Finite Group Databasebeta

← D61⋊#1912 #30615790

Subgroups — D61⋊#1912 #30615790

Order 15616.

1694 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840830C211yesmaximal, characteristic, complemented
2D61⋊#421C212yesmaximal
3C2×D61⋊#101C211yesmaximal, characteristic, complemented
4#21461867C214yesmaximal, complemented
5#21461866C214yesmaximal, complemented
6#21461489C212yesmaximal, complemented
7#536×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2212yes—
9#16840830—24no—
10#16840830—22no—
11#16840830C2214yescomplemented
12C2×Q8×#6923—24no—
13C2×#14519856—22no—
14C2×#14519856C2212yes—
15#16840761C2214yes—
16#16840760C2214yes—
17#16840759—24no—
18D61⋊#104—24no—
19D61⋊#101—22no—
20D61⋊#101C2214yescomplemented
21C2×#14519838—22no—
22#16840629C2218yescomplemented
23C2×#41×D61C2211yescharacteristic
24C22×Q8×D61C2211yescharacteristic
25#16840338C411yescharacteristic, complemented
26#16840337C2212yes—
27#16840336C411yescharacteristic, complemented
28#16840335—22no—
29#16840334—22no—
30#16840333C411yescharacteristic, complemented
31#16840332C2212yes—
32C4⋊Q8×D61C412yescomplemented
33#124×D61C412yescomplemented
34#123×D61—22no—
35C22⋊Q8×D61—22no—
36#117×D61C2211yescharacteristic
37C61×#536C411yescharacteristic, complemented
38C2×#11758213—22no—
39D61⋊M16—22no—
40D61⋊M16—22no—
41D61⋊M16—22no—
42D61⋊M16C2314yes—
43C2×#11758217—22no—
44C2×D61⋊C8—22no—
45Q8×#6923—216no—
46Q8×#6923—216no—
47#14519856—28no—
48#14519856—28no—
49#14519856—22no—
50#14519856—22no—
51#14519856D414yes—
52#14519856C2314yes—
53C2×C4⋊#6923—24no—
54C2×C4⋊#6923—22no—
55C2×C22⋊#6923—22no—
56C2×C4×#6923—24no—
57C2×C4×#6923D414yes—
58#14519842—28no—
59#14519841—28no—
60#14519838—28no—
61C2×#11758226C2311yescharacteristic
62C2×#11758222—21no—
63C2×#11758222C2×C411yescharacteristic
64C2×#11758224—21no—
65C2×#11758224C2311yescharacteristic
66#41×D61—22no—
67#41×D61—22no—
68#41×D61—22no—
69#41×D61—22no—
70C22×Dic122C2×C411yescharacteristic
71C22×Dic122D412yes—
72C2×Q8×D61—22no—
73C2×Q8×D61—22no—
74C2×Q8×D61—22no—
75C2×Q8×D61—22no—
76C2×Q8×D61C2×C411yescharacteristic
77C2×Q8×D61C2×C412yes—
78C2×Q8×D61D412yes—
79C2×Q8×D61D412yes—
80C2×Q8×D61C2312yes—
81C2×D4×D61—21no—
82C2×D4×D61C2×C411yescharacteristic
83C22×C4×D61—21no—
84C22×C4×D61D412yes—
85C22×C4×D61C2311yescharacteristic
86#14519777C2×C412yescomplemented
87#14519776—24no—
88#14519775C2×C412yescomplemented
89#14519775C2×C412yescomplemented
90#14519766C2×C412yescomplemented
91#14519765C2×C412yescomplemented
92#14519764C2×C412yescomplemented
93#14519763—24no—
94#14519762—24no—
95#14519761—22no—
96#14519760—22no—
97#14519759—24no—
98#14519758—22no—
99#14519757—22no—
100#14519756—22no—
101#14519755—24no—
102#14519754—24no—
103#14519753—22no—
104#14519736C2×C411yescharacteristic
105#14519735—21no—
106#14519734C2312yes—
107#14519733—21no—
108C4⋊C4×D61—24no—
109C4⋊C4×D61—21no—
110C22⋊C4×D61—21no—
111C22⋊C4×D61—24no—
112C42×D61C2×C412yes—
113C2×#41×C61C2×C411yescharacteristic
114C22×Q8×C61C2×C411yescharacteristic
115C4⋊Q8×C61C2×C412yescomplemented
116#124×C61C2×C412yescomplemented
117#123×C61—22no—
118C22⋊Q8×C61—22no—
119#117×C61C2×C411yescharacteristic
120C23×D61—21no—
121C23×D61C2×D411yescharacteristic
122C22×#6923—24no—
123#41×C61—22no—
124#41×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61C22⋊C412yes—
128C2×Q8×C61C22×C411yescharacteristic
129C2×Q8×C61C22×C412yes—
130C2×D4×C61—21no—
131C2×D4×C61C22×C411yescharacteristic
132C22×C4×C61—21no—
133C22×C4×C61C22⋊C412yes—
134C22×C4×C61C22×C411yescharacteristic
135C2×C22⋊D61—21no—
136C2×C22⋊D61—21no—
137C2×C22⋊D61—21no—
138C22×Dic61—21no—
139C22×Dic61C22⋊C412yes—
140C22×Dic61C2×D411yescharacteristic
141#11758226—22no—
142#11758226—22no—
143Q8×D61—24no—
144Q8×D61—24no—
145Q8×D61—24no—
146Q8×D61—24no—
147Q8×D61—24no—
148Q8×D61—24no—
149Q8×D61—24no—
150Q8×D61—24no—
151Q8×D61—22no—
152Q8×D61—22no—
153#11758224—22no—
154#11758224—22no—
155#11758224—42no—
156#11758224—42no—
157D4×D61—22no—
158D4×D61—22no—
159D4×D61—41no—
160D4×D61—41no—
161D4×D61—41no—
162D4×D61—41no—
163#11758222—22no—
164#11758222—22no—
165#11758222—42no—
166#11758222—42no—
167C2×D244—21no—
168C2×D244C22×C411yescharacteristic
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—21no—
174C2×C4×D61—22no—
175C2×C4×D61—21no—
176C2×C4×D61—21no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61C22⋊C414yes—
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C2×D414yes—
182C2×C4×D61C2×D412yes—
183C2×C4×D61C2×D412yes—
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×C4×D61#4112yes—
187C2×C4×D61#4112yes—
188C2×C4×D61C2411yescharacteristic
189C2×Dic122—22no—
190C2×Dic122—22no—
191C2×Dic122—24no—
192C2×Dic122—24no—
193C2×Dic122—21no—
194C2×Dic122C22⋊C414yes—
195C2×Dic122C22⋊C412yes—
196C2×Dic122C22×C411yescharacteristic
197C2×Dic122C22×C412yes—
198C2×Dic122C2×D414yes—
199C22⋊#6923—24no—
200C22⋊#6923—24no—
201#11758217—24no—
202#11758217—42no—
203C2×#92420—42no—
204C4⋊#6923—216no—
205C4⋊#6923—24no—
206C4⋊#6923—24no—
207C4×#6923—216no—
208C4×#6923—24no—
209C4×#6923C2×D418yes—
210#11758213—24no—
211#11758213—42no—
212D61⋊C8—24no—
213D61⋊C8—42no—
214C4⋊C4×C61—21no—
215C4⋊C4×C61—24no—
216C22⋊C4×C61—21no—
217C22⋊C4×C61—24no—
218C42×C61C22×C412yes—
219C22⋊Dic61—21no—
220C22⋊Dic61—24no—
221#11758198—22no—
222#11758198—24no—
223#11758198—24no—
224C4⋊Dic61—21no—
225C4⋊Dic61—24no—
226#11758196—24no—
227#11758196—24no—
228#11758196—22no—
229C4×Dic61—22no—
230C4×Dic61C22×C412yes—
231C23×C61—21no—
232C23×C61C2×C22⋊C411yescharacteristic
233C22×D61—21no—
234C22×D61—42no—
235C22×D61—44no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61#12011yescharacteristic
240C22×D61#12011yescharacteristic
241C22×D61#12311yescharacteristic
242C22×D61#12311yescharacteristic
243C22×D61C22×D411yescharacteristic
244C2×#6923—28no—
245C2×#6923—48no—
246Q8×C61—24no—
247Q8×C61—24no—
248Q8×C61—22no—
249D4×C61—22no—
250D4×C61—41no—
251D4×C61—41no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—42no—
257C2×C4×C61C2×C22⋊C414yes—
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C4×D412yes—
260C2×C4×C61C4×D412yes—
261C2×C4×C61C23×C411yescharacteristic
262C22⋊D61—41no—
263C22⋊D61—41no—
264C22⋊D61—41no—
265C22⋊D61—41no—
266C22⋊D61—42no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—42no—
272C2×Dic61C2×C22⋊C414yes—
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61#12112yes—
275C2×Dic61#12112yes—
276C2×Dic61C22×D411yescharacteristic
277D244—22no—
278D244—41no—
279D244—41no—
280C4×D61—24no—
281C4×D61—24no—
282C4×D61—24no—
283C4×D61—24no—
284C4×D61—44no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—42no—
288C4×D61—42no—
289C4×D61C4×D412yes—
290C4×D61C4×D412yes—
291C4×D61#12012yes—
292C4×D61#12112yes—
293C4×D61#12112yes—
294C4×D61#12312yes—
295C4×D61C22×D412yes—
296C4×D61C2×#4112yes—
297Dic122—24no—
298Dic122—24no—
299Dic122—28no—
300Dic122—28no—
301Dic122—22no—
302Dic122—42no—
303#92420—44no—
304C2×#1912—611nomaximal, Sylow
305C22×C61—41no—
306C22×C61—42no—
307C22×C61—41no—
308C22×C61#38611yescharacteristic
309C22×C61#38611yescharacteristic
310C22×C61C22×C22⋊C411yescharacteristic
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—42no—
318C2×D61—41no—
319C2×D61—82no—
320C2×D61—82no—
321C2×D61#38611yescharacteristic
322C2×D61#38611yescharacteristic
323C2×D61C2×#12011yescharacteristic
324C2×D61C2×#12311yescharacteristic
325#6923—48no—
326C4×C61—24no—
327C4×C61—42no—
328C4×C61—41no—
329C4×C61#38612yes—
330C4×C61C2×C4×D412yes—
331Dic61—24no—
332Dic61—42no—
333Dic61—41no—
334Dic61#38612yes—
335Dic61C2×#12112yes—
336C2×#536—611no—
337C2×#519—611no—
338C2×#421—612no—
339C22×#101—611no—
340#1912—614no—
341#1912—614no—
342C2×H(3,ℤ/4)—612no—
343C2×C61—21no—
344C2×C61—21no—
345C2×C61—21no—
346C2×C61—81no—
347C2×C61C2×#38611yescharacteristic
348D61—21no—
349D61—21no—
350D61—21no—
351D61—81no—
352D61#191212yescomplemented
353C22×#41—611no—
354C23×Q8—611no—
355C22×M16—612no—
356#536—611no—
357#536—611no—
358#536—612no—
359#536—612no—
360#536—611no—
361#536—1222no—
362#536—1222no—
363#536#692311yescharacteristic, complemented
364C2×C4⋊Q8—612no—
365C2×#124—612no—
366C2×#123—1222no—
367C2×C22⋊Q8—1222no—
368#519—614no—
369#519—1224no—
370#519—1222no—
371C2×C4×Q8—1224no—
372C2×#117—612no—
373C2×#117—611no—
374C2×#117—1222no—
375#421—614no—
376#421—614no—
377#421—1224no—
378C2×#104—1224no—
379C2×#101—614no—
380C2×#101—1222no—
381C2×#101—1222no—
382H(3,ℤ/4)—618no—
383C61C2×#191211yescharacteristic, complemented, Sylow
384C2×#41—611no—
385C2×#41—611no—
386C2×#41—611no—
387C2×#41—1221no—
388C2×#41—1221no—
389C2×#41—1222no—
390C2×#41—1222no—
391C2×#41—1222no—
392C2×#41—1222no—
393C2×#41C2×#692311yescharacteristic
394C22×Q8—612no—
395C22×Q8—612no—
396C22×Q8—612no—
397C22×Q8—612no—
398C22×Q8—612no—
399C22×Q8—611no—
400C22×Q8—611no—
401C22×Q8—1222no—
402C22×Q8—1222no—
403C22×Q8—1222no—
404C22×Q8—1222no—
405C22×Q8C2×#692311yescharacteristic
406C22×D4—611no—
407C22×D4—1221no—
408C23×C4—611no—
409C23×C4—612no—
410C23×C4—1221no—
411C2×M16—614no—
412C2×M16—1222no—
413C2×M16—1222no—
414C2×M16—1222no—
415C2×M16—1222no—
416C2×M16—1222no—
417C22×C8—1222no—
418C4⋊Q8—612no—
419C4⋊Q8—612no—
420C4⋊Q8—612no—
421C4⋊Q8—1224no—
422C4⋊Q8C2×#692312yescomplemented
423#124—612no—
424#124—612no—
425#124—612no—
426#124—1224no—
427#124C2×#692312yescomplemented
428#123—22no—
429#123—1224no—
430#123—1222no—
431#123—1224no—
432#123—1222no—
433#123—1222no—
434C22⋊Q8—22no—
435C22⋊Q8—1224no—
436C22⋊Q8—1224no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439C22⋊Q8—1222no—
440C4×Q8—12216no—
441C4×Q8—12216no—
442#117—614no—
443#117—614no—
444#117—611no—
445#117—612no—
446#117—1221no—
447#117—1221no—
448#117—1222no—
449#117—1222no—
450#117—1228no—
451#117—1228no—
452#117C2×#692311yescharacteristic
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1224no—
455C2×C4⋊C4—1222no—
456C2×C4⋊C4—1224no—
457C2×C22⋊C4—1224no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1222no—
460C2×C42—614no—
461C2×C42—612no—
462C2×C42—1224no—
463#104—1228no—
464#104—1228no—
465#101—1228no—
466C24—611no—
467C24—1221no—
468#41—22no—
469#41—22no—
470#41—1222no—
471#41—1222no—
472#41—1222no—
473#41—1222no—
474#41—1222no—
475#41—1222no—
476#41—2442no—
477#41—2442no—
478#41—2442no—
479#41—2442no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—614no—
483C2×Q8—612no—
484C2×Q8—614no—
485C2×Q8—612no—
486C2×Q8—611no—
487C2×Q8—1224no—
488C2×Q8—1224no—
489C2×Q8—1221no—
490C2×Q8—1224no—
491C2×Q8—1224no—
492C2×Q8—1224no—
493C2×Q8—1224no—
494C2×Q8—1224no—
495C2×Q8—1224no—
496C2×Q8—1224no—
497C2×Q8—1224no—
498C2×Q8—1222no—
499C2×Q8—1222no—
500C2×Q8—1222no—
501C2×Q8—1222no—
502C2×Q8C22⋊#692312yes—
503C2×Q8C22×#692311yescharacteristic
504C2×Q8C22×#692312yes—
505C2×D4—21no—
506C2×D4—611no—
507C2×D4—1221no—
508C2×D4—1222no—
509C2×D4—1222no—
510C2×D4—1221no—
511C2×D4—1221no—
512C2×D4—1221no—
513C2×D4—2441no—
514C2×D4—2441no—
515C2×D4—2441no—
516C2×D4—2441no—
517C2×D4C22×#692311yescharacteristic
518C22×C4—21no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—612no—
523C22×C4—612no—
524C22×C4—612no—
525C22×C4—612no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—612no—
529C22×C4—614no—
530C22×C4—614no—
531C22×C4—1221no—
532C22×C4—1222no—
533C22×C4—1222no—
534C22×C4—1222no—
535C22×C4—1222no—
536C22×C4—1222no—
537C22×C4—1221no—
538C22×C4—1221no—
539C22×C4—1224no—
540C22×C4—1221no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4C22⋊#692312yes—
544C22×C4C22×#692311yescharacteristic
545M16—1224no—
546M16—1224no—
547M16—2442no—
548M16—2442no—
549C2×C8—1224no—
550C2×C8—2442no—
551C2×C8—2442no—
552C4⋊C4—21no—
553C4⋊C4—24no—
554C4⋊C4—1222no—
555C4⋊C4—1224no—
556C4⋊C4—1224no—
557C4⋊C4—1224no—
558C4⋊C4—1224no—
559C4⋊C4—12216no—
560C4⋊C4—1224no—
561C4⋊C4—1221no—
562C22⋊C4—21no—
563C22⋊C4—24no—
564C22⋊C4—1224no—
565C22⋊C4—1222no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C22⋊C4—1224no—
569C22⋊C4—1221no—
570C22⋊C4—1224no—
571C42—618no—
572C42—612no—
573C42—1222no—
574C42—1224no—
575C42—12216no—
576C42C22×#692312yes—
577C23—21no—
578C23—611no—
579C23—611no—
580C23—611no—
581C23—611no—
582C23—611no—
583C23—611no—
584C23—1221no—
585C23—2441no—
586C23—2441no—
587C23—2442no—
588C23—2444no—
589C23C2×C22⋊#692311yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—22no—
593Q8—1228no—
594Q8—1228no—
595Q8—1224no—
596Q8—1224no—
597Q8—1222no—
598Q8—2442no—
599D4—22no—
600D4—41no—
601D4—41no—
602D4—1222no—
603D4—2441no—
604D4—2441no—
605D4—2442no—
606D4—2441no—
607D4—2441no—
608D4—2441no—
609D4—2441no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—42no—
615C2×C4—612no—
616C2×C4—612no—
617C2×C4—611no—
618C2×C4—611no—
619C2×C4—612no—
620C2×C4—612no—
621C2×C4—612no—
622C2×C4—612no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—612no—
626C2×C4—612no—
627C2×C4—614no—
628C2×C4—1224no—
629C2×C4—1224no—
630C2×C4—1224no—
631C2×C4—1224no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1228no—
635C2×C4—1222no—
636C2×C4—1221no—
637C2×C4—2442no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2444no—
641C2×C4—2442no—
642C2×C4—2448no—
643C2×C4—2442no—
644C2×C4C2×C22⋊#692314yes—
645C2×C4C2×C22⋊#692311yescharacteristic
646C2×C4D4×#692312yes—
647C2×C4D4×#692312yes—
648C2×C4C23×#692311yescharacteristic
649C8—2444no—
650C22—41no—
651C22—42no—
652C22—41no—
653C22—611no—
654C22—611no—
655C22—611no—
656C22—611no—
657C22—1222no—
658C22—1222no—
659C22—1222no—
660C22—4882no—
661C22—4882no—
662C22—2441no—
663C22—2441no—
664C22—2441no—
665C22—2442no—
666C22—2441no—
667C22#1684071011yescharacteristic
668C22#1684071011yescharacteristic
669C22C22×C22⋊#692311yescharacteristic
670C4—24no—
671C4—42no—
672C4—41no—
673C4—612no—
674C4—612no—
675C4—1224no—
676C4—2442no—
677C4—2448no—
678C4—2441no—
679C4#1684071012yes—
680C4C2×D4×#692312yes—
681C2—21no—
682C2—21no—
683C2—21no—
684C2—81no—
685C2—612no—
686C2—1221no—
687C2—1221no—
688C2—1221no—
689C2—4881no—
690C2C2×#1684071011yescharacteristic
subgroupG/H
centerC2C2×#16840710
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC2×C61C2×#386
prime radicalC2×#41×D61C22
cosocleC2×C4×D61C24
hypercenter#536#6923
hypocenterC61C2×#1912
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16840830C2semidirect
3. C2×D61⋊#101C2semidirect
4. #21461867C2semidirect
5. #21461866C2semidirect
6. #21461489C2semidirect
11. #16840830C22semidirect
20. D61⋊#101C22semidirect
22. #16840629C22semidirect
25. #16840338C4semidirect
27. #16840336C4semidirect
30. #16840333C4semidirect
32. C4⋊Q8×D61C4semidirect
33. #124×D61C4semidirect
37. C61×#536C4semidirect
86. #14519777C2×C4semidirect
88. #14519775C2×C4semidirect
89. #14519775C2×C4semidirect
90. #14519766C2×C4semidirect
91. #14519765C2×C4semidirect
92. #14519764C2×C4semidirect
115. C4⋊Q8×C61C2×C4semidirect
116. #124×C61C2×C4semidirect
352. D61#1912semidirect
363. #536#6923semidirect
383. C61C2×#1912semidirect
422. C4⋊Q8C2×#6923semidirect
427. #124C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168408302
D61⋊#4212
C2×D61⋊#1012
#214618672
#214618662
#214614892
#536×D612
C2×#191261