Finite Group Databasebeta

← D61×#1856 #29680813

Subgroups — D61×#1856 #29680813

Order 15616.

1706 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
1#518×D61C211yesmaximal, characteristic
2#428×D61C212yesmaximal, complemented
3C2×#100×D61C211yesmaximal, characteristic
4#390×D61C211yesmaximal, characteristic
5#21446031C211yesmaximal, characteristic, complemented
6#21446030C212yesmaximal, complemented
7#21446029C211yesmaximal, characteristic, complemented
8#21446028C211yesmaximal, characteristic, complemented
9#21446027C212yesmaximal, complemented
10#339×D61C212yesmaximal, complemented
11C61×#1856C211yesmaximal, characteristic, complemented
12C22×D4×D61C2211yescharacteristic
13C2×M16×D61C2212yes—
14#16840237C2211yescharacteristic
15#16840236C2212yes—
16#16840235—22no—
17#16840234—22no—
18#16840234—22no—
19C4×D4×D61—22no—
20C4×D4×D61—22no—
21#117×D61C2212yes—
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C22⋊C4×D61—22no—
24C2×C22⋊C4×D61—22no—
25C2×C42×D61C2211yescharacteristic
26#16839805C2212yescomplemented
27#16839804C2212yescomplemented
28#16839803—22no—
29#16839802C2212yescomplemented
30#16839801—22no—
31#107×D61—22no—
32#106×D61—22no—
33C2×#14519694C2211yescharacteristic
34C2×#14519693C2212yes—
35#100×D61C418yescomplemented
36#16839665C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839662—22no—
40#16839523C2212yescomplemented
41#16839522C2212yescomplemented
42#16839521C2212yescomplemented
43C61×#518C2211yescharacteristic
44C61×#428C2212yescomplemented
45C2×#100×C61C2211yescharacteristic
46C61×#390C2211yescharacteristic
47C61×#339C2212yescomplemented
48C24×D61D412yes—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61D412yes—
51C2×D4×D61—22no—
52C2×D4×D61—22no—
53C2×D4×D61—24no—
54C2×D4×D61C2×C414yes—
55C2×D4×D61Q814yes—
56C22×C4×D61—21no—
57C22×C4×D61—22no—
58C22×C4×D61—22no—
59C22×C4×D61—21no—
60C22×C4×D61C2311yescharacteristic
61C2×#11758194C2312yes—
62C2×#11758189—22no—
63M16×D61—24no—
64M16×D61C2×C418yescomplemented
65C2×C8×D61—22no—
66C4×D244—22no—
67C4×D244—22no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70#14519739—22no—
71#14519739—22no—
72C4×C22⋊D61—24no—
73C4×C22⋊D61—24no—
74#14519737—24no—
75#14519737—24no—
76#14519736C2312yes—
77#14519735D412yes—
78#14519735D412yes—
79#14519734—22no—
80#14519733—22no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61D412yes—
86C4⋊C4×D61D412yes—
87C2×C22⋊Dic61—22no—
88C2×C22⋊Dic61—22no—
89C2×#11758198—22no—
90C2×#11758198—22no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C22⋊C4×D61—24no—
94C22⋊C4×D61—24no—
95C22⋊C4×D61—22no—
96C22⋊C4×D61—24no—
97C22⋊C4×D61—48no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—22no—
101C42×D61—24no—
102#14519711—22no—
103#14519710—22no—
104#14519710—22no—
105#14519709—22no—
106#14519708—22no—
107#14519708—22no—
108#14519694C2×C414yescomplemented
109#14519693C2×C418yescomplemented
110C22×D4×C61C2311yescharacteristic
111C2×M16×C61C2312yes—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114#117×C61C2312yes—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—22no—
117C2×C22⋊C4×C61—22no—
118C2×C42×C61C2311yescharacteristic
119#107×C61—22no—
120#106×C61—22no—
121#100×C61C2×C414yescomplemented
122C24×C61C2×D412yes—
123C23×D61—22no—
124C23×D61—22no—
125C23×D61—24no—
126C23×D61—44no—
127C23×D61—44no—
128C23×D61C4⋊C418yes—
129C23×D61C2×D411yescharacteristic
130C23×D61C2×D412yes—
131C2×D4×C61—22no—
132C2×D4×C61—22no—
133C2×D4×C61C22×C412yes—
134C2×D4×C61C2×Q812yes—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—24no—
142C2×C22⋊D61C4⋊C418yes—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148D4×D61—28no—
149D4×D61—28no—
150D4×D61—28no—
151D4×D61—28no—
152C2×D244—22no—
153C2×D244—22no—
154C2×D244C22×C412yes—
155C2×D244C2×Q812yes—
156C2×C4×D61—24no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—44no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×C4×D61—84no—
167C2×C4×D61—84no—
168C2×C4×D61C22×C412yes—
169C2×C4×D61C2×D412yes—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×Q812yes—
172C2×C4×D61#4114yes—
173M16×C61—22no—
174M16×C61C22×C414yescomplemented
175C2×C8×C61—22no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61C2×D412yes—
178C4⋊C4×C61C2×D412yes—
179C22⋊C4×C61—24no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—44no—
182C42×C61—22no—
183C42×C61—22no—
184C22⋊Dic61—24no—
185C22⋊Dic61—22no—
186C22⋊Dic61—44no—
187#11758198—24no—
188#11758198—24no—
189#11758198—22no—
190#11758198—22no—
191#11758198—44no—
192#11758198—44no—
193C4⋊Dic61—22no—
194C4⋊Dic61C2×D412yes—
195C4⋊Dic61C2×D412yes—
196#11758196—22no—
197#11758196—22no—
198#11758196—42no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61—22no—
202C4×Dic61—22no—
203C4×Dic61—42no—
204#11758194—22no—
205#11758194C22×C414yescomplemented
206C2×#92418—22no—
207#11758189—28no—
208C8×D61—28no—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61C2×C4⋊C414yes—
214C23×C61C22×D411yescharacteristic
215C22×D61—22no—
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—28no—
220C22×D61—28no—
221C22×D61—416no—
222C22×D61—44no—
223C22×D61—44no—
224C22×D61—42no—
225C22×D61—42no—
226C22×D61—88no—
227C22×D61—82no—
228C22×D61—82no—
229C22×D61C2×C4⋊C412yes—
230C22×D61C2×C4⋊C414yes—
231C22×D61#12011yescharacteristic
232C22×D61#12011yescharacteristic
233C22×D61#12312yes—
234D4×C61—24no—
235D4×C61—24no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—41no—
243C2×C4×C61—82no—
244C2×C4×C61—82no—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D412yes—
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4112yes—
249C22⋊D61—216no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—42no—
256C2×Dic61—41no—
257C2×Dic61—82no—
258C2×Dic61—82no—
259C2×Dic61C2×C4⋊C412yes—
260C2×Dic61#12012yes—
261C2×Dic61#12312yes—
262D244—24no—
263D244—24no—
264C4×D61—48no—
265C4×D61—44no—
266C4×D61—44no—
267C4×D61—84no—
268C4×D61—84no—
269C4×D61—82no—
270C4×D61—82no—
271C4×D61C4×D414yes—
272C4×D61C4×D414yes—
273C4×D61C22⋊Q814yes—
274C4×D61C22⋊Q814yes—
275C8×C61—24no—
276#92418—24no—
277C2×#1856—611nomaximal, Sylow
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—24no—
281C22×C61—44no—
282C22×C61—42no—
283C22×C61—42no—
284C22×C61—82no—
285C22×C61C22×C4⋊C411yescharacteristic
286C22×C61C2×#12011yescharacteristic
287C22×C61C2×#12311yescharacteristic
288C2×D61—21no—
289C2×D61—21no—
290C2×D61—24no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—416no—
296C2×D61—44no—
297C2×D61—88no—
298C2×D61—82no—
299C2×D61#38512yes—
300C2×D61#38512yes—
301C2×D61#45312yes—
302C2×D61#45512yes—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61—81no—
307C4×C61C2×C4×D412yes—
308C4×C61C2×C22⋊Q812yes—
309Dic61—42no—
310Dic61—42no—
311Dic61—42no—
312Dic61—81no—
313Dic61#38514yes—
314C2×#518—611no—
315C2×#428—612no—
316C22×#100—611no—
317C2×#390—611no—
318#1856—612no—
319#1856—611no—
320#1856—611no—
321#1856—612no—
322#1856—611no—
323#1856D6111yescharacteristic, direct product factor
324C2×#339—612no—
325C2×C61—21no—
326C2×C61—21no—
327C2×C61—44no—
328C2×C61C2×#38511yescharacteristic
329C2×C61C2×#45311yescharacteristic
330C2×C61C2×#45511yescharacteristic
331D61—22no—
332D61—44no—
333D61#185614yesdirect product factor
334C23×D4—611no—
335C22×M16—612no—
336#518—611no—
337#518—612no—
338#518—1222no—
339#518—1222no—
340#518—1222no—
341#518C2×D6111yescharacteristic
342C2×C4×D4—1222no—
343C2×C4×D4—1222no—
344C2×#117—612no—
345C22×C4⋊C4—611no—
346C22×C22⋊C4—1222no—
347C22×C22⋊C4—1222no—
348C22×C42—611no—
349#428—612no—
350#428—612no—
351#428—612no—
352#428—1222no—
353#428—1222no—
354#428C2×D6112yescomplemented
355C2×#107—1222no—
356C2×#106—1222no—
357C2×#100—611no—
358C2×#100—612no—
359C2×#100—618no—
360C2×#100C2×D6111yescharacteristic
361#390—611no—
362#390—611no—
363#390—611no—
364#390—1222no—
365#390C2×D6111yescharacteristic
366#339—612no—
367#339—612no—
368#339—612no—
369#339C2×D6112yescomplemented
370C61C2×#185611yescharacteristic, complemented, Sylow
371C25—612no—
372C22×D4—612no—
373C22×D4—614no—
374C22×D4—614no—
375C22×D4—611no—
376C22×D4—1222no—
377C22×D4—1222no—
378C22×D4—1224no—
379C22×D4C22×D6111yescharacteristic
380C23×C4—611no—
381C23×C4—1222no—
382C23×C4—1222no—
383C23×C4—1221no—
384C23×C4—1221no—
385C2×M16—612no—
386C2×M16—618no—
387C2×M16—1222no—
388C2×M16—1224no—
389C2×M16C22×D6112yes—
390C22×C8—1222no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1224no—
396C4×D4—1224no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1224no—
400C4×D4—1224no—
401C4×D4—1222no—
402C4×D4—1222no—
403#117—612no—
404#117—612no—
405#117—612no—
406#117—1222no—
407#117—1222no—
408#117C22×D6112yes—
409C2×C4⋊C4—612no—
410C2×C4⋊C4—612no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1224no—
420C2×C22⋊C4—1224no—
421C2×C22⋊C4—1224no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1222no—
425C2×C22⋊C4—1222no—
426C2×C22⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C22⋊C4—2448no—
429C2×C42—611no—
430C2×C42—1222no—
431C2×C42—1224no—
432C2×C42—1221no—
433C2×C42C22×D6111yescharacteristic
434#107—22no—
435#107—1222no—
436#107—1222no—
437#107—1222no—
438#106—22no—
439#106—1222no—
440#106—1222no—
441#106—1222no—
442#100—614no—
443#100—618no—
444#100C4×D6114yescomplemented
445C24—611no—
446C24—612no—
447C24—618no—
448C24—1222no—
449C24—1224no—
450C24—1222no—
451C24—2444no—
452C24—2444no—
453C24D4×D6112yes—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—618no—
457C2×D4—612no—
458C2×D4—612no—
459C2×D4—1228no—
460C2×D4—1228no—
461C2×D4—1228no—
462C2×D4—1228no—
463C2×D4—1224no—
464C2×D4—1224no—
465C2×D4—1222no—
466C2×D4—1222no—
467C2×D4C2×C4×D6112yes—
468C2×D4Q8×D6112yes—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—611no—
474C22×C4—612no—
475C22×C4—612no—
476C22×C4—612no—
477C22×C4—612no—
478C22×C4—614no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1224no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—4884no—
489C22×C4—4884no—
490C22×C4—2441no—
491C22×C4—2441no—
492C22×C4—2444no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4C23×D6111yescharacteristic
496M16—22no—
497M16—614no—
498M16—1228no—
499M16—1222no—
500M16C2×C4×D6114yescomplemented
501C2×C8—22no—
502C2×C8—1228no—
503C2×C8—1222no—
504C4⋊C4—22no—
505C4⋊C4—612no—
506C4⋊C4—612no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—2442no—
511C4⋊C4D4×D6112yes—
512C4⋊C4D4×D6112yes—
513C22⋊C4—24no—
514C22⋊C4—22no—
515C22⋊C4—44no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1224no—
519C22⋊C4—1224no—
520C22⋊C4—1224no—
521C22⋊C4—1222no—
522C22⋊C4—2444no—
523C22⋊C4—2444no—
524C22⋊C4—2444no—
525C42—22no—
526C42—22no—
527C42—1222no—
528C42—1222no—
529C42—1222no—
530C42—1222no—
531C42—2442no—
532C23—22no—
533C23—22no—
534C23—42no—
535C23—42no—
536C23—611no—
537C23—612no—
538C23—611no—
539C23—612no—
540C23—614no—
541C23—1222no—
542C23—1222no—
543C23—1228no—
544C23—1228no—
545C23—1222no—
546C23—1222no—
547C23—4882no—
548C23—4882no—
549C23—4888no—
550C23—2442no—
551C23—2442no—
552C23—2444no—
553C23—2444no—
554C23—24416no—
555C23C4⋊C4×D6114yes—
556C23C2×D4×D6111yescharacteristic
557D4—24no—
558D4—24no—
559D4—12216no—
560D4—1224no—
561D4—1224no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—614no—
575C2×C4—614no—
576C2×C4—614no—
577C2×C4—614no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—4884no—
582C2×C4—4884no—
583C2×C4—4882no—
584C2×C4—4882no—
585C2×C4—4882no—
586C2×C4—4882no—
587C2×C4—2444no—
588C2×C4—2444no—
589C2×C4—2448no—
590C2×C4—2442no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C22×C4×D6111yescharacteristic
595C2×C4C2×D4×D6112yes—
596C2×C4C2×Q8×D6111yescharacteristic
597C2×C4#41×D6112yes—
598C8—24no—
599C8—1224no—
600C22—21no—
601C22—21no—
602C22—24no—
603C22—44no—
604C22—42no—
605C22—42no—
606C22—82no—
607C22—612no—
608C22—612no—
609C22—612no—
610C22—612no—
611C22—1224no—
612C22—1224no—
613C22—1221no—
614C22—1221no—
615C22—1224no—
616C22—4882no—
617C22—4888no—
618C22—2444no—
619C22—24416no—
620C22—2442no—
621C22—2442no—
622C22C2×C4⋊C4×D6111yescharacteristic
623C22#120×D6111yescharacteristic
624C22#123×D6111yescharacteristic
625C4—42no—
626C4—42no—
627C4—42no—
628C4—81no—
629C4—614no—
630C4—4881no—
631C4—2442no—
632C4—2442no—
633C4—2442no—
634C4C4×D4×D6112yes—
635C4C22⋊Q8×D6112yes—
636C2—21no—
637C2—21no—
638C2—44no—
639C2—614no—
640C2—1222no—
641C2—2444no—
642C2#385×D6111yescharacteristic
643C2#453×D6111yescharacteristic
644C2#455×D6111yescharacteristic
subgroupG/H
centerC22#120×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1856C2
socleC22×C61C2×#120
prime radicalC22×D4×D61C22
cosocleC22×C4×C61C24
hypercenter#1856D61
hypocenterC61C2×#1856
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1856

Complements

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

NH ≅ G/Ntype
2. #428×D61C2semidirect
5. #21446031C2semidirect
6. #21446030C2semidirect
7. #21446029C2semidirect
8. #21446028C2semidirect
9. #21446027C2semidirect
10. #339×D61C2semidirect
11. C61×#1856C2semidirect
26. #16839805C22semidirect
27. #16839804C22semidirect
29. #16839802C22semidirect
35. #100×D61C4semidirect
40. #16839523C22semidirect
41. #16839522C22semidirect
42. #16839521C22semidirect
44. C61×#428C22semidirect
47. C61×#339C22semidirect
64. M16×D61C2×C4semidirect
108. #14519694C2×C4semidirect
109. #14519693C2×C4semidirect
121. #100×C61C2×C4semidirect
174. M16×C61C22×C4semidirect
205. #11758194C22×C4semidirect
323. #1856D61direct (N × H)
333. D61#1856direct (N × H)
354. #428C2×D61semidirect
369. #339C2×D61semidirect
370. C61C2×#1856semidirect
444. #100C4×D61semidirect
500. M16C2×C4×D61semidirect

Maximal subgroups

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

subgroupindex
#518×D612
#428×D612
C2×#100×D612
#390×D612
#214460312
#214460302
#214460292
#214460282
#214460272
#339×D612
C61×#18562
C2×#185661