Finite Group Databasebeta

← D61×#2061 #29708114

Subgroups — D61×#2061 #29708114

Order 15616.

1130 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#544×D61C211yesmaximal, characteristic
2#428×D61C212yesmaximal
3#382×D61C211yesmaximal, characteristic
4#21447170C211yesmaximal, characteristic, complemented
5#21447169C211yesmaximal, characteristic, complemented
6#21447168C212yesmaximal, complemented
7#21447167C212yesmaximal, complemented
8#21447166C211yesmaximal, characteristic, complemented
9#343×D61C211yesmaximal, characteristic
10#339×D61C212yesmaximal
11C61×#2061C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840393C2211yescharacteristic
14#16840392—22no—
15#16840391—22no—
16#16840390C2212yes—
17#16840389—22no—
18C4⋊Q8×D61—22no—
19#125×D61—22no—
20C22⋊Q8×D61—22no—
21#117×D61C2212yes—
22C2×C4⋊C4×D61C2211yescharacteristic
23C2×C4⋊C4×D61C2211yescharacteristic
24C2×C42×D61C2211yescharacteristic
25#16839805C2212yes—
26#16839804C2212yes—
27#16839803—22no—
28#16839802C2212yes—
29#16839801—22no—
30#107×D61—22no—
31#106×D61—22no—
32#16839628C2211yescharacteristic
33#16839627C2211yescharacteristic
34#16839627C2211yescharacteristic
35#16839626—22no—
36#95×D61—22no—
37#16839531C2211yescharacteristic
38#16839530C2212yes—
39#16839523C2212yes—
40#16839522C2212yes—
41#16839521C2212yes—
42C61×#544C2211yescharacteristic
43C61×#428C2212yes—
44C61×#382C2211yescharacteristic
45C61×#343C2211yescharacteristic
46C61×#339C2212yes—
47C2×Q8×D61—22no—
48C22×C4×D61—21no—
49C22×C4×D61—21no—
50C22×C4×D61D412yes—
51C22×C4×D61C2311yescharacteristic
52C2×#11758194C2312yes—
53C2×#11758189—22no—
54M16×D61—24no—
55M16×D61Q818yescomplemented
56C2×C8×D61—22no—
57#14519777—22no—
58#14519776—24no—
59#14519776—22no—
60#14519776—22no—
61#14519775—24no—
62#14519769—22no—
63#14519768—22no—
64#14519768—22no—
65#14519767—24no—
66#14519767—24no—
67#14519757—22no—
68#14519756—22no—
69#14519755—24no—
70#14519754—24no—
71#14519753—22no—
72#14519736C2312yes—
73#14519735D412yes—
74#14519735D412yes—
75#14519734—22no—
76#14519733—22no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196D412yes—
81C4⋊C4×D61—22no—
82C4⋊C4×D61—24no—
83C4⋊C4×D61—22no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61D412yes—
88C4⋊C4×D61D412yes—
89C22⋊C4×D61—22no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61C2311yescharacteristic
92C42×D61—22no—
93C42×D61—22no—
94C42×D61Q814yescomplemented
95#14519711—22no—
96#14519710—22no—
97#14519710—22no—
98#14519709—22no—
99#14519708—22no—
100#14519708—22no—
101#14519684—22no—
102#14519684—22no—
103#14519684—22no—
104C2×M16×C61C2312yes—
105C4⋊Q8×C61—22no—
106#125×C61—22no—
107C22⋊Q8×C61—22no—
108#117×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#107×C61—22no—
113#106×C61—22no—
114#95×C61—22no—
115C23×D61C2×D411yescharacteristic
116C2×Q8×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61C2×D412yes—
124C22×Dic61C2×D411yescharacteristic
125Q8×D61—84no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—24no—
130C2×C4×D61—22no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61—41no—
136C2×C4×D61—41no—
137C2×C4×D61C2×D412yes—
138C2×C4×D61C2×D412yes—
139C2×C4×D61C2×Q814yes—
140C2×C4×D61C2×Q812yes—
141C2×C4×D61#4112yes—
142C2×Dic122—24no—
143C2×Dic122—22no—
144M16×C61—22no—
145M16×C61C2×Q814yescomplemented
146C2×C8×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61C2×D412yes—
152C4⋊C4×C61C2×D412yes—
153C22⋊C4×C61—22no—
154C42×C61—22no—
155C42×C61—21no—
156C42×C61C2×Q812yescomplemented
157C22⋊Dic61—22no—
158#11758198—22no—
159#11758198—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—24no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C2×D412yes—
165C4⋊Dic61C2×D412yes—
166#11758196—24no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—22no—
171#11758196—22no—
172#11758196—42no—
173C4×Dic61—22no—
174C4×Dic61—21no—
175C4×Dic61—22no—
176C4×Dic61—22no—
177C4×Dic61—24no—
178C4×Dic61C2×Q812yescomplemented
179#11758194—22no—
180#11758194C2×Q814yescomplemented
181C2×#92418—22no—
182#11758189—28no—
183C8×D61—28no—
184C23×C61C22×D411yescharacteristic
185C22×D61—22no—
186C22×D61—22no—
187C22×D61#12011yescharacteristic
188C22×D61#12011yescharacteristic
189C22×D61C22⋊Q812yes—
190C22×D61C4⋊Q812yes—
191Q8×C61—82no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61—41no—
201C2×C4×C61C22×D412yes—
202C2×C4×C61C22×Q812yes—
203C2×C4×C61C22×Q811yescharacteristic
204C2×C4×C61C2×#4111yescharacteristic
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—41no—
214C2×Dic61#12012yes—
215C2×Dic61C22⋊Q812yes—
216C2×Dic61C4⋊Q812yes—
217C4×D61—48no—
218C4×D61—44no—
219C4×D61—44no—
220C4×D61—44no—
221C4×D61—82no—
222C4×D61—82no—
223C4×D61—82no—
224C4×D61C22⋊Q814yes—
225C4×D61C22⋊Q814yes—
226C4×D61C4⋊Q814yes—
227C4×D61C4⋊Q814yes—
228Dic122—82no—
229Dic122—84no—
230C8×C61—24no—
231#92418—24no—
232C2×#2061—611nomaximal, Sylow
233C22×C61—21no—
234C22×C61—21no—
235C22×C61C2×#12011yescharacteristic
236C22×C61C2×C22⋊Q811yescharacteristic
237C22×C61C2×C4⋊Q811yescharacteristic
238C2×D61—21no—
239C2×D61—21no—
240C2×D61—24no—
241C2×D61—24no—
242C2×D61#39512yes—
243C2×D61#39512yes—
244C2×D61#45412yes—
245C2×D61#45512yes—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—42no—
249C4×C61—42no—
250C4×C61—81no—
251C4×C61C2×C22⋊Q812yes—
252C4×C61C2×C4⋊Q812yes—
253Dic61—42no—
254Dic61—42no—
255Dic61—42no—
256Dic61—42no—
257Dic61—81no—
258Dic61#39514yes—
259C2×#544—611no—
260C2×#428—612no—
261C2×#382—611no—
262#2061—612no—
263#2061—611no—
264#2061—611no—
265#2061—612no—
266#2061—611no—
267#2061D6111yescharacteristic, direct product factor
268C2×#343—611no—
269C2×#339—612no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61C2×#39511yescharacteristic
273C2×C61C2×#45511yescharacteristic
274C2×C61C2×#45411yescharacteristic
275D61—22no—
276D61#206114yesdirect product factor
277C22×M16—612no—
278#544—612no—
279#544—611no—
280#544—1222no—
281#544—1222no—
282#544—1222no—
283#544C2×D6111yescharacteristic
284C2×C4⋊Q8—1222no—
285C2×#125—1222no—
286C2×C22⋊Q8—1222no—
287C2×#117—612no—
288C22×C4⋊C4—611no—
289C22×C4⋊C4—611no—
290C22×C42—611no—
291#428—612no—
292#428—612no—
293#428—612no—
294#428—1222no—
295#428—1222no—
296#428C2×D6112yes—
297C2×#107—1222no—
298C2×#106—1222no—
299#382—611no—
300#382—611no—
301#382—611no—
302#382—1222no—
303#382C2×D6111yescharacteristic
304C2×#95—1222no—
305#343—612no—
306#343—611no—
307#343C2×D6111yescharacteristic
308#339—612no—
309#339—612no—
310#339—612no—
311#339C2×D6112yes—
312C61C2×#206111yescharacteristic, complemented, Sylow
313C22×Q8—1222no—
314C23×C4—611no—
315C23×C4—612no—
316C23×C4—1221no—
317C23×C4—1221no—
318C2×M16—612no—
319C2×M16—618no—
320C2×M16—1222no—
321C2×M16—1224no—
322C2×M16C22×D6112yes—
323C22×C8—1222no—
324C4⋊Q8—22no—
325C4⋊Q8—1222no—
326C4⋊Q8—1222no—
327C4⋊Q8—1224no—
328C4⋊Q8—1224no—
329C4⋊Q8—1222no—
330#125—22no—
331#125—1222no—
332#125—1222no—
333#125—1224no—
334#125—1224no—
335#125—1222no—
336C22⋊Q8—22no—
337C22⋊Q8—1224no—
338C22⋊Q8—1224no—
339C22⋊Q8—1222no—
340C22⋊Q8—1222no—
341C22⋊Q8—1222no—
342#117—612no—
343#117—612no—
344#117—612no—
345#117—1222no—
346#117—1222no—
347#117C22×D6112yes—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—612no—
351C2×C4⋊C4—611no—
352C2×C4⋊C4—611no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1222no—
355C2×C4⋊C4—1222no—
356C2×C4⋊C4—1224no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4C22×D6111yescharacteristic
361C2×C4⋊C4C22×D6111yescharacteristic
362C2×C22⋊C4—1222no—
363C2×C42—611no—
364C2×C42—614no—
365C2×C42—1222no—
366C2×C42—1222no—
367C2×C42—1221no—
368C2×C42C22×D6111yescharacteristic
369#107—22no—
370#107—1222no—
371#107—1222no—
372#107—1222no—
373#106—22no—
374#106—1222no—
375#106—1222no—
376#106—1222no—
377#95—22no—
378#95—1222no—
379#95—1222no—
380#95—1222no—
381C24—611no—
382C2×Q8—22no—
383C2×Q8—1224no—
384C2×Q8—1222no—
385C2×Q8—4884no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—611no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—614no—
393C22×C4—612no—
394C22×C4—612no—
395C22×C4—1222no—
396C22×C4—1224no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—2441no—
403C22×C4—2441no—
404C22×C4—2444no—
405C22×C4—2444no—
406C22×C4—2442no—
407C22×C4—2442no—
408C22×C4D4×D6112yes—
409C22×C4C23×D6111yescharacteristic
410M16—22no—
411M16—614no—
412M16—1228no—
413M16—1222no—
414M16Q8×D6114yescomplemented
415C2×C8—22no—
416C2×C8—1228no—
417C2×C8—1222no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—612no—
423C4⋊C4—612no—
424C4⋊C4—1224no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C4⋊C4—1224no—
428C4⋊C4—1224no—
429C4⋊C4—1224no—
430C4⋊C4—1222no—
431C4⋊C4—1224no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—2442no—
435C4⋊C4D4×D6112yes—
436C4⋊C4D4×D6112yes—
437C22⋊C4—22no—
438C22⋊C4—1222no—
439C22⋊C4—1222no—
440C22⋊C4—1222no—
441C42—22no—
442C42—21no—
443C42—612no—
444C42—1224no—
445C42—1222no—
446C42—1222no—
447C42—1222no—
448C42—1221no—
449C42Q8×D6112yescomplemented
450C23—611no—
451C23—611no—
452C23—612no—
453C23—612no—
454C23—1222no—
455C23—1222no—
456C23C2×D4×D6111yescharacteristic
457Q8—82no—
458Q8—4882no—
459Q8—4884no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—612no—
470C2×C4—612no—
471C2×C4—612no—
472C2×C4—614no—
473C2×C4—614no—
474C2×C4—614no—
475C2×C4—614no—
476C2×C4—1222no—
477C2×C4—1222no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—4882no—
481C2×C4—4882no—
482C2×C4—4882no—
483C2×C4—2444no—
484C2×C4—2444no—
485C2×C4—2444no—
486C2×C4—2448no—
487C2×C4—2442no—
488C2×C4—2441no—
489C2×C4—2441no—
490C2×C4—2441no—
491C2×C4—2442no—
492C2×C4C2×D4×D6112yes—
493C2×C4C2×Q8×D6112yes—
494C2×C4C2×Q8×D6111yescharacteristic
495C2×C4#41×D6111yescharacteristic
496C8—24no—
497C8—1224no—
498C22—21no—
499C22—21no—
500C22—612no—
501C22—612no—
502C22—612no—
503C22—612no—
504C22—1224no—
505C22—1224no—
506C22—1221no—
507C22—1221no—
508C22#120×D6111yescharacteristic
509C22C22⋊Q8×D6111yescharacteristic
510C22C4⋊Q8×D6111yescharacteristic
511C4—42no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—81no—
516C4—614no—
517C4—4881no—
518C4—2442no—
519C4—2442no—
520C4—2442no—
521C4—2442no—
522C4C22⋊Q8×D6112yes—
523C4C4⋊Q8×D6112yes—
524C2—21no—
525C2—21no—
526C2—614no—
527C2—1222no—
528C2#395×D6111yescharacteristic
529C2#455×D6111yescharacteristic
530C2#454×D6111yescharacteristic
subgroupG/H
centerC22#120×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2061C2
socleC22×C61C2×#120
prime radicalC23×D61C2×D4
hypercenter#2061D61
hypocenterC61C2×#2061
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
4. #21447170C2semidirect
5. #21447169C2semidirect
6. #21447168C2semidirect
7. #21447167C2semidirect
8. #21447166C2semidirect
11. C61×#2061C2semidirect
55. M16×D61Q8semidirect
94. C42×D61Q8semidirect
145. M16×C61C2×Q8semidirect
156. C42×C61C2×Q8semidirect
178. C4×Dic61C2×Q8semidirect
180. #11758194C2×Q8semidirect
267. #2061D61direct (N × H)
276. D61#2061direct (N × H)
312. C61C2×#2061semidirect
414. M16Q8×D61semidirect
449. C42Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#544×D612
#428×D612
#382×D612
#214471702
#214471692
#214471682
#214471672
#214471662
#343×D612
#339×D612
C61×#20612
C2×#206161