427C2×C4⋊C4—1228no—
428C2×C4⋊C4—1228no—
429C2×C4⋊C4—1228no—
430C2×C4⋊C4—1224no—
431C2×C4⋊C4—1228no—
432C2×C4⋊C4—1228no—
433C2×C4⋊C4—1224no—
434C2×C4⋊C4C22×D6112yes—
435C2×C4⋊C4C22×D6112yes—
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6112yes—
438C2×C22⋊C4—612no—
439C2×C22⋊C4—612no—
440C2×C22⋊C4—612no—
441C2×C22⋊C4—612no—
442C2×C22⋊C4—614no—
443C2×C22⋊C4—1228no—
444C2×C22⋊C4—1228no—
445C2×C22⋊C4C22×D6112yes—
446C2×C22⋊C4C22×D6112yes—
447C2×C42—614no—
448C2×C42—612no—
449C2×C42—1228no—
450C2×C42C22×D6112yes—
451#95—614no—
452#95—612no—
453#95—618no—
454#95—618no—
455#95—618no—
456#95—618no—
457#95—618no—
458#95—618no—
459#95C22×D6112yes—
460#95C22×D6118yes—
461#95C22×D6118yes—
462C24—611no—
463C24—614no—
464C24—618no—
465C24—611no—
466C24—1224no—
467C24—1224no—
468C24C23×D6111yescharacteristic
469C2×Q8—22no—
470C2×Q8—24no—
471C2×Q8—1224no—
472C2×Q8—1228no—
473C2×Q8—1222no—
474C2×Q8—1224no—
475C2×Q8—24416no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—611no—
480C22×C4—614no—
481C22×C4—618no—
482C22×C4—614no—
483C22×C4—611no—
484C22×C4—614no—
485C22×C4—614no—
486C22×C4—614no—
487C22×C4—614no—
488C22×C4—612no—
489C22×C4—612no—
490C22×C4—614no—
491C22×C4—1224no—
492C22×C4—12232no—
493C22×C4—1224no—
494C22×C4—1228no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1228no—
498C22×C4—1228no—
499C22×C4—1228no—
500C22×C4—12216no—
501C22×C4—12216no—
502C22×C4—1228no—
503C22×C4—1228no—
504C22×C4—1228no—
505C22×C4—1224no—
506C22×C4—1224no—
507C22×C4—1224no—
508C22×C4—1228no—
509C22×C4—24432no—
510C22×C4D4×D6114yes—
511C22×C4C23×D6114yes—
512C22×C4C23×D6112yes—
513C22×C4C23×D6112yes—
514C22×C4C23×D6114yes—
515C22×C4C23×D6111yescharacteristic
516C22×C4C23×D6111yescharacteristic
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—1228no—
522C4⋊C4—1228no—
523C4⋊C4—1224no—
524C4⋊C4—1224no—
525C4⋊C4—1224no—
526C4⋊C4—1224no—
527C4⋊C4—1224no—
528C4⋊C4—1224no—
529C4⋊C4—1222no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—1228no—
533C22⋊C4—1224no—
534C22⋊C4—1224no—
535C22⋊C4—1224no—
536C22⋊C4—1224no—
537C42—24no—
538C42—1228no—
539C42—1224no—
540C23—22no—
541C23—22no—
542C23—612no—
543C23—614no—
544C23—614no—
545C23—614no—
546C23—6116no—
547C23—612no—
548C23—614no—
549C23—1228no—
550C23—12232no—
551C23—1228no—
552C23—1222no—
553C23—1222no—
554C23C2×D4×D6112yes—
555C23#41×D6114yes—
556C23C24×D6111yescharacteristic
557Q8—44no—
558Q8—2448no—
559Q8—2444no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—48no—
572C2×C4—612no—
573C2×C4—614no—
574C2×C4—6116no—
575C2×C4—612no—
576C2×C4—1228no—
577C2×C4—1228no—
578C2×C4—1228no—
579C2×C4—1222no—
580C2×C4—1228no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1224no—
584C2×C4—1224no—
585C2×C4—1224no—
586C2×C4—1228no—
587C2×C4—1224no—
588C2×C4—1224no—
589C2×C4—1224no—
590C2×C4—24416no—
591C2×C4—2448no—
592C2×C4—24416no—
593C2×C4—24416no—
594C2×C4—2448no—
595C2×C4—2448no—
596C2×C4C2×D4×D6112yes—
597C2×C4C2×D4×D6112yes—
598C2×C4#41×D6114yes—
599C22—22no—
600C22—28no—
601C22—22no—
602C22—614no—
603C22—614no—
604C22—6116no—
605C22—614no—
606C22—614no—
607C22—12232no—
608C22—1222no—
609C22—1228no—
610C22—1222no—
611C22#124×D6114yes—
612C22C22×D4×D6111yescharacteristic
613C22C2×#41×D6112yes—
614C22C2×#41×D6112yes—
615C222−1+4×D6112yes—
616C4—22no—
617C4—42no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—614no—
622C4—1222no—
623C4—2442no—
624C4—2444no—
625C4—2444no—
626C4—2442no—
627C4#123×D6114yes—
628C2—24no—
629C2—618no—
630C2—1224no—
631C2C2×#123×D6111yescharacteristic
632C2C2×#124×D6111yescharacteristic
633C2#536×D6111yescharacteristic
634C2#552×D6114yes—
subgroupG/H
center, FrattiniC23C24×D61
commutator, socle, cosocleC23×C61C25
FittingC61×#2503C2
prime radicalC24×D61C23
hypercenter#2503D61
hypocenterC61C2×#2503
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #2503

Complements

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

NH ≅ G/Ntype
3. #21450305C2semidirect
4. #21450304C2semidirect
5. #21450303C2semidirect
6. #21450302C2semidirect
7. #21450301C2semidirect
8. #21450300C2semidirect
9. #21450299C2semidirect
10. #400×D61C2semidirect
11. #391×D61C2semidirect
12. #382×D61C2semidirect
15. C61×#2503C2semidirect
37. #16839704C22semidirect
38. #16839703C22semidirect
39. #16839702C22semidirect
40. #16839701C22semidirect
41. #16839700C22semidirect
42. #16839669C22semidirect
43. #16839668C22semidirect
44. #16839667C22semidirect
45. #16839666C22semidirect
46. #16839628C22semidirect
47. #16839627C22semidirect
48. #16839627C22semidirect
49. #16839626C22semidirect
64. C61×#400C22semidirect
65. C61×#391C22semidirect
66. C61×#382C22semidirect
320. #2503D61direct (N × H)
332. D61#2503direct (N × H)
361. #400C2×D61semidirect
366. #391C2×D61semidirect
371. #382C2×D61semidirect
386. C61C2×#2503semidirect

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
#214503052
#214503042
#214503032
#214503022
#214503012
#214503002
#214502992
#400×D612
#391×D612
#382×D612
#381×D612
C4×C22⋊C4×D612
C61×#25032
C2×#250361