Finite Group Databasebeta

← D61⋊#1506 #30610952

Subgroups — D61⋊#1506 #30610952

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#432C211yesmaximal, characteristic
2C2×#16840698C211yesmaximal, characteristic
3#21461213C212yesmaximal, complemented
4#21461212C214yesmaximal, complemented
5#21461211C212yesmaximal, complemented
6D61⋊#328C214yesmaximal
7#533×D61C211yesmaximal, characteristic
8C22×C4×#6923—21no—
9#16840785C2212yes—
10#16840784C2212yes—
11#16840783—21no—
12#16840782—21no—
13D61⋊(C4⋊C8)—21no—
14D61⋊(C22⋊C8)C2212yes—
15#16840721—21no—
16#16840698—21no—
17#16840698—21no—
18#16840698C2212yes—
19#16840698C2212yes—
20C2×(C2×#6923)⋊C4—21no—
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—
48#14519844—22no—
49C4⋊#92420—22no—
50#14519834C2314yes—
51C22⋊#92420—22no—
52#14519832—22no—
53#14519832—22no—
54(C2×#6923)⋊C4—22no—
55(C2×#6923)⋊C4—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4C2314yes—
58C2×Q8×D61—21no—
59C2×Q8×D61C2×C412yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519777C2×C411yescharacteristic, complemented
64#14519777C2×C411yescharacteristic, complemented
65#14519776—21no—
66#14519776—21no—
67#14519776—21no—
68#14519776C2×C412yescomplemented
69#14519775—21no—
70#14519775C2×C412yescomplemented
71#14519769—21no—
72#14519768—21no—
73#14519768—21no—
74#14519767—22no—
75#14519767—22no—
76#14519757—21no—
77#14519757C2312yes—
78#14519756—21no—
79#14519756C2×C412yes—
80#14519755—22no—
81#14519755—22no—
82#14519754—22no—
83#14519754—22no—
84#14519753—21no—
85#14519753C2312yes—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yescomplemented
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122C2×C412yescomplemented
92#14519742—22no—
93#14519742—22no—
94#14519742—21no—
95#14519742C2×C412yescomplemented
96#14519736—21no—
97#14519735—21no—
98#14519735—21no—
99#14519734—22no—
100#14519733—22no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61C2×C412yes—
107C22⋊C4×D61—22no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C42×D61—22no—
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61C2311yescharacteristic
119C42×D61C2311yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic, complemented
121C4⋊Q8×C61C2×C411yescharacteristic, complemented
122#125×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61C2×C412yes—
125C4×Q8×C61—21no—
126C4×Q8×C61C2×C412yescomplemented
127#117×C61—21no—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C22×#6923—28no—
131C22×#6923—24no—
132C22×#6923—44no—
133C22×#6923—44no—
134C2×Q8×C61—21no—
135C2×Q8×C61C22×C412yes—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22×C411yescharacteristic
139C22×Dic61C22⋊C411yescharacteristic
140C22×Dic61C22⋊C411yescharacteristic
141C22×Dic61C2×D411yescharacteristic
142Q8×D61—44no—
143Q8×D61—44no—
144C2×C4×D61—24no—
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×D61C22×C411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122C22×C412yes—
165C2×#92420—24no—
166C4×#6923—28no—
167C4×#6923—28no—
168D61⋊C8—44no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C22×C412yes—
175C22⋊C4×C61—22no—
176C42×C61—22no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22×C411yescharacteristic
180C42×C61C22×C411yescharacteristic
181C22⋊Dic61—22no—
182#11758198—22no—
183#11758198—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C22×C411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C23×C61C2×C22⋊C411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61C2×C22⋊C411yescharacteristic
217C22×D61#13714yes—
218C22×D61C22×D411yescharacteristic
219C2×#6923—28no—
220C2×#6923—44no—
221C2×#6923—416no—
222C2×#6923—44no—
223Q8×C61—42no—
224Q8×C61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—41no—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C2×C22⋊C411yescharacteristic
235C2×C4×C61C23×C411yescharacteristic
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C2×C22⋊C411yescharacteristic
246C2×Dic61C22×D411yescharacteristic
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—41no—
257C4×D61—41no—
258C4×D61—44no—
259C4×D61—44no—
260C4×D61#10414yes—
261Dic122—44no—
262Dic122—42no—
263Dic122—44no—
264Dic122—42no—
265#92420—44no—
266C2×#1506—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61#41912yes—
270C22×C61C22×C22⋊C411yescharacteristic
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—28no—
274C2×D61#41012yes—
275C2×D61#41912yes—
276C2×D61C2×#10412yes—
277C2×D61C2×#13712yes—
278#6923—48no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—21no—
284C4×C61—42no—
285C4×C61—42no—
286C4×C61C2×#10412yes—
287Dic61—21no—
288Dic61—21no—
289Dic61—21no—
290Dic61—21no—
291Dic61—21no—
292Dic61—42no—
293Dic61—42no—
294Dic61C2×#10412yes—
295C2×#533—611no—
296C2×#432—611no—
297C2×#376—611no—
298#1506—612no—
299#1506—614no—
300#1506—612no—
301C2×#328—614no—
302C2×C61—22no—
303C2×C61C2×#41011yescharacteristic
304C2×C61C2×#41911yescharacteristic
305C2×C61C22×#10411yescharacteristic
306D61—22no—
307D61#150614yescomplemented
308#533—611no—
309#533—611no—
310#533—612no—
311#533—612no—
312#533—611no—
313#533—1221no—
314#533—1221no—
315#533—1221no—
316#533—1221no—
317#533#692311yescharacteristic, complemented
318C2×C4⋊Q8—611no—
319C2×C4⋊Q8—611no—
320C2×#125—1221no—
321C2×C22⋊Q8—612no—
322C2×C22⋊Q8—1221no—
323C2×C4×Q8—612no—
324C2×C4×Q8—1221no—
325C2×#117—1221no—
326C22×C42—611no—
327C22×C42—1221no—
328#432—612no—
329#432—612no—
330#432—1221no—
331#432—1221no—
332C2×C4⋊C8—1221no—
333C2×C22⋊C8—612no—
334C2×#97—1221no—
335#376—612no—
336#376—612no—
337#376—1221no—
338#376—1221no—
339C2×#95—612no—
340C2×#95—1221no—
341#328—618no—
342#328—618no—
343C61C2×#150611yescharacteristic, complemented, Sylow
344C22×Q8—612no—
345C22×Q8—1221no—
346C23×C4—611no—
347C23×C4—611no—
348C23×C4—611no—
349C23×C4—1222no—
350C22×C8—1222no—
351C4⋊Q8—612no—
352C4⋊Q8—612no—
353C4⋊Q8—611no—
354C4⋊Q8—611no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8C2×#692311yescharacteristic, complemented
360C4⋊Q8C2×#692311yescharacteristic, complemented
361#125—21no—
362#125—1222no—
363#125—1222no—
364#125—1221no—
365#125—1221no—
366#125—1221no—
367C22⋊Q8—21no—
368C22⋊Q8—612no—
369C22⋊Q8—612no—
370C22⋊Q8—612no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1222no—
374C22⋊Q8—1222no—
375C22⋊Q8—1221no—
376C22⋊Q8—1221no—
377C22⋊Q8—1221no—
378C22⋊Q8C2×#692312yes—
379C4×Q8—21no—
380C4×Q8—612no—
381C4×Q8—612no—
382C4×Q8—612no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×Q8—1221no—
387C4×Q8—1222no—
388C4×Q8—1221no—
389C4×Q8—1221no—
390C4×Q8C2×#692312yescomplemented
391#117—21no—
392#117—1222no—
393#117—1222no—
394#117—1221no—
395#117—1221no—
396#117—1221no—
397C2×C4⋊C4—612no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1222no—
402C2×C4⋊C4—1221no—
403C2×C22⋊C4—1222no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—1222no—
416C2×C42—1222no—
417C2×C42—1222no—
418C2×C42—1228no—
419C2×C42C2×#692311yescharacteristic
420C4⋊C8—1222no—
421C4⋊C8—1222no—
422C22⋊C8—614no—
423C22⋊C8—1222no—
424#97—1222no—
425#97—1222no—
426#95—614no—
427#95—1222no—
428#95—1222no—
429#95—1222no—
430C24—611no—
431C2×Q8—21no—
432C2×Q8—612no—
433C2×Q8—1222no—
434C2×Q8—1222no—
435C2×Q8—1221no—
436C2×Q8—2444no—
437C2×Q8—2444no—
438C2×Q8C22×#692312yes—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1224no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1228no—
460C22×C4—2442no—
461C22×C4—2444no—
462C22×C4—2444no—
463C22×C4C22⋊#692311yescharacteristic
464C22×C4C22⋊#692311yescharacteristic
465C22×C4C22×#692311yescharacteristic
466C2×C8—1224no—
467C2×C8—2444no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—612no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4C22×#692312yes—
488C22⋊C4—22no—
489C22⋊C4—1222no—
490C22⋊C4—1222no—
491C22⋊C4—1222no—
492C42—22no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—1222no—
506C42—1222no—
507C42—1228no—
508C42—1228no—
509C42—1222no—
510C42C22⋊#692311yescharacteristic
511C42C22⋊#692311yescharacteristic
512C42C22×#692311yescharacteristic
513C42C22×#692311yescharacteristic
514C23—611no—
515C23—614no—
516C23—611no—
517C23—1222no—
518C23—1222no—
519C23C2×C22⋊#692311yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—2442no—
523Q8—2444no—
524Q8—2444no—
525Q8—2442no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—41no—
531C2×C4—611no—
532C2×C4—614no—
533C2×C4—611no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1228no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2441no—
557C2×C4—2444no—
558C2×C4—24416no—
559C2×C4—2444no—
560C2×C4C2×C22⋊#692311yescharacteristic
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4C2×C22⋊#692311yescharacteristic
563C2×C4C2×C22⋊#692311yescharacteristic
564C2×C4C2×C22⋊#692311yescharacteristic
565C2×C4C23×#692311yescharacteristic
566C8—2444no—
567C22—21no—
568C22—21no—
569C22—612no—
570C22—612no—
571C22—612no—
572C22—612no—
573C22—1228no—
574C22—1221no—
575C22—1221no—
576C22#1684075512yes—
577C22C22×C22⋊#692311yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—42no—
584C4—42no—
585C4—612no—
586C4—1221no—
587C4—1221no—
588C4—1221no—
589C4—1221no—
590C4—1221no—
591C4—2448no—
592C4—2442no—
593C4—2442no—
594C4D61⋊#10412yes—
595C2—22no—
596C2—614no—
597C2—1222no—
598C2D61⋊#41011yescharacteristic
599C2C2×#1684075511yescharacteristic
600C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC22C22×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×#1506
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461213C2semidirect
4. #21461212C2semidirect
5. #21461211C2semidirect
24. #16840319C4semidirect
32. #16840314C4semidirect
33. C4⋊Q8×D61C4semidirect
34. C4⋊Q8×D61C4semidirect
39. C4×Q8×D61C4semidirect
42. C61×#533C4semidirect
63. #14519777C2×C4semidirect
64. #14519777C2×C4semidirect
68. #14519776C2×C4semidirect
70. #14519775C2×C4semidirect
87. Q8×Dic61C2×C4semidirect
91. C4×Dic122C2×C4semidirect
95. #14519742C2×C4semidirect
120. C4⋊Q8×C61C2×C4semidirect
121. C4⋊Q8×C61C2×C4semidirect
126. C4×Q8×C61C2×C4semidirect
307. D61#1506semidirect
317. #533#6923semidirect
343. C61C2×#1506semidirect
359. C4⋊Q8C2×#6923semidirect
360. C4⋊Q8C2×#6923semidirect
390. C4×Q8C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4322
C2×#168406982
#214612132
#214612122
#214612112
D61⋊#3282
#533×D612
C2×#150661