Finite Group Databasebeta

← D61⋊#1503 #30610914

Subgroups — D61⋊#1503 #30610914

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#431C211yesmaximal, characteristic
2C2×C4×C4⋊#6923C211yesmaximal, characteristic
3#21461203C212yesmaximal, complemented
4#21461202C212yesmaximal, complemented
5#21461201C214yesmaximal, complemented
6D61⋊#328C214yesmaximal
7#533×D61C211yesmaximal, characteristic
8C22×C4⋊#6923—21no—
9C22×C4×#6923—21no—
10#16840781C2212yes—
11#16840780C2212yes—
12#16840779—21no—
13#16840778—21no—
14D61⋊(C4⋊C8)—21no—
15D61⋊(C22⋊C8)C2212yes—
16C4×D61⋊C8—21no—
17C4×C4⋊#6923—21no—
18C4×C4⋊#6923—21no—
19C4×C4⋊#6923C2212yes—
20C4×C4⋊#6923C2212yes—
21C2×(C2×#6923)⋊C4C2212yes—
22#16840616C2218yes—
23#16840615C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318—21no—
26#16840317C2211yescharacteristic
27#16840317C2211yescharacteristic
28#16840316—21no—
29#16840316C2212yes—
30#16840315—21no—
31#16840314—21no—
32#16840314C412yescomplemented
33C4⋊Q8×D61C411yescharacteristic, complemented
34C4⋊Q8×D61C411yescharacteristic, complemented
35#125×D61—21no—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61C2212yes—
38C4×Q8×D61—21no—
39C4×Q8×D61C412yescomplemented
40#117×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C23×#6923—22no—
44C2×D61⋊C8—22no—
45C2×C4⋊#6923—28no—
46C2×C4⋊#6923—22no—
47C2×C4⋊#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50C2×C4×#6923—44no—
51#14519844—22no—
52C4⋊#92420—22no—
53#14519834C2314yes—
54C22⋊#92420—22no—
55C4×#92420—22no—
56C4×#92420—22no—
57(C2×#6923)⋊C4—22no—
58(C2×#6923)⋊C4C2314yes—
59C2×Q8×D61—21no—
60C2×Q8×D61C2×C412yes—
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63C22×C4×D61C2311yescharacteristic
64#14519777C2×C411yescharacteristic, complemented
65#14519777C2×C411yescharacteristic, complemented
66#14519776—21no—
67#14519776—21no—
68#14519776—21no—
69#14519776C2×C412yescomplemented
70#14519775—21no—
71#14519775C2×C412yescomplemented
72#14519769—21no—
73#14519768—21no—
74#14519768—21no—
75#14519767—22no—
76#14519767—22no—
77#14519757—21no—
78#14519757C2312yes—
79#14519756—21no—
80#14519756C2×C412yes—
81#14519755—22no—
82#14519755—22no—
83#14519754—22no—
84#14519754—22no—
85#14519753—21no—
86#14519753C2312yes—
87Q8×Dic61—21no—
88Q8×Dic61C2×C412yescomplemented
89C4×Dic122—21no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92C4×Dic122C2×C412yescomplemented
93#14519742—22no—
94#14519742—22no—
95#14519742—21no—
96#14519742C2×C412yescomplemented
97#14519736—21no—
98#14519735—21no—
99#14519735—21no—
100#14519734—22no—
101#14519733—22no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—21no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61C2×C412yes—
108C22⋊C4×D61—22no—
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C42×D61—22no—
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61C2311yescharacteristic
120C42×D61C2311yescharacteristic
121C4⋊Q8×C61C2×C411yescharacteristic, complemented
122C4⋊Q8×C61C2×C411yescharacteristic, complemented
123#125×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61C2×C412yes—
126C4×Q8×C61—21no—
127C4×Q8×C61C2×C412yescomplemented
128#117×C61—21no—
129C2×C42×C61C2×C411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C22×#6923—28no—
132C22×#6923—24no—
133C22×#6923—44no—
134C22×#6923—44no—
135C2×Q8×C61—21no—
136C2×Q8×C61C22×C412yes—
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×Dic61C22⋊C411yescharacteristic
141C22×Dic61C22⋊C411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143Q8×D61—44no—
144Q8×D61—44no—
145C2×C4×D61—24no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—42no—
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C22⋊C411yescharacteristic
153C2×C4×D61SD1614yes—
154C2×C4×D61Q1614yes—
155C2×C4×D61C22×C411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2411yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—22no—
165C2×Dic122—22no—
166C2×Dic122C22×C412yes—
167C2×#92420—24no—
168C4⋊#6923—28no—
169C4⋊#6923—28no—
170C4×#6923—44no—
171C4×#6923—44no—
172D61⋊C8—44no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61C22×C412yes—
179C22⋊C4×C61—22no—
180C42×C61—22no—
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C22⋊Dic61—22no—
186#11758198—22no—
187#11758198—22no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193C4⋊Dic61C22×C412yes—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22×C411yescharacteristic
212C4×Dic61C22×C411yescharacteristic
213C4×Dic61C2×D411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C23×C61C2×C22⋊C411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61C2×C22⋊C411yescharacteristic
221C22×D61C2×SD1612yes—
222C22×D61C2×Q1612yes—
223C22×D61C22×D411yescharacteristic
224C2×#6923—28no—
225C2×#6923—44no—
226C2×#6923—416no—
227C2×#6923—44no—
228Q8×C61—42no—
229Q8×C61—42no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—41no—
234C2×C4×C61#10314yes—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C23×C411yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—41no—
245C2×Dic61#10314yes—
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C411yescharacteristic
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C22×D411yescharacteristic
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—41no—
258C4×D61—41no—
259C4×D61—44no—
260C4×D61—44no—
261C4×D61#10314yes—
262C4×D61#10314yes—
263C4×D61#10414yes—
264C4×D61C2×SD1612yes—
265C4×D61C2×SD1612yes—
266C4×D61C2×Q1612yes—
267C4×D61C2×Q1612yes—
268Dic122—44no—
269Dic122—42no—
270Dic122—44no—
271Dic122—42no—
272#92420—44no—
273C2×#1503—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61C2×#10312yes—
277C22×C61C22×C22⋊C411yescharacteristic
278C2×D61—21no—
279C2×D61—21no—
280C2×D61—24no—
281C2×D61—24no—
282C2×D61#41012yes—
283C2×D61C2×#10312yes—
284C2×D61C2×#10412yes—
285C2×D61C22×SD1611yescharacteristic
286C2×D61C22×Q1611yescharacteristic
287#6923—48no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61C2×#10312yes—
294C4×C61C2×#10312yes—
295C4×C61C2×#10412yes—
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—42no—
300Dic61—42no—
301Dic61C2×#10312yes—
302Dic61C2×#10312yes—
303Dic61C2×#10412yes—
304C2×#533—611no—
305C2×#431—611no—
306C2×C4×C4⋊C4—611no—
307#1503—614no—
308#1503—612no—
309#1503—612no—
310C2×#328—614no—
311C2×C61—22no—
312C2×C61C2×#41011yescharacteristic
313C2×C61C22×#10311yescharacteristic
314C2×C61C22×#10411yescharacteristic
315D61—21no—
316D61—21no—
317D61#150314yescomplemented
318#533—611no—
319#533—611no—
320#533—612no—
321#533—612no—
322#533—611no—
323#533—1221no—
324#533—1221no—
325#533—1221no—
326#533—1221no—
327#533#692311yescharacteristic, complemented
328C2×C4⋊Q8—611no—
329C2×C4⋊Q8—611no—
330C2×#125—1221no—
331C2×C22⋊Q8—612no—
332C2×C22⋊Q8—1221no—
333C2×C4×Q8—612no—
334C2×C4×Q8—1221no—
335C2×#117—1221no—
336C22×C4⋊C4—1221no—
337C22×C42—611no—
338C22×C42—1221no—
339#431—612no—
340#431—612no—
341#431—1221no—
342#431—1221no—
343C2×C4⋊C8—1221no—
344C2×C22⋊C8—612no—
345C2×C4×C8—1221no—
346C4×C4⋊C4—612no—
347C4×C4⋊C4—612no—
348C4×C4⋊C4—1221no—
349C4×C4⋊C4—1221no—
350C2×#95—612no—
351#328—618no—
352#328—618no—
353C61C2×#150311yescharacteristic, complemented, Sylow
354C22×Q8—612no—
355C22×Q8—1221no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—1222no—
360C22×C8—1222no—
361C4⋊Q8—612no—
362C4⋊Q8—612no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8C2×#692311yescharacteristic, complemented
370C4⋊Q8C2×#692311yescharacteristic, complemented
371#125—21no—
372#125—1222no—
373#125—1222no—
374#125—1221no—
375#125—1221no—
376#125—1221no—
377C22⋊Q8—21no—
378C22⋊Q8—612no—
379C22⋊Q8—612no—
380C22⋊Q8—612no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1221no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388C22⋊Q8C2×#692312yes—
389C4×Q8—21no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×Q8—1221no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8—1221no—
400C4×Q8C2×#692312yescomplemented
401#117—21no—
402#117—1222no—
403#117—1222no—
404#117—1221no—
405#117—1221no—
406#117—1221no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1222no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1222no—
414C2×C4⋊C4—1222no—
415C2×C4⋊C4—1228no—
416C2×C22⋊C4—1222no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—1222no—
429C2×C42—1222no—
430C2×C42—1222no—
431C2×C42—2444no—
432C2×C42C2×#692311yescharacteristic
433C4⋊C8—1222no—
434C4⋊C8—1222no—
435C22⋊C8—614no—
436C22⋊C8—1222no—
437C4×C8—1222no—
438C4×C8—1222no—
439#95—614no—
440#95—1222no—
441C24—611no—
442C2×Q8—21no—
443C2×Q8—612no—
444C2×Q8—1222no—
445C2×Q8—1222no—
446C2×Q8—1221no—
447C2×Q8—2444no—
448C2×Q8—2444no—
449C2×Q8C22×#692312yes—
450C22×C4—611no—
451C22×C4—614no—
452C22×C4—614no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1224no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1224no—
471C22×C4—1228no—
472C22×C4—2442no—
473C22×C4—2444no—
474C22×C4—2444no—
475C22×C4C22⋊#692311yescharacteristic
476C22×C4C22⋊#692311yescharacteristic
477C22×C4C22×#692311yescharacteristic
478C2×C8—1224no—
479C2×C8—2444no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—612no—
486C4⋊C4—1222no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1228no—
495C4⋊C4—1228no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4C22×#692312yes—
502C22⋊C4—22no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C42—22no—
507C42—611no—
508C42—611no—
509C42—611no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—1222no—
520C42—1222no—
521C42—1222no—
522C42—2444no—
523C42—2444no—
524C42C22⋊#692311yescharacteristic
525C42C22⋊#692311yescharacteristic
526C42C22×#692311yescharacteristic
527C42C22×#692311yescharacteristic
528C23—611no—
529C23—612no—
530C23—612no—
531C23—611no—
532C23—1222no—
533C23—1222no—
534C23C2×C22⋊#692311yescharacteristic
535Q8—42no—
536Q8—42no—
537Q8—2442no—
538Q8—2444no—
539Q8—2444no—
540Q8—2442no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—614no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—614no—
550C2×C4—614no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—614no—
556C2×C4—611no—
557C2×C4—611no—
558C2×C4—611no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1222no—
565C2×C4—1228no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2444no—
571C2×C4—2444no—
572C2×C4—2441no—
573C2×C4—2444no—
574C2×C4—24416no—
575C2×C4—2444no—
576C2×C4#1451984014yes—
577C2×C4C2×C22⋊#692311yescharacteristic
578C2×C4C2×C22⋊#692311yescharacteristic
579C2×C4C2×C22⋊#692311yescharacteristic
580C2×C4C2×C22⋊#692311yescharacteristic
581C2×C4C2×C22⋊#692311yescharacteristic
582C2×C4C23×#692311yescharacteristic
583C8—2444no—
584C22—21no—
585C22—21no—
586C22—612no—
587C22—612no—
588C22—612no—
589C22—611no—
590C22—611no—
591C22—1224no—
592C22—1224no—
593C22—1221no—
594C22—1221no—
595C22C2×#1451984012yes—
596C22C22×C22⋊#692311yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—42no—
601C4—42no—
602C4—612no—
603C4—612no—
604C4—612no—
605C4—1221no—
606C4—1221no—
607C4—1221no—
608C4—2448no—
609C4—2442no—
610C4—2442no—
611C4C2×#1451984012yes—
612C4C2×#1451984012yes—
613C4D61⋊#10412yes—
614C2—22no—
615C2—614no—
616C2—1221no—
617C2—1221no—
618C2D61⋊#41011yescharacteristic
619C2C22×#1451984011yescharacteristic
620C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61C2×#1503
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461203C2semidirect
4. #21461202C2semidirect
5. #21461201C2semidirect
24. #16840319C4semidirect
32. #16840314C4semidirect
33. C4⋊Q8×D61C4semidirect
34. C4⋊Q8×D61C4semidirect
39. C4×Q8×D61C4semidirect
42. C61×#533C4semidirect
64. #14519777C2×C4semidirect
65. #14519777C2×C4semidirect
69. #14519776C2×C4semidirect
71. #14519775C2×C4semidirect
88. Q8×Dic61C2×C4semidirect
92. C4×Dic122C2×C4semidirect
96. #14519742C2×C4semidirect
121. C4⋊Q8×C61C2×C4semidirect
122. C4⋊Q8×C61C2×C4semidirect
127. C4×Q8×C61C2×C4semidirect
317. D61#1503semidirect
327. #533#6923semidirect
353. C61C2×#1503semidirect
369. C4⋊Q8C2×#6923semidirect
370. C4⋊Q8C2×#6923semidirect
400. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4312
C2×C4×C4⋊#69232
#214612032
#214612022
#214612012
D61⋊#3282
#533×D612
C2×#150361