Finite Group Databasebeta

← D61×#1552 #29648380

Subgroups — D61×#1552 #29648380

Order 15616.

1030 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#533×D61C211yesmaximal, characteristic
2#432×D61C211yesmaximal, characteristic
3C2×C4⋊C8×D61C211yesmaximal, characteristic
4#21444520C211yesmaximal, characteristic, complemented
5#21444519C211yesmaximal, characteristic, complemented
6#21444518C212yesmaximal, complemented
7#21444517C211yesmaximal, characteristic, complemented
8#21444516C212yesmaximal, complemented
9#332×D61C212yesmaximal, complemented
10#332×D61C212yesmaximal, complemented
11C61×#1552C211yesmaximal, characteristic, complemented
12C22×C8×D61—21no—
13#16840319C2211yescharacteristic
14#16840318C412yes—
15#16840317C2211yescharacteristic
16#16840317C2211yescharacteristic
17#16840316—21no—
18#16840316—21no—
19#16840315C412yes—
20#16840314—21no—
21#16840314—21no—
22C4⋊Q8×D61C2211yescharacteristic
23C4⋊Q8×D61C2211yescharacteristic
24#125×D61C412yes—
25C22⋊Q8×D61—21no—
26C22⋊Q8×D61—21no—
27C4×Q8×D61—21no—
28C4×Q8×D61—21no—
29#117×D61C412yes—
30C2×C42×D61C2211yescharacteristic
31#16839819C2211yescharacteristic
32#16839818C2212yes—
33#16839817—21no—
34#16839816—21no—
35C2×C4⋊#92418C2211yescharacteristic
36C2×#14519706—21no—
37C4⋊C8×D61—21no—
38C4⋊C8×D61—21no—
39C4⋊C8×D61C2212yes—
40C4⋊C8×D61C2212yes—
41C4⋊C8×D61C2212yes—
42C22⋊C8×D61—21no—
43#97×D61—21no—
44#16839509C2212yescomplemented
45#16839509C2212yescomplemented
46#16839508C2212yescomplemented
47#16839508C2212yescomplemented
48#16839507C2212yescomplemented
49#16839507C2212yescomplemented
50C61×#533C2211yescharacteristic
51C61×#432C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#332C2212yescomplemented
55C2×Q8×D61—21no—
56C2×Q8×D61—21no—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C22×#92418—21no—
61C2×C8×D61—22no—
62C2×C8×D61—22no—
63C2×C8×D61—22no—
64C2×C8×D61—44no—
65#14519777C2311yescharacteristic
66#14519777C2311yescharacteristic
67#14519776—21no—
68#14519776—21no—
69#14519776C2×C412yes—
70#14519776C2×C412yes—
71#14519775—21no—
72#14519775—21no—
73#14519769C2×C412yes—
74#14519768C2×C412yes—
75#14519768C2×C412yes—
76#14519767—22no—
77#14519767—22no—
78#14519757—21no—
79#14519757—21no—
80#14519756—21no—
81#14519756—21no—
82#14519755—22no—
83#14519755—22no—
84#14519754—22no—
85#14519754—22no—
86#14519753—21no—
87#14519753—21no—
88Q8×Dic61—21no—
89Q8×Dic61—21no—
90C4×Dic122—21no—
91C4×Dic122—21no—
92C4×Dic122—22no—
93C4×Dic122—22no—
94#14519742—22no—
95#14519742—22no—
96#14519742—21no—
97#14519742—21no—
98#14519736C2×C412yes—
99#14519735C2×C412yes—
100#14519735C2×C412yes—
101#14519734—22no—
102#14519733—22no—
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—21no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61C2×C412yes—
108C4⋊C4×D61C2×C412yes—
109C22⋊C4×D61—22no—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C42×D61—22no—
114C42×D61C2×C411yescharacteristic
115C42×D61C2×C411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61C2311yescharacteristic
121C42×D61C2311yescharacteristic
122C4⋊#92418—21no—
123C4⋊#92418C2312yes—
124C4⋊#92418C2312yes—
125#14519706—22no—
126#14519706—22no—
127#14519706—22no—
128C22⋊#92418—21no—
129#14519689—22no—
130#14519688—21no—
131#14519687—22no—
132C22×C8×C61—21no—
133C4⋊Q8×C61C2311yescharacteristic
134C4⋊Q8×C61C2311yescharacteristic
135#125×C61C2×C412yes—
136C22⋊Q8×C61—21no—
137C22⋊Q8×C61—21no—
138C4×Q8×C61—21no—
139C4×Q8×C61—21no—
140#117×C61C2×C412yes—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61C2312yes—
144C4⋊C8×C61C2312yes—
145C22⋊C8×C61—21no—
146#97×C61—21no—
147C23×D61C22⋊C411yescharacteristic
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61C22⋊C411yescharacteristic
154C22×Dic61C22⋊C411yescharacteristic
155C22×Dic61C22⋊C411yescharacteristic
156Q8×D61—44no—
157Q8×D61—42no—
158Q8×D61—42no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61C22⋊C411yescharacteristic
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22×C411yescharacteristic
172C2×C4×D61C22×C411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C2×C8×C61—42no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C412yes—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—22no—
191C42×C61—22no—
192C42×C61C22×C411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2411yescharacteristic
196C22⋊Dic61—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22×C412yes—
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C2×#92418—22no—
229C2×#92418—22no—
230C2×#92418—42no—
231C8×D61—44no—
232C8×D61—44no—
233C8×D61—44no—
234C23×C61C2×C22⋊C411yescharacteristic
235C22×D61—22no—
236C22×D61—22no—
237C22×D61#10114yes—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61C2×C22⋊C411yescharacteristic
240Q8×C61—42no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—44no—
278C4×D61—44no—
279C4×D61#13514yes—
280C4×D61AGL(1,ℤ/8)12yes—
281C4×D61#13712yes—
282Dic122—42no—
283Dic122—41no—
284Dic122—41no—
285Dic122—44no—
286Dic122—44no—
287C8×C61—42no—
288C8×C61—42no—
289#92418—42no—
290#92418—42no—
291C2×#1552—611nomaximal, Sylow
292C22×C61—21no—
293C22×C61—21no—
294C22×C61C2×#10112yes—
295C22×C61C22×C22⋊C411yescharacteristic
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—28no—
299C2×D61C2×#10112yes—
300C2×D61C2×#10112yes—
301C2×D61#41612yes—
302C2×D61#41712yes—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—42no—
308C4×C61—42no—
309C4×C61C2×#13512yes—
310C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
311C4×C61C2×#13711yescharacteristic
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—42no—
317Dic61—42no—
318Dic61#41612yes—
319Dic61#41712yes—
320C2×#533—611no—
321C2×#432—611no—
322C22×C4⋊C8—611no—
323#1552—611no—
324#1552—611no—
325#1552—612no—
326#1552—612no—
327#1552—611no—
328#1552D6111yescharacteristic, direct product factor
329C2×#332—612no—
330C2×#332—612no—
331C2×C61—22no—
332C2×C61C22×#10111yescharacteristic
333C2×C61C2×#41611yescharacteristic
334C2×C61C2×#41711yescharacteristic
335D61—22no—
336D61#155214yesdirect product factor
337C23×C8—1221no—
338#533—611no—
339#533—611no—
340#533—612no—
341#533—612no—
342#533—611no—
343#533—1221no—
344#533—1221no—
345#533—1221no—
346#533—1221no—
347#533C2×D6111yescharacteristic
348C2×C4⋊Q8—611no—
349C2×C4⋊Q8—611no—
350C2×#125—612no—
351C2×C22⋊Q8—1221no—
352C2×C22⋊Q8—1221no—
353C2×C4×Q8—1221no—
354C2×C4×Q8—1221no—
355C2×#117—612no—
356C22×C42—611no—
357#432—611no—
358#432—612no—
359#432—1221no—
360#432—1221no—
361#432C2×D6111yescharacteristic
362C2×C4⋊C8—611no—
363C2×C4⋊C8—612no—
364C2×C4⋊C8—612no—
365C2×C4⋊C8—612no—
366C2×C4⋊C8—1221no—
367C2×C4⋊C8—1221no—
368C2×C4⋊C8—1221no—
369C2×C4⋊C8C2×D6111yescharacteristic
370C2×C22⋊C8—1221no—
371C2×#97—1221no—
372#332—612no—
373#332—612no—
374#332—612no—
375#332—612no—
376#332—612no—
377#332—612no—
378#332C2×D6112yescomplemented
379#332C2×D6112yescomplemented
380C61C2×#155211yescharacteristic, complemented, Sylow
381C22×Q8—1221no—
382C22×Q8—1221no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C22×C8—21no—
387C22×C8—1222no—
388C22×C8—1222no—
389C22×C8—1222no—
390C22×C8—1221no—
391C22×C8—2444no—
392C4⋊Q8—612no—
393C4⋊Q8—612no—
394C4⋊Q8—611no—
395C4⋊Q8—611no—
396C4⋊Q8—1221no—
397C4⋊Q8—1221no—
398C4⋊Q8—1221no—
399C4⋊Q8—1221no—
400C4⋊Q8C22×D6111yescharacteristic
401C4⋊Q8C22×D6111yescharacteristic
402#125—612no—
403#125—612no—
404#125—612no—
405#125—1222no—
406#125—1222no—
407#125C4×D6112yes—
408C22⋊Q8—21no—
409C22⋊Q8—21no—
410C22⋊Q8—1222no—
411C22⋊Q8—1222no—
412C22⋊Q8—1222no—
413C22⋊Q8—1222no—
414C22⋊Q8—1221no—
415C22⋊Q8—1221no—
416C22⋊Q8—1221no—
417C22⋊Q8—1221no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420C4×Q8—21no—
421C4×Q8—21no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×Q8—1221no—
425C4×Q8—1221no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—1221no—
429C4×Q8—1221no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432#117—612no—
433#117—612no—
434#117—612no—
435#117—1222no—
436#117—1222no—
437#117C4×D6112yes—
438C2×C4⋊C4—612no—
439C2×C4⋊C4—612no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C22⋊C4—1222no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—1222no—
457C2×C42C22×D6111yescharacteristic
458C4⋊C8—21no—
459C4⋊C8—612no—
460C4⋊C8—612no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463C4⋊C8—1222no—
464C4⋊C8—1221no—
465C4⋊C8C22×D6112yes—
466C4⋊C8C22×D6112yes—
467C22⋊C8—21no—
468C22⋊C8—1222no—
469C22⋊C8—1221no—
470#97—21no—
471#97—1222no—
472#97—1221no—
473C24—611no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—1222no—
477C2×Q8—1222no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—2444no—
481C2×Q8—2442no—
482C2×Q8—2442no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—1222no—
499C22×C4—1224no—
500C22×C4—1222no—
501C22×C4—1224no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4D4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—42no—
510C2×C8—1222no—
511C2×C8—1222no—
512C2×C8—2444no—
513C2×C8—2444no—
514C2×C8—2444no—
515C2×C8—2442no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—612no—
521C4⋊C4—612no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1222no—
528C4⋊C4—1222no—
529C4⋊C4—1222no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4C2×C4×D6112yes—
535C4⋊C4C2×C4×D6112yes—
536C22⋊C4—22no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C42—22no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—1222no—
554C42—1222no—
555C42—1222no—
556C42C2×C4×D6111yescharacteristic
557C42D4×D6111yescharacteristic
558C42D4×D6111yescharacteristic
559C42C23×D6111yescharacteristic
560C23—611no—
561C23—611no—
562C23—614no—
563C23—1222no—
564C23—1222no—
565C23C22⋊C4×D6111yescharacteristic
566Q8—42no—
567Q8—41no—
568Q8—41no—
569Q8—2444no—
570Q8—2444no—
571Q8—2442no—
572Q8—2441no—
573Q8—2441no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—614no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—2444no—
606C2×C4—2444no—
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4C22⋊C4×D6111yescharacteristic
609C2×C4C22⋊C4×D6111yescharacteristic
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C8—42no—
614C8—42no—
615C8—2442no—
616C8—2442no—
617C22—21no—
618C22—21no—
619C22—612no—
620C22—612no—
621C22—612no—
622C22—612no—
623C22—1228no—
624C22—1221no—
625C22—1221no—
626C22#101×D6112yes—
627C22C2×C22⋊C4×D6111yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—42no—
633C4—42no—
634C4—612no—
635C4—612no—
636C4—1221no—
637C4—1221no—
638C4—1221no—
639C4—1221no—
640C4—2442no—
641C4—2442no—
642C4#135×D6112yes—
643C4AGL(1,ℤ/8)×D6111yescharacteristic
644C4#137×D6111yescharacteristic
645C2—22no—
646C2—614no—
647C2—1222no—
648C2C2×#101×D6111yescharacteristic
649C2#416×D6111yescharacteristic
650C2#417×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1552C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC42×C61C24
hypercenter#1552D61
hypocenterC61C2×#1552
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
4. #21444520C2semidirect
5. #21444519C2semidirect
6. #21444518C2semidirect
7. #21444517C2semidirect
8. #21444516C2semidirect
9. #332×D61C2semidirect
10. #332×D61C2semidirect
11. C61×#1552C2semidirect
44. #16839509C22semidirect
45. #16839509C22semidirect
46. #16839508C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839507C22semidirect
53. C61×#332C22semidirect
54. C61×#332C22semidirect
328. #1552D61direct (N × H)
336. D61#1552direct (N × H)
378. #332C2×D61semidirect
379. #332C2×D61semidirect
380. C61C2×#1552semidirect

Maximal subgroups

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

subgroupindex
#533×D612
#432×D612
C2×C4⋊C8×D612
#214445202
#214445192
#214445182
#214445172
#214445162
#332×D612
C61×#15522
C2×#155261