Finite Group Databasebeta

← #30616335

Subgroups — #30616335

Order 15616.

1694 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#423C211yesmaximal, characteristic, complemented
2C2×#16840755C212yesmaximal, complemented
3C2×#16840754C212yesmaximal
4C4×#14519856C211yesmaximal, characteristic, complemented
5#21461987C218yesmaximal, complemented
6C2×#124×D61C211yesmaximal, characteristic
7C2×D61⋊M16C2212yes—
8C2×#14519856C2212yes—
9#16840767—21no—
10#16840766—21no—
11#16840765—22no—
12D61⋊(C4⋊C8)C2214yescomplemented
13#16840755—28no—
14#16840754—28no—
15C2×#14519840C2218yescomplemented
16C2×#14519839C2218yes—
17C4×C4⋊#6923C2214yescomplemented
18C4×C22⋊#6923—22no—
19#16840698—21no—
20C42×#6923—21no—
21C22×Q8×D61C2211yescharacteristic
22C22×D4×D61C2211yescharacteristic
23C2×#14519766C411yescharacteristic, complemented
24C2×#14519765C411yescharacteristic, complemented
25C2×#14519764C411yescharacteristic, complemented
26C2×#14519763—22no—
27#124×D61—21no—
28#124×D61—21no—
29#124×D61—21no—
30#124×D61—21no—
31#124×D61C414yescomplemented
32#124×D61C2214yes—
33C2×C22⋊C4×D61—22no—
34C2×C42×D61C2211yescharacteristic
35C2×#124×C61C411yescharacteristic, complemented
36C2×#11758213—22no—
37D61⋊M16—24no—
38D61⋊M16—24no—
39C2×#11758217—22no—
40C2×D61⋊C8C2314yes—
41#14519856—24no—
42#14519856D418yes—
43C2×C4⋊#6923C2314yes—
44C2×C22⋊#6923—22no—
45C2×C4×#6923—22no—
46C2×C4×#6923—24no—
47#14519844—24no—
48C4⋊#92420—24no—
49#14519840—216no—
50#14519839—216no—
51(C2×#6923)⋊C4—24no—
52C24×D61—21no—
53C22×Dic122—21no—
54C22×Dic122C2×C411yescharacteristic
55C2×Q8×D61—21no—
56C2×Q8×D61—21no—
57C2×Q8×D61—24no—
58C2×Q8×D61—24no—
59C2×Q8×D61C2×C412yes—
60C2×Q8×D61C2312yes—
61C22×D244C2×C411yescharacteristic
62C22×C22⋊D61—21no—
63C2×D4×D61—24no—
64C2×D4×D61—21no—
65C2×D4×D61—21no—
66C2×D4×D61—24no—
67C2×D4×D61C2×C412yes—
68C2×D4×D61C2312yes—
69C22×C4×D61—21no—
70C22×C4×D61D412yes—
71C22×C4×D61C2311yescharacteristic
72#14519766—21no—
73#14519766—21no—
74#14519766C2×C414yes—
75#14519765—21no—
76#14519765—21no—
77#14519765C2×C414yes—
78#14519764—21no—
79#14519764—21no—
80#14519764C2×C414yes—
81#14519763—24no—
82#14519763—24no—
83#14519763—24no—
84#14519763—24no—
85C2×C22⋊Dic61—22no—
86C2×#11758198—22no—
87C2×#11758198—22no—
88C22⋊C4×D61—24no—
89C22⋊C4×D61—24no—
90C22⋊C4×D61—24no—
91C22⋊C4×D61—24no—
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61D412yes—
94C42×D61C2×C412yes—
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61D411yescharacteristic
98C42×D61D411yescharacteristic
99C42×D61C2312yes—
100C22×Q8×C61C2×C411yescharacteristic
101C22×D4×C61C2×C411yescharacteristic
102#124×C61—21no—
103#124×C61—21no—
104#124×C61C2×C414yes—
105C2×C22⋊C4×C61—22no—
106C2×C42×C61C2×C411yescharacteristic
107C24×C61—21no—
108C23×D61—21no—
109C23×D61—22no—
110C23×D61—22no—
111C23×D61—48no—
112C23×D61—44no—
113C23×D61C2×D411yescharacteristic
114C22×#6923—24no—
115C2×Q8×C61—21no—
116C2×Q8×C61—24no—
117C2×Q8×C61C22×C412yes—
118C2×D4×C61—21no—
119C2×D4×C61—24no—
120C2×D4×C61C22×C412yes—
121C22×C4×C61—21no—
122C22×C4×C61C22⋊C412yes—
123C22×C4×C61C22×C411yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—42no—
127C2×C22⋊D61—42no—
128C22×Dic61—21no—
129C22×Dic61C22⋊C412yes—
130C22×Dic61C2×D411yescharacteristic
131Q8×D61—28no—
132Q8×D61—24no—
133Q8×D61—24no—
134Q8×D61—44no—
135Q8×D61—42no—
136Q8×D61—42no—
137D4×D61—28no—
138D4×D61—24no—
139D4×D61—24no—
140D4×D61—44no—
141D4×D61—42no—
142D4×D61—42no—
143C2×D244—24no—
144C2×D244—21no—
145C2×D244C22×C412yes—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—44no—
149C2×C4×D61—44no—
150C2×C4×D61C22⋊C412yes—
151C2×C4×D61C22⋊C412yes—
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C2×D412yes—
154C2×C4×D61C2×D412yes—
155C2×C4×D61C2×D414yes—
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61#4114yes—
159C2×C4×D61C2411yescharacteristic
160C2×Dic122—24no—
161C2×Dic122—21no—
162C2×Dic122—22no—
163C2×Dic122—22no—
164C2×Dic122—42no—
165C2×Dic122—42no—
166C2×Dic122C22×C412yes—
167C22⋊#6923—28no—
168#11758217—44no—
169C2×#92420—24no—
170C4⋊#6923—24no—
171C4⋊#6923C2×D418yes—
172C4×#6923—28no—
173C4×#6923—48no—
174#11758213—44no—
175D61⋊C8—24no—
176D61⋊C8—24no—
177C22⋊C4×C61—24no—
178C22⋊C4×C61—24no—
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22×C412yes—
182C22⋊Dic61—24no—
183C22⋊Dic61—24no—
184#11758198—24no—
185#11758198—24no—
186#11758198—24no—
187#11758198—24no—
188C4×Dic61C22⋊C412yes—
189C4×Dic61C22⋊C412yes—
190C4×Dic61C22⋊C411yescharacteristic
191C4×Dic61C22⋊C411yescharacteristic
192C4×Dic61C22×C412yes—
193C4×Dic61C2×D412yes—
194C4×Dic61C2×D412yes—
195C23×C61—22no—
196C23×C61—44no—
197C23×C61—42no—
198C23×C61C2×C22⋊C411yescharacteristic
199C22×D61—24no—
200C22×D61—22no—
201C22×D61—48no—
202C22×D61—44no—
203C22×D61—42no—
204C22×D61—44no—
205C22×D61—48no—
206C22×D61—44no—
207C22×D61—816no—
208C22×D61C2×C22⋊C411yescharacteristic
209C22×D61#12114yes—
210C22×D61C22×D411yescharacteristic
211C2×#6923—28no—
212C2×#6923—48no—
213Q8×C61—24no—
214Q8×C61—42no—
215D4×C61—24no—
216D4×C61—42no—
217C2×C4×C61—22no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C22⋊C412yes—
221C2×C4×C61C2×C22⋊C412yes—
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C4×D414yes—
224C2×C4×C61C23×C411yescharacteristic
225C22⋊D61—42no—
226C22⋊D61—42no—
227C22⋊D61—44no—
228C2×Dic61—22no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C412yes—
233C2×Dic61C2×C22⋊C411yescharacteristic
234C2×Dic61#12412yes—
235C2×Dic61#12712yes—
236C2×Dic61C22×D411yescharacteristic
237D244—24no—
238D244—42no—
239C4×D61—24no—
240C4×D61—24no—
241C4×D61—42no—
242C4×D61—42no—
243C4×D61—44no—
244C4×D61—44no—
245C4×D61C4×D414yes—
246C4×D61#12114yes—
247C4×D61#12412yes—
248C4×D61#12712yes—
249C4×D61C22×D412yes—
250C4×D61C2×#4112yes—
251Dic122—24no—
252Dic122—42no—
253Dic122—42no—
254Dic122—42no—
255Dic122—44no—
256#92420—44no—
257#21261—611nomaximal, Sylow
258C22×C61—22no—
259C22×C61—44no—
260C22×C61—42no—
261C22×C61—84no—
262C22×C61#39012yes—
263C22×C61C22×C22⋊C411yescharacteristic
264C2×D61—22no—
265C2×D61—28no—
266C2×D61—44no—
267C2×D61—42no—
268C2×D61—88no—
269C2×D61—84no—
270C2×D61—88no—
271C2×D61#39012yes—
272C2×D61C2×#12112yes—
273C2×D61#57514yes—
274#6923—48no—
275C4×C61—22no—
276C4×C61—22no—
277C4×C61—42no—
278C4×C61#39012yes—
279C4×C61C2×C4×D412yes—
280Dic61—22no—
281Dic61—22no—
282Dic61—42no—
283Dic61#39012yes—
284Dic61C2×#12411yescharacteristic
285Dic61C2×#12711yescharacteristic
286C22×#124—611no—
287C2×#423—611no—
288C2×#419—612no—
289C2×#418—612no—
290C4×#117—611no—
291#1983—618no—
292C2×C61—22no—
293C2×C61—82no—
294C2×C61C2×#39011yescharacteristic
295C2×C61#289612yes—
296D61—22no—
297D61—82no—
298D61#304014yes—
299C23×Q8—611no—
300C23×D4—611no—
301C22×M16—612no—
302C2×#124—611no—
303C2×#124—611no—
304C2×#124—614no—
305C2×#124—614no—
306C2×#124—611no—
307C2×#124—1222no—
308C2×#124—1221no—
309C2×#124—1221no—
310C2×#124—1221no—
311C2×#124—1221no—
312C2×#124#692311yescharacteristic, complemented
313C2×#117—612no—
314C22×C22⋊C4—1222no—
315C22×C42—611no—
316#423—1222no—
317#423—1221no—
318#423—1221no—
319C2×C4⋊C8—614no—
320#419—1228no—
321#418—1228no—
322C2×#103—618no—
323C2×#102—618no—
324C4×C4⋊C4—614no—
325C4×C22⋊C4—1222no—
326#376—1221no—
327C43—1221no—
328C61#2126111yescharacteristic, complemented, Sylow
329C25—1221no—
330C22×Q8—612no—
331C22×Q8—612no—
332C22×Q8—611no—
333C22×Q8—1221no—
334C22×Q8—1221no—
335C22×Q8—1221no—
336C22×Q8—1224no—
337C22×Q8—1224no—
338C22×Q8C2×#692311yescharacteristic
339C22×D4—612no—
340C22×D4—612no—
341C22×D4—611no—
342C22×D4—1224no—
343C22×D4—1224no—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4—1221no—
347C22×D4C2×#692311yescharacteristic
348C23×C4—611no—
349C23×C4—612no—
350C23×C4—1221no—
351C2×M16—1222no—
352C2×M16—1224no—
353C2×M16—1224no—
354C2×M16—1222no—
355C22×C8—614no—
356#124—21no—
357#124—21no—
358#124—614no—
359#124—614no—
360#124—614no—
361#124—1224no—
362#124—1224no—
363#124—1224no—
364#124—1224no—
365#124—1221no—
366#124—1221no—
367#124—1221no—
368#124—1221no—
369#124—1221no—
370#124—1221no—
371#124C2×#692314yes—
372#117—618no—
373#117—1224no—
374C2×C4⋊C4—614no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—1224no—
377C2×C22⋊C4—1224no—
378C2×C22⋊C4—1224no—
379C2×C22⋊C4—1224no—
380C2×C22⋊C4—1222no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C22⋊C4—1222no—
384C2×C42—612no—
385C2×C42—611no—
386C2×C42—611no—
387C2×C42—611no—
388C2×C42—611no—
389C2×C42—611no—
390C2×C42—612no—
391C2×C42—612no—
392C2×C42—1222no—
393C2×C42—1224no—
394C2×C42C2×#692311yescharacteristic
395C4⋊C8—1224no—
396C4⋊C8—1224no—
397#103—12216no—
398#102—12216no—
399#95—1224no—
400C24—21no—
401C24—611no—
402C24—1222no—
403C24—1222no—
404C24—1221no—
405C24—2444no—
406C24—2448no—
407C2×Q8—21no—
408C2×Q8—24no—
409C2×Q8—612no—
410C2×Q8—1222no—
411C2×Q8—1222no—
412C2×Q8—1224no—
413C2×Q8—1228no—
414C2×Q8—1224no—
415C2×Q8—1221no—
416C2×Q8—1224no—
417C2×Q8—2442no—
418C2×Q8—2442no—
419C2×Q8—2444no—
420C2×Q8—2442no—
421C2×Q8—2442no—
422C2×Q8C22×#692312yes—
423C2×D4—21no—
424C2×D4—24no—
425C2×D4—612no—
426C2×D4—1224no—
427C2×D4—1228no—
428C2×D4—1224no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4—1221no—
432C2×D4—1224no—
433C2×D4—2444no—
434C2×D4—2442no—
435C2×D4—2442no—
436C2×D4—2442no—
437C2×D4—2442no—
438C2×D4C22×#692312yes—
439C22×C4—21no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—614no—
444C22×C4—614no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—612no—
450C22×C4—612no—
451C22×C4—612no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1224no—
455C22×C4—1221no—
456C22×C4—2444no—
457C22×C4—2444no—
458C22×C4C22⋊#692312yes—
459C22×C4C22×#692311yescharacteristic
460M16—2444no—
461M16—2444no—
462C2×C8—1224no—
463C2×C8—1224no—
464C2×C8—1224no—
465C4⋊C4—618no—
466C4⋊C4—1224no—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—1224no—
470C22⋊C4—1224no—
471C22⋊C4—1224no—
472C22⋊C4—1224no—
473C22⋊C4—1228no—
474C22⋊C4—1224no—
475C22⋊C4—1224no—
476C42—612no—
477C42—612no—
478C42—612no—
479C42—612no—
480C42—611no—
481C42—611no—
482C42—612no—
483C42—1228no—
484C42—2448no—
485C42C22⋊#692311yescharacteristic
486C42C22⋊#692311yescharacteristic
487C42C22×#692312yes—
488C23—22no—
489C23—44no—
490C23—42no—
491C23—611no—
492C23—614no—
493C23—611no—
494C23—1224no—
495C23—1222no—
496C23—48816no—
497C23—2444no—
498C23—2444no—
499C23—2444no—
500C23—2448no—
501C23—2448no—
502C23—2442no—
503C23C2×C22⋊#692311yescharacteristic
504Q8—24no—
505Q8—42no—
506Q8—1224no—
507Q8—2442no—
508Q8—2442no—
509Q8—2444no—
510Q8—2442no—
511D4—24no—
512D4—42no—
513D4—1224no—
514D4—2442no—
515D4—2442no—
516D4—2444no—
517D4—2442no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—612no—
522C2×C4—612no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—614no—
526C2×C4—612no—
527C2×C4—612no—
528C2×C4—614no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—612no—
532C2×C4—612no—
533C2×C4—1224no—
534C2×C4—1224no—
535C2×C4—1228no—
536C2×C4—1222no—
537C2×C4—2442no—
538C2×C4—2442no—
539C2×C4—2444no—
540C2×C4—2444no—
541C2×C4—2442no—
542C2×C4—2448no—
543C2×C4—2442no—
544C2×C4C2×C22⋊#692312yes—
545C2×C4C2×C22⋊#692312yes—
546C2×C4C2×C22⋊#692311yescharacteristic
547C2×C4D4×#692314yes—
548C2×C4C23×#692311yescharacteristic
549C8—2444no—
550C22—22no—
551C22—42no—
552C22—44no—
553C22—84no—
554C22—612no—
555C22—614no—
556C22—612no—
557C22—1228no—
558C22—1222no—
559C22—4884no—
560C22—4888no—
561C22—4888no—
562C22—2442no—
563C22—2444no—
564C22#1684071412yes—
565C22C22×C22⋊#692311yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—611no—
570C4—611no—
571C4—612no—
572C4—1222no—
573C4—1222no—
574C4—2448no—
575C4—2442no—
576C4#1684071412yes—
577C4C2×D4×#692312yes—
578C2—22no—
579C2—82no—
580C2—614no—
581C2—1222no—
582C2—4882no—
583C2C2×#1684071411yescharacteristic
584C2#2146243912yes—
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#124×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D4×D61C22
cosocleC2×C4×D61C24
hypercenterC2×#124#6923
hypocenterC61#21261
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. D61⋊#423C2semidirect
2. C2×#16840755C2semidirect
4. C4×#14519856C2semidirect
5. #21461987C2semidirect
12. D61⋊(C4⋊C8)C22semidirect
15. C2×#14519840C22semidirect
17. C4×C4⋊#6923C22semidirect
23. C2×#14519766C4semidirect
24. C2×#14519765C4semidirect
25. C2×#14519764C4semidirect
31. #124×D61C4semidirect
35. C2×#124×C61C4semidirect
312. C2×#124#6923semidirect
328. C61#21261semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4232
C2×#168407552
C2×#168407542
C4×#145198562
#214619872
C2×#124×D612
#2126161