Finite Group Databasebeta

← D61×#1852 #29680715

Subgroups — D61×#1852 #29680715

Order 15616.

1966 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×#107×D61C212yesmaximal, complemented
3C2×C22⋊C8×D61C211yesmaximal, characteristic
4#380×D61C211yesmaximal, characteristic
5#21446009C211yesmaximal, characteristic, complemented
6#21446008C212yesmaximal, complemented
7#21446007C211yesmaximal, characteristic, complemented
8#21446006C211yesmaximal, characteristic, complemented
9#21446005C212yesmaximal, complemented
10#340×D61C212yesmaximal, complemented
11C61×#1852C211yesmaximal, characteristic, complemented
12C23×C4×D61—21no—
13C23×C4×D61C2211yescharacteristic
14C22×C8×D61C2212yes—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196—21no—
17C2×C4⋊C4×D61—28no—
18C2×C4⋊C4×D61—21no—
19C2×C4⋊C4×D61—21no—
20C2×C4⋊C4×D61C2212yes—
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C4⋊C4×D61C2212yes—
24C2×C22⋊C4×D61—21no—
25C2×#14519711C2212yescomplemented
26C2×#14519710C2212yescomplemented
27C2×#14519710C2212yescomplemented
28#107×D61—28no—
29C2×C22⋊#92418C2211yescharacteristic
30C2×#14519689C2212yes—
31C22⋊C8×D61C418yescomplemented
32#16839622C2211yescharacteristic
33#16839621—21no—
34#16839620—21no—
35#16839619C2211yescharacteristic
36#16839619C2211yescharacteristic
37#95×D61—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C22×C4⋊C4×C61C2211yescharacteristic
42C2×#107×C61C2212yescomplemented
43C2×C22⋊C8×C61C2211yescharacteristic
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C23×Dic61—21no—
48C23×Dic61D411yescharacteristic
49C22×C4×D61—24no—
50C22×C4×D61—24no—
51C22×C4×D61—28no—
52C22×C4×D61—22no—
53C22×C4×D61—22no—
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—44no—
57C22×C4×D61—44no—
58C22×C4×D61C2×C414yes—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61Q814yes—
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2312yes—
65C2×C8×D61—24no—
66C2×C8×D61—24no—
67C2×C8×D61C2×C418yescomplemented
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—24no—
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—28no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C4⋊C4×D61—28no—
79C4⋊C4×D61—28no—
80C4⋊C4×D61—24no—
81C4⋊C4×D61—24no—
82C4⋊C4×D61—24no—
83C4⋊C4×D61—24no—
84C4⋊C4×D61—24no—
85C4⋊C4×D61D418yes—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C22⋊C4×D61—44no—
90#14519711—24no—
91#14519710—24no—
92#14519710—24no—
93C22⋊#92418C2×C414yescomplemented
94#14519689C2×C418yescomplemented
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98C23×C4×C61—21no—
99C23×C4×C61C2311yescharacteristic
100C22×C8×C61C2312yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—24no—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C4⋊C4×C61C2312yes—
106C2×C22⋊C4×C61—21no—
107#107×C61—24no—
108C22⋊C8×C61C2×C414yescomplemented
109#95×C61—21no—
110C24×C61C2×D411yescharacteristic
111C23×D61—22no—
112C23×D61—22no—
113C23×D61—22no—
114C23×D61—22no—
115C23×D61C4⋊C414yes—
116C23×D61D814yes—
117C23×D61Q1614yes—
118C23×D61C2×D411yescharacteristic
119C23×D61C2×D411yescharacteristic
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×Q812yes—
131C22×C4×C61C2411yescharacteristic
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—24no—
135C22×Dic61—22no—
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—42no—
139C22×Dic61—42no—
140C22×Dic61C4⋊C414yes—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—216no—
150C2×C4×D61—216no—
151C2×C4×D61—28no—
152C2×C4×D61—28no—
153C2×C4×D61—44no—
154C2×C4×D61—432no—
155C2×C4×D61—44no—
156C2×C4×D61—44no—
157C2×C4×D61—44no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61C4⋊C414yes—
165C2×C4×D61C4⋊C414yes—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D414yes—
169C2×C4×D61C2×Q812yes—
170C2×C4×D61C2×Q812yes—
171C2×C4×D61#4114yes—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61C22×C414yescomplemented
175C4⋊C4×C61—24no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61C2×D414yes—
180C22⋊C4×C61—42no—
181C22⋊Dic61—42no—
182#11758198—42no—
183#11758198—42no—
184C4⋊Dic61—24no—
185C4⋊Dic61—22no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4⋊Dic61C2×D414yes—
189#11758196—28no—
190#11758196—28no—
191#11758196—42no—
192#11758196—42no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418C22×C414yescomplemented
196C8×D61—216no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61C2×C4⋊C412yes—
202C23×C61C2×D812yes—
203C23×C61C2×Q1612yes—
204C23×C61C22×D411yescharacteristic
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—28no—
210C22×D61—28no—
211C22×D61—28no—
212C22×D61—28no—
213C22×D61—28no—
214C22×D61—28no—
215C22×D61#107116yescomplemented
216C22×D61C2×C4⋊C412yes—
217C22×D61C2×C4⋊C412yes—
218C22×D61#12012yes—
219C22×D61#12312yes—
220C22×D61C2×D812yes—
221C22×D61C2×D812yes—
222C22×D61C2×Q1612yes—
223C22×D61C2×Q1612yes—
224C22×D61AGL(1,ℤ/8)12yes—
225C22×D61#13712yes—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—48no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C2×C4⋊C412yes—
241C2×C4×C61C23×C411yescharacteristic
242C2×C4×C61C22×D412yes—
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4112yes—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—28no—
248C2×Dic61—24no—
249C2×Dic61—48no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61C2×C4⋊C412yes—
259C2×Dic61C2×C4⋊C412yes—
260C2×Dic61#12012yes—
261C2×Dic61#12312yes—
262C4×D61—28no—
263C4×D61—24no—
264C4×D61—24no—
265C4×D61—28no—
266C4×D61—416no—
267C4×D61—416no—
268C4×D61—84no—
269C4×D61—84no—
270C4×D61C4×D418yes—
271C4×D61C22⋊Q818yes—
272C8×C61—24no—
273#92418—24no—
274C2×#1852—611nomaximal, Sylow
275C22×C61—22no—
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—22no—
280C22×C61—22no—
281C22×C61C2×#10714yescomplemented
282C22×C61C22×C4⋊C411yescharacteristic
283C22×C61C2×#12011yescharacteristic
284C22×C61C2×#12311yescharacteristic
285C22×C61C22×D811yescharacteristic
286C22×C61C22×Q1611yescharacteristic
287C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
288C22×C61C2×#13711yescharacteristic
289C2×D61—22no—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—232no—
296C2×D61#38514yes—
297C2×D61C2×#10714yes—
298C2×D61C2×#10714yescomplemented
299C2×D61#42814yes—
300C2×D61#44714yes—
301C2×D61#45114yes—
302C2×D61#48014yes—
303C2×D61#48414yes—
304C4×C61—22no—
305C4×C61—44no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C4×D412yes—
309C4×C61C2×C22⋊Q812yes—
310Dic61—22no—
311Dic61—44no—
312Dic61—81no—
313Dic61—81no—
314Dic61#38514yes—
315C23×C4⋊C4—611no—
316C22×#107—612no—
317C22×C22⋊C8—611no—
318C2×#380—611no—
319#1852—612no—
320#1852—611no—
321#1852—611no—
322#1852—612no—
323#1852—611no—
324#1852D6111yescharacteristic, direct product factor
325C2×#340—612no—
326C2×C61—24no—
327C2×C61C2×#38511yescharacteristic
328C2×C61C22×#10711yescharacteristic
329C2×C61C2×#42811yescharacteristic
330C2×C61C2×#44711yescharacteristic
331C2×C61C2×#45111yescharacteristic
332C2×C61C2×#48011yescharacteristic
333C2×C61C2×#48411yescharacteristic
334D61—24no—
335D61#185218yesdirect product factor
336C24×C4—611no—
337C24×C4—1221no—
338C23×C8—612no—
339C22×C4⋊C4—612no—
340C22×C4⋊C4—611no—
341C22×C4⋊C4—612no—
342C22×C4⋊C4—611no—
343C22×C4⋊C4—611no—
344C22×C4⋊C4—1228no—
345C22×C4⋊C4—1221no—
346C22×C4⋊C4—1221no—
347C22×C4⋊C4—1221no—
348C22×C4⋊C4C2×D6111yescharacteristic
349C22×C22⋊C4—1221no—
350C2×#107—612no—
351C2×#107—612no—
352C2×#107—612no—
353C2×#107—1228no—
354C2×#107C2×D6112yescomplemented
355C2×C22⋊C8—611no—
356C2×C22⋊C8—612no—
357C2×C22⋊C8—618no—
358C2×C22⋊C8C2×D6111yescharacteristic
359#380—611no—
360#380—611no—
361#380—611no—
362#380—1221no—
363#380—1221no—
364#380C2×D6111yescharacteristic
365C2×#95—1221no—
366#340—612no—
367#340—612no—
368#340—612no—
369#340C2×D6112yescomplemented
370C61C2×#185211yescharacteristic, complemented, Sylow
371C25—611no—
372C23×C4—21no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4—614no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—614no—
379C23×C4—611no—
380C23×C4—1224no—
381C23×C4—1224no—
382C23×C4—1222no—
383C23×C4—1222no—
384C23×C4—1228no—
385C23×C4—1221no—
386C23×C4—1221no—
387C23×C4—1221no—
388C23×C4—2444no—
389C23×C4—2444no—
390C23×C4C22×D6111yescharacteristic
391C22×C8—612no—
392C22×C8—618no—
393C22×C8—1224no—
394C22×C8—1224no—
395C22×C8C22×D6112yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—618no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
402C2×C4⋊C4—1224no—
403C2×C4⋊C4—1224no—
404C2×C4⋊C4—1228no—
405C2×C4⋊C4—1228no—
406C2×C4⋊C4—1224no—
407C2×C4⋊C4—1224no—
408C2×C4⋊C4—1224no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1228no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1224no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6112yes—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C22⋊C4—2444no—
424#107—24no—
425#107—1224no—
426#107—1224no—
427#107—1224no—
428C22⋊C8—614no—
429C22⋊C8—618no—
430C22⋊C8C4×D6114yescomplemented
431#95—21no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435C24—611no—
436C24—611no—
437C24—614no—
438C24—614no—
439C24—614no—
440C24—1222no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24D4×D6111yescharacteristic
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—611no—
454C22×C4—612no—
455C22×C4—614no—
456C22×C4—612no—
457C22×C4—614no—
458C22×C4—611no—
459C22×C4—614no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—12216no—
467C22×C4—12216no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1228no—
471C22×C4—1228no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—2442no—
481C22×C4—2442no—
482C22×C4—2442no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4—2442no—
486C22×C4—2444no—
487C22×C4—2444no—
488C22×C4—2444no—
489C22×C4—24432no—
490C22×C4—2444no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4C2×C4×D6112yes—
494C22×C4D4×D6111yescharacteristic
495C22×C4Q8×D6112yes—
496C22×C4C23×D6111yescharacteristic
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—614no—
500C2×C8—12216no—
501C2×C8—1222no—
502C2×C8—1222no—
503C2×C8C2×C4×D6114yescomplemented
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—614no—
509C4⋊C4—1228no—
510C4⋊C4—1228no—
511C4⋊C4—1224no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4D4×D6114yes—
518C22⋊C4—42no—
519C22⋊C4—2442no—
520C22⋊C4—2442no—
521C22⋊C4—2442no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—612no—
527C23—612no—
528C23—612no—
529C23—612no—
530C23—612no—
531C23—612no—
532C23—612no—
533C23—612no—
534C23—612no—
535C23—612no—
536C23—6116no—
537C23—1228no—
538C23—1228no—
539C23—1228no—
540C23—1228no—
541C23—1228no—
542C23—1228no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1221no—
547C23C4⋊C4×D6112yes—
548C23D8×D6112yes—
549C23Q16×D6112yes—
550C23C2×D4×D6111yescharacteristic
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—48no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—618no—
569C2×C4—618no—
570C2×C4—612no—
571C2×C4—1228no—
572C2×C4—1228no—
573C2×C4—1228no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—4884no—
580C2×C4—4884no—
581C2×C4—24416no—
582C2×C4—24416no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2448no—
590C2×C4—2442no—
591C2×C4—2442no—
592C2×C4C4⋊C4×D6112yes—
593C2×C4C22×C4×D6111yescharacteristic
594C2×C4C2×D4×D6112yes—
595C2×C4C2×Q8×D6111yescharacteristic
596C2×C4#41×D6112yes—
597C8—24no—
598C8—1224no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—614no—
606C22—614no—
607C22—614no—
608C22—614no—
609C22—614no—
610C22—614no—
611C22—614no—
612C22—614no—
613C22—12232no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22#107×D6114yescomplemented
621C22C2×C4⋊C4×D6111yescharacteristic
622C22#120×D6111yescharacteristic
623C22#123×D6111yescharacteristic
624C22C2×D8×D6111yescharacteristic
625C22C2×Q16×D6111yescharacteristic
626C22AGL(1,ℤ/8)×D6111yescharacteristic
627C22#137×D6111yescharacteristic
628C4—22no—
629C4—44no—
630C4—81no—
631C4—81no—
632C4—614no—
633C4—1222no—
634C4—4881no—
635C4—4881no—
636C4—2444no—
637C4C4×D4×D6112yes—
638C4C22⋊Q8×D6112yes—
639C2—24no—
640C2—618no—
641C2—1224no—
642C2#385×D6111yescharacteristic
643C2C2×#107×D6111yescharacteristic
644C2#428×D6111yescharacteristic
645C2#447×D6111yescharacteristic
646C2#451×D6111yescharacteristic
647C2#480×D6111yescharacteristic
648C2#484×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1852C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1852D61
hypocenterC61C2×#1852
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#107×D61C2semidirect
5. #21446009C2semidirect
6. #21446008C2semidirect
7. #21446007C2semidirect
8. #21446006C2semidirect
9. #21446005C2semidirect
10. #340×D61C2semidirect
11. C61×#1852C2semidirect
25. C2×#14519711C22semidirect
26. C2×#14519710C22semidirect
27. C2×#14519710C22semidirect
31. C22⋊C8×D61C4semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
42. C2×#107×C61C22semidirect
45. C61×#340C22semidirect
67. C2×C8×D61C2×C4semidirect
93. C22⋊#92418C2×C4semidirect
94. #14519689C2×C4semidirect
108. C22⋊C8×C61C2×C4semidirect
174. C2×C8×C61C22×C4semidirect
195. C2×#92418C22×C4semidirect
215. C22×D61#107semidirect
281. C22×C61C2×#107semidirect
298. C2×D61C2×#107semidirect
324. #1852D61direct (N × H)
335. D61#1852direct (N × H)
354. C2×#107C2×D61semidirect
369. #340C2×D61semidirect
370. C61C2×#1852semidirect
430. C22⋊C8C4×D61semidirect
503. C2×C8C2×C4×D61semidirect
620. C22#107×D61semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C2×#107×D612
C2×C22⋊C8×D612
#380×D612
#214460092
#214460082
#214460072
#214460062
#214460052
#340×D612
C61×#18522
C2×#185261