Finite Group Databasebeta

← D61×#2128 #29832799

Subgroups — D61×#2128 #29832799

Order 15616.

1422 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#535×D61C211yesmaximal, characteristic
2#410×D61C212yesmaximal, complemented
3#21447491C211yesmaximal, characteristic, complemented
4#21447490C212yesmaximal, complemented
5#21447489C214yesmaximal, complemented
6#353×D61C214yesmaximal, complemented
7C61×#2128C211yesmaximal, characteristic, complemented
8C2×#41×D61C2211yescharacteristic
9C22×D4×D61C411yescharacteristic, complemented
10#16840331C411yescharacteristic, complemented
11#16840330C412yescomplemented
12#16840329C411yescharacteristic, complemented
13#16840328—22no—
14#16840327—22no—
15#16840326C2211yescharacteristic
16#16840325C2212yes—
17#127×D61C2212yes—
18#124×D61C412yescomplemented
19#121×D61—22no—
20#120×D61—22no—
21#117×D61—22no—
22#117×D61C411yescharacteristic, complemented
23#16839736C2212yescomplemented
24#16839735—22no—
25#16839734—22no—
26#16839734C2214yescomplemented
27#99×D61—22no—
28#99×D61C2214yescomplemented
29#16839555C2214yescomplemented
30#16839554C2218yescomplemented
31C61×#535C2211yescharacteristic
32C61×#410C2212yescomplemented
33C61×#353C2214yescomplemented
34C24×D61—21no—
35C2×#11758226C2×C411yescharacteristic
36C2×#11758222C2×C411yescharacteristic
37C2×#11758222D412yes—
38C2×#11758224D412yes—
39C2×#11758224C2311yescharacteristic
40#41×D61—42no—
41#41×D61—42no—
42C2×Q8×D61C2×C411yescharacteristic
43C22×D244C2×C411yescharacteristic, complemented
44C22×C22⋊D61—21no—
45C2×D4×D61—24no—
46C2×D4×D61—21no—
47C2×D4×D61—21no—
48C2×D4×D61—42no—
49C2×D4×D61—42no—
50C2×D4×D61C2×C412yes—
51C2×D4×D61C2×C412yescomplemented
52C2×D4×D61D412yes—
53C2×D4×D61C2311yescharacteristic
54C22×C4×D61C2×C411yescharacteristic
55C22×C4×D61D412yes—
56#14519774C2×C412yes—
57#14519773—24no—
58#14519773C2×C412yes—
59#14519773C2312yes—
60#14519766C2×C412yescomplemented
61#14519765C2×C412yescomplemented
62#14519764C2×C412yescomplemented
63#14519763—24no—
64#14519752—22no—
65#14519751—22no—
66#14519750—24no—
67#14519749—24no—
68#14519748—22no—
69#14519747—22no—
70#14519747—24no—
71#14519746—22no—
72#14519745—22no—
73#14519745—24no—
74#14519736—22no—
75#14519736C2×C411yescharacteristic, complemented
76#14519735—21no—
77#14519735—42no—
78#14519734—42no—
79#14519734C2×C412yescomplemented
80#14519733—21no—
81#14519733—24no—
82C4⋊C4×D61—21no—
83C4⋊C4×D61—42no—
84C22⋊C4×D61—21no—
85C22⋊C4×D61—24no—
86C22⋊C4×D61—24no—
87C42×D61—42no—
88C42×D61C2×C412yes—
89#14519692—22no—
90#14519692C2314yescomplemented
91#14519691—24no—
92#14519691—24no—
93C2×#41×C61C2311yescharacteristic
94C22×D4×C61C2×C411yescharacteristic, complemented
95#127×C61C2312yes—
96#124×C61C2×C412yescomplemented
97#121×C61—22no—
98#120×C61—22no—
99#117×C61—22no—
100#117×C61C2×C411yescharacteristic, complemented
101#99×C61—22no—
102#99×C61C2314yescomplemented
103C24×C61—21no—
104C23×D61—21no—
105C23×D61—22no—
106C23×D61—22no—
107C23×D61—42no—
108C23×D61—42no—
109C23×D61—84no—
110C23×D61C22⋊C411yescharacteristic
111C23×D61C2×D412yes—
112#41×C61—42no—
113C2×Q8×C61C22×C411yescharacteristic
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—42no—
117C2×D4×C61C22×C412yes—
118C2×D4×C61C2×D412yes—
119C2×D4×C61C2411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C2×D412yes—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—44no—
127C2×C22⋊D61C22⋊C412yes—
128C2×C22⋊D61C2×D412yes—
129C2×C22⋊D61C2×D412yes—
130C22×Dic61C22⋊C411yescharacteristic
131C22×Dic61C2×D412yes—
132#11758226—42no—
133Q8×D61—41no—
134Q8×D61—41no—
135#11758224—42no—
136#11758224—42no—
137#11758224—42no—
138D4×D61—42no—
139D4×D61—42no—
140D4×D61—41no—
141D4×D61—41no—
142D4×D61—44no—
143D4×D61—42no—
144D4×D61—42no—
145D4×D61—44no—
146D4×D61—42no—
147D4×D61—42no—
148D4×D61—88no—
149#11758222—42no—
150#11758222—42no—
151#11758222—42no—
152C2×D244—21no—
153C2×D244—24no—
154C2×D244—42no—
155C2×D244C22⋊C412yes—
156C2×D244C22×C412yes—
157C2×D244C22×C411yescharacteristic
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—44no—
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C412yes—
168C2×C4×D61C22⋊C412yes—
169C2×C4×D61C22⋊C412yes—
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C22×C411yescharacteristic
173C2×Dic122C22⋊C412yes—
174C2×Dic122C22×C411yescharacteristic
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—24no—
179C22⋊C4×C61—24no—
180C42×C61—42no—
181C42×C61C22×C412yes—
182C22⋊Dic61—21no—
183C22⋊Dic61—24no—
184C22⋊Dic61—24no—
185#11758198—22no—
186#11758198—24no—
187#11758198—24no—
188#11758198—44no—
189C4⋊Dic61—21no—
190C4⋊Dic61—42no—
191#11758196—22no—
192#11758196—44no—
193C4×Dic61—22no—
194C4×Dic61—44no—
195C4×Dic61—42no—
196C4×Dic61C22×C412yes—
197C23×C61—22no—
198C23×C61—42no—
199C23×C61—82no—
200C23×C61C2×C22⋊C411yescharacteristic
201C23×C61C22×D412yes—
202C22×D61—22no—
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—22no—
206C22×D61—22no—
207C22×D61—42no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61—48no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C22×D61—84no—
215C22×D61—82no—
216C22×D61—88no—
217C22×D61—88no—
218C22×D61C2×C22⋊C411yescharacteristic
219C22×D61C2×C22⋊C411yescharacteristic
220C22×D61C2×C22⋊C412yes—
221Q8×C61—41no—
222D4×C61—42no—
223D4×C61—41no—
224D4×C61—42no—
225D4×C61—42no—
226D4×C61—84no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—44no—
232C2×C4×C61C2×C22⋊C412yes—
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C22⋊D61—42no—
236C22⋊D61—42no—
237C22⋊D61—42no—
238C22⋊D61—44no—
239C22⋊D61—44no—
240C22⋊D61—84no—
241C22⋊D61—84no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—44no—
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×C22⋊C411yescharacteristic
250D244—42no—
251D244—41no—
252D244—42no—
253D244—42no—
254D244—84no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—21no—
258C4×D61—21no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—44no—
266C4×D61—88no—
267C4×D61#9912yes—
268C4×D61#9912yes—
269Dic122—42no—
270Dic122—41no—
271C2×#2128—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—21no—
274C22×C61—42no—
275C22×C61—41no—
276C22×C61—44no—
277C22×C61—42no—
278C22×C61—84no—
279C22×C61C22×C22⋊C411yescharacteristic
280C2×D61—22no—
281C2×D61—21no—
282C2×D61—22no—
283C2×D61—42no—
284C2×D61—44no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—41no—
288C2×D61—44no—
289C2×D61—42no—
290C2×D61—82no—
291C2×D61—84no—
292C2×D61—88no—
293C2×D61—84no—
294C2×D61—84no—
295C2×D61C2×#9911yescharacteristic
296C2×D61C2×#9911yescharacteristic
297C4×C61—21no—
298C4×C61—22no—
299C4×C61—42no—
300C4×C61—84no—
301C4×C61C2×#9912yes—
302Dic61—22no—
303Dic61—21no—
304Dic61—42no—
305Dic61—84no—
306Dic61C2×#9912yes—
307C2×#535—611no—
308C2×#410—612no—
309#2128—612no—
310#2128—614no—
311#2128—611no—
312#2128D6111yescharacteristic, direct product factor
313C2×#353—614no—
314C2×C61—21no—
315C2×C61—42no—
316C2×C61—41no—
317C2×C61—82no—
318C2×C61C22×#9911yescharacteristic
319D61—21no—
320D61—41no—
321D61—42no—
322D61—82no—
323D61#212812yesdirect product factor
324C22×#41—611no—
325C23×D4—611no—
326#535—611no—
327#535—611no—
328#535—612no—
329#535—612no—
330#535—611no—
331#535—1222no—
332#535—1222no—
333#535C2×D6111yescharacteristic
334C2×#127—612no—
335C2×#124—612no—
336C2×#121—1222no—
337C2×#120—1222no—
338C2×#117—611no—
339C2×#117—1222no—
340#410—612no—
341#410—614no—
342#410—1222no—
343#410—1222no—
344#410C2×D6112yescomplemented
345C2×#99—614no—
346C2×#99—1222no—
347#353—614no—
348#353—618no—
349#353C2×D6114yescomplemented
350C61C2×#212811yescharacteristic, complemented, Sylow
351C25—1221no—
352C2×#41—611no—
353C2×#41—612no—
354C2×#41—611no—
355C2×#41—611no—
356C2×#41—612no—
357C2×#41—2442no—
358C2×#41—2442no—
359C2×#41C22×D6111yescharacteristic
360C22×Q8—611no—
361C22×D4—612no—
362C22×D4—611no—
363C22×D4—612no—
364C22×D4—612no—
365C22×D4—611no—
366C22×D4—1221no—
367C22×D4—1221no—
368C22×D4—1221no—
369C22×D4—1224no—
370C22×D4—2442no—
371C22×D4—2442no—
372C22×D4C4×D6111yescharacteristic, complemented
373C23×C4—611no—
374C23×C4—612no—
375#127—612no—
376#127—612no—
377#127—612no—
378#127—1224no—
379#127C22×D6112yes—
380#124—612no—
381#124—612no—
382#124—612no—
383#124—1224no—
384#124C4×D6112yescomplemented
385#121—22no—
386#121—1224no—
387#121—1222no—
388#121—1222no—
389#121—1224no—
390#121—1222no—
391#120—22no—
392#120—1222no—
393#120—1224no—
394#120—1224no—
395#120—1222no—
396#120—1222no—
397#117—22no—
398#117—611no—
399#117—612no—
400#117—1221no—
401#117—1221no—
402#117—1224no—
403#117—1222no—
404#117—2442no—
405#117—2442no—
406#117C4×D6111yescharacteristic, complemented
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—2442no—
409C2×C22⋊C4—1224no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1224no—
412C2×C42—612no—
413C2×C42—2442no—
414#99—22no—
415#99—614no—
416#99—1224no—
417#99—1224no—
418#99—1222no—
419#99C22×D6114yescomplemented
420C24—21no—
421C24—612no—
422C24—611no—
423C24—1222no—
424C24—1222no—
425C24—1221no—
426C24—4884no—
427C24—2442no—
428C24—2442no—
429#41—42no—
430#41—2442no—
431#41—2442no—
432#41—2442no—
433#41—2442no—
434#41—2442no—
435#41—2442no—
436#41—2442no—
437C2×Q8—611no—
438C2×Q8—612no—
439C2×Q8—2441no—
440C2×Q8—2441no—
441C2×Q8C2×C4×D6111yescharacteristic
442C2×D4—24no—
443C2×D4—21no—
444C2×D4—42no—
445C2×D4—612no—
446C2×D4—612no—
447C2×D4—612no—
448C2×D4—611no—
449C2×D4—612no—
450C2×D4—612no—
451C2×D4—1224no—
452C2×D4—1222no—
453C2×D4—1222no—
454C2×D4—1224no—
455C2×D4—1224no—
456C2×D4—1221no—
457C2×D4—4888no—
458C2×D4—2442no—
459C2×D4—2442no—
460C2×D4—2442no—
461C2×D4—2441no—
462C2×D4—2441no—
463C2×D4—2442no—
464C2×D4—2442no—
465C2×D4—2442no—
466C2×D4—2442no—
467C2×D4—2444no—
468C2×D4—2444no—
469C2×D4—2444no—
470C2×D4C2×C4×D6112yes—
471C2×D4D4×D6112yes—
472C2×D4C23×D6111yescharacteristic
473C22×C4—611no—
474C22×C4—612no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—611no—
481C22×C4—612no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1222no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—2444no—
490C22×C4C2×C4×D6111yescharacteristic
491C22×C4D4×D6112yes—
492C4⋊C4—21no—
493C4⋊C4—42no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—2442no—
497C4⋊C4—2444no—
498C22⋊C4—21no—
499C22⋊C4—24no—
500C22⋊C4—24no—
501C22⋊C4—1224no—
502C22⋊C4—1222no—
503C22⋊C4—1224no—
504C22⋊C4—1221no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—2444no—
508C42—42no—
509C42—612no—
510C42—1222no—
511C42—2442no—
512C42—2444no—
513C42C2×C4×D6112yes—
514C23—22no—
515C23—42no—
516C23—82no—
517C23—611no—
518C23—611no—
519C23—612no—
520C23—1222no—
521C23—1222no—
522C23—1221no—
523C23—1221no—
524C23—1222no—
525C23—4882no—
526C23—4884no—
527C23—4888no—
528C23—4888no—
529C23—2442no—
530C23—2442no—
531C23—2442no—
532C23—2448no—
533C23—2444no—
534C23—2444no—
535C23—2442no—
536C23C22⋊C4×D6111yescharacteristic
537C23C2×D4×D6112yes—
538Q8—41no—
539Q8—2441no—
540Q8—2442no—
541D4—42no—
542D4—41no—
543D4—42no—
544D4—42no—
545D4—84no—
546D4—4884no—
547D4—4884no—
548D4—4884no—
549D4—2441no—
550D4—2442no—
551D4—2442no—
552D4—2444no—
553D4—2442no—
554D4—2442no—
555D4—2442no—
556D4—2444no—
557D4—2442no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—44no—
563C2×C4—611no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—611no—
567C2×C4—612no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—4888no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4C22⋊C4×D6112yes—
586C2×C4C22⋊C4×D6111yescharacteristic
587C2×C4C22×C4×D6111yescharacteristic
588C22—22no—
589C22—21no—
590C22—42no—
591C22—41no—
592C22—44no—
593C22—42no—
594C22—84no—
595C22—611no—
596C22—611no—
597C22—1222no—
598C22—1222no—
599C22—1221no—
600C22—4882no—
601C22—4884no—
602C22—4884no—
603C22—4884no—
604C22—4888no—
605C22—2444no—
606C22—2442no—
607C22—2441no—
608C22—2444no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×C22⋊C4×D6111yescharacteristic
613C4—21no—
614C4—22no—
615C4—42no—
616C4—84no—
617C4—612no—
618C4—1221no—
619C4—1222no—
620C4—4884no—
621C4—2442no—
622C4#99×D6112yes—
623C2—21no—
624C2—42no—
625C2—41no—
626C2—82no—
627C2—612no—
628C2—1221no—
629C2—4882no—
630C2—2441no—
631C2—2442no—
632C2C2×#99×D6111yescharacteristic
subgroupG/H
centerC2C2×#99×D61
commutatorC2×C4×C61C23×C4
FrattiniC2×D4C23×D61
FittingC61×#2128C2
socleC2×C61C22×#99
prime radical#535×D61C2
cosocleC2×D4×C61C24
hypercenter#2128D61
hypocenterC61C2×#2128
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2128

Complements

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

NH ≅ G/Ntype
2. #410×D61C2semidirect
3. #21447491C2semidirect
4. #21447490C2semidirect
5. #21447489C2semidirect
6. #353×D61C2semidirect
7. C61×#2128C2semidirect
9. C22×D4×D61C4semidirect
10. #16840331C4semidirect
11. #16840330C4semidirect
12. #16840329C4semidirect
18. #124×D61C4semidirect
22. #117×D61C4semidirect
23. #16839736C22semidirect
26. #16839734C22semidirect
28. #99×D61C22semidirect
29. #16839555C22semidirect
30. #16839554C22semidirect
32. C61×#410C22semidirect
33. C61×#353C22semidirect
43. C22×D244C2×C4semidirect
51. C2×D4×D61C2×C4semidirect
60. #14519766C2×C4semidirect
61. #14519765C2×C4semidirect
62. #14519764C2×C4semidirect
75. #14519736C2×C4semidirect
79. #14519734C2×C4semidirect
90. #14519692C23semidirect
94. C22×D4×C61C2×C4semidirect
96. #124×C61C2×C4semidirect
100. #117×C61C2×C4semidirect
102. #99×C61C23semidirect
312. #2128D61direct (N × H)
323. D61#2128direct (N × H)
344. #410C2×D61semidirect
349. #353C2×D61semidirect
350. C61C2×#2128semidirect
372. C22×D4C4×D61semidirect
384. #124C4×D61semidirect
406. #117C4×D61semidirect
419. #99C22×D61semidirect

Maximal subgroups

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

subgroupindex
#535×D612
#410×D612
#214474912
#214474902
#214474892
#353×D612
C61×#21282
C2×#212861