Finite Group Databasebeta

← #30606326

Subgroups — #30606326

Order 15616.

766 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#413C211yesmaximal, characteristic
2D61⋊#408C211yesmaximal, characteristic
3#21460918C214yesmaximal, complemented
4#410×D61C211yesmaximal, characteristic
5C2×D61⋊M16—21no—
6C2×D61⋊M16C411yescharacteristic, complemented
7C22×D61⋊C8C411yescharacteristic, complemented
8#16840749—21no—
9#16840749C2212yes—
10#16840748—21no—
11#16840748C412yescomplemented
12D61⋊#101C412yescomplemented
13D61⋊#100—21no—
14#16840737C412yescomplemented
15#16840736C2212yes—
16#16840735—21no—
17#16840735—21no—
18D61⋊(C22⋊C8)—21no—
19D61⋊(C22⋊C8)—21no—
20C2×#41×D61C2211yescharacteristic
21#117×D61—21no—
22#16839736C411yescharacteristic
23#16839735—21no—
24#16839734—21no—
25#16839734C2212yes—
26#99×D61—21no—
27#99×D61C412yes—
28C61×#410C411yescharacteristic
29C2×#11758213—41no—
30C2×#11758213C2×C412yescomplemented
31D61⋊M16—22no—
32D61⋊M16—22no—
33D61⋊M16—41no—
34D61⋊M16—41no—
35D61⋊M16—41no—
36D61⋊M16—41no—
37C2×#11758217—21no—
38C2×#11758217—22no—
39C2×D61⋊C8—21no—
40C2×D61⋊C8—21no—
41C2×D61⋊C8—41no—
42C2×D61⋊C8—41no—
43C2×D61⋊C8—41no—
44C2×D61⋊C8C2×C412yescomplemented
45C22×#92420—21no—
46#14519838—22no—
47#14519838C2×C414yescomplemented
48#14519837—22no—
49#14519837—22no—
50#14519834—22no—
51#14519834—22no—
52C22⋊#92420—22no—
53C22⋊#92420—22no—
54C2×#11758226C2×C411yescharacteristic
55C2×#11758222C2×C411yescharacteristic
56C2×#11758222D411yescharacteristic
57C2×#11758222Q811yescharacteristic
58C2×#11758224D411yescharacteristic
59C2×#11758224D411yescharacteristic
60C2×#11758224C2311yescharacteristic
61#41×D61—41no—
62#41×D61—41no—
63#41×D61—41no—
64#41×D61—41no—
65C2×Q8×D61C2×C411yescharacteristic
66C2×D4×D61C2×C411yescharacteristic
67C2×D4×D61D411yescharacteristic
68C2×D4×D61Q811yescharacteristic
69C22×C4×D61C2×C411yescharacteristic
70C22×C4×D61D411yescharacteristic
71C22×C4×D61D411yescharacteristic
72#14519736—21no—
73#14519735—41no—
74#14519734—41no—
75#14519733—22no—
76C4⋊C4×D61—41no—
77C22⋊C4×D61—22no—
78C42×D61—41no—
79#14519692—21no—
80#14519692C2×C412yes—
81#14519691—22no—
82#14519691—22no—
83C2×#41×C61C2×C411yescharacteristic
84#117×C61—21no—
85#99×C61—21no—
86#99×C61C2×C412yes—
87C23×D61C22⋊C411yescharacteristic
88C23×D61C4⋊C411yescharacteristic
89C23×D61C4⋊C411yescharacteristic
90#41×C61—41no—
91#41×C61—41no—
92C2×Q8×C61C4211yescharacteristic
93C2×D4×C61C22⋊C411yescharacteristic
94C2×D4×C61C4⋊C411yescharacteristic
95C2×D4×C61C22×C411yescharacteristic
96C22×C4×C61C4211yescharacteristic
97C22×C4×C61C22⋊C411yescharacteristic
98C22×C4×C61C4⋊C411yescharacteristic
99C2×C22⋊D61C22⋊C411yescharacteristic
100C2×C22⋊D61C22⋊C411yescharacteristic
101C2×C22⋊D61C4⋊C411yescharacteristic
102C2×C22⋊D61C4⋊C411yescharacteristic
103C2×C22⋊D61C2×D411yescharacteristic
104C2×C22⋊D61C2×Q811yescharacteristic
105C22×Dic61C22⋊C411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108#11758226—41no—
109#11758226—41no—
110Q8×D61—41no—
111Q8×D61—41no—
112#11758224—41no—
113#11758224—41no—
114#11758224—41no—
115#11758224—41no—
116#11758224—41no—
117#11758224—41no—
118D4×D61—41no—
119D4×D61—41no—
120D4×D61—41no—
121D4×D61—41no—
122D4×D61—41no—
123D4×D61—41no—
124#11758222—41no—
125#11758222—41no—
126#11758222—41no—
127#11758222—41no—
128#11758222—41no—
129#11758222—41no—
130C2×D244C4211yescharacteristic
131C2×D244C4⋊C411yescharacteristic
132C2×D244C4⋊C411yescharacteristic
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×C4×D61—21no—
141C2×C4×D61—21no—
142C2×C4×D61—21no—
143C2×C4×D61—21no—
144C2×C4×D61—21no—
145C2×C4×D61—42no—
146C2×C4×D61C4211yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22⋊C411yescharacteristic
152C2×C4×D61C4⋊C411yescharacteristic
153C2×C4×D61C4⋊C411yescharacteristic
154C2×C4×D61C4⋊C411yescharacteristic
155C2×C4×D61C22×C411yescharacteristic
156C2×Dic122C22⋊C411yescharacteristic
157C2×Dic122C4⋊C411yescharacteristic
158C2×Dic122C22×C411yescharacteristic
159#11758217—24no—
160#11758217—42no—
161#11758217—42no—
162C2×#92420—22no—
163C2×#92420—22no—
164C2×#92420—42no—
165C2×#92420—42no—
166#11758213—42no—
167#11758213—42no—
168#11758213—42no—
169D61⋊C8—42no—
170D61⋊C8—42no—
171D61⋊C8—41no—
172D61⋊C8—41no—
173D61⋊C8—41no—
174D61⋊C8—41no—
175D61⋊C8—42no—
176C4⋊C4×C61—41no—
177C22⋊C4×C61—22no—
178C42×C61—41no—
179C22⋊Dic61—22no—
180#11758198—42no—
181C4⋊Dic61—41no—
182#11758196—42no—
183C4×Dic61—42no—
184C4×Dic61—41no—
185C23×C61#9511yescharacteristic
186C23×C61C2×C22⋊C411yescharacteristic
187C23×C61C2×C4⋊C411yescharacteristic
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—42no—
195C22×D61—42no—
196C22×D61—42no—
197C22×D61#9511yescharacteristic
198C22×D61#9511yescharacteristic
199C22×D61#9511yescharacteristic
200C22×D61C2×C4⋊C411yescharacteristic
201Q8×C61—41no—
202D4×C61—41no—
203D4×C61—41no—
204D4×C61—41no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—42no—
212C2×C4×C61#9511yescharacteristic
213C2×C4×C61#9511yescharacteristic
214C2×C4×C61#9511yescharacteristic
215C2×C4×C61C2×C4211yescharacteristic
216C22⋊D61—41no—
217C22⋊D61—41no—
218C22⋊D61—41no—
219C22⋊D61—41no—
220C22⋊D61—41no—
221C22⋊D61—41no—
222C2×Dic61—21no—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—42no—
229C2×Dic61#9511yescharacteristic
230C2×Dic61C2×C22⋊C411yescharacteristic
231C2×Dic61C2×C22⋊C411yescharacteristic
232C2×Dic61C2×C4⋊C411yescharacteristic
233D244—41no—
234D244—41no—
235D244—41no—
236C4×D61—21no—
237C4×D61—21no—
238C4×D61—21no—
239C4×D61—21no—
240C4×D61—21no—
241C4×D61—21no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—21no—
246C4×D61—21no—
247C4×D61—21no—
248C4×D61—21no—
249C4×D61—21no—
250C4×D61—21no—
251C4×D61—21no—
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—84no—
255C4×D61#9911yescharacteristic
256C4×D61#9911yescharacteristic
257C4×D61#9911yescharacteristic
258C4×D61#9911yescharacteristic
259Dic122—41no—
260Dic122—41no—
261Dic122—41no—
262#92420—44no—
263#92420—42no—
264#92420—42no—
265C2×#1395—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61C2×#9511yescharacteristic
273C2×D61—22no—
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—42no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61#34211yescharacteristic
287C2×D61#34211yescharacteristic
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—82no—
292C4×C61#34211yescharacteristic
293C4×C61#34211yescharacteristic
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—82no—
298Dic61C2×#9911yescharacteristic
299Dic61C2×#9911yescharacteristic
300C2×#413—611no—
301C2×#410—611no—
302C2×#408—611no—
303#1395—614no—
304C2×C61—21no—
305C2×C61—41no—
306C2×C61—41no—
307C2×C61—41no—
308C2×C61C2×#34211yescharacteristic
309D61—21no—
310D61—41no—
311D61—41no—
312D61—41no—
313D61#139512yescomplemented
314C22×#41—611no—
315C22×M16—611no—
316C22×M16—1221no—
317C23×C8—611no—
318C2×#117—1221no—
319#413—612no—
320#413—612no—
321#413—1221no—
322#413—1221no—
323C2×#101—612no—
324C2×#100—1221no—
325#410—611no—
326#410—612no—
327#410—1221no—
328#410—1221no—
329#410#692311yescharacteristic
330C2×#99—612no—
331C2×#99—1221no—
332#408—612no—
333#408—612no—
334#408—1221no—
335#408—1221no—
336C2×C22⋊C8—1221no—
337C2×C22⋊C8—1221no—
338C61C2×#139511yescharacteristic, complemented, Sylow
339C2×#41—611no—
340C2×#41—611no—
341C2×#41—611no—
342C2×#41—611no—
343C2×#41—611no—
344C2×#41—611no—
345C2×#41—611no—
346C2×#41—2441no—
347C2×#41—2441no—
348C2×#41—2441no—
349C2×#41—2441no—
350C2×#41C2×#692311yescharacteristic
351C22×Q8—611no—
352C22×D4—611no—
353C22×D4—611no—
354C22×D4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C2×M16—612no—
359C2×M16—1221no—
360C2×M16—1222no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16—2441no—
364C2×M16—2441no—
365C2×M16—2441no—
366C2×M16—2441no—
367C2×M16—2441no—
368C22×C8—612no—
369C22×C8—1221no—
370C22×C8—1221no—
371C22×C8—1221no—
372C22×C8—2441no—
373C22×C8—2441no—
374C22×C8—2441no—
375#117—21no—
376#117—1222no—
377#117—1221no—
378#117—2441no—
379#117—2441no—
380C2×C4⋊C4—2441no—
381C2×C22⋊C4—1222no—
382C2×C42—2441no—
383#101—614no—
384#101—1222no—
385#100—1222no—
386#100—1222no—
387#99—21no—
388#99—612no—
389#99—1222no—
390#99—1222no—
391#99—1221no—
392#99C2×#692312yes—
393C22⋊C8—1222no—
394C22⋊C8—1222no—
395C22⋊C8—1222no—
396C22⋊C8—1222no—
397C24—611no—
398C24—611no—
399C24—611no—
400#41—41no—
401#41—41no—
402#41—2441no—
403#41—2441no—
404#41—2441no—
405#41—2441no—
406#41—2441no—
407#41—2441no—
408#41—2441no—
409#41—2441no—
410#41—2441no—
411#41—2441no—
412#41—2441no—
413#41—2441no—
414#41—2441no—
415#41—2441no—
416C2×Q8—611no—
417C2×Q8—611no—
418C2×Q8—611no—
419C2×Q8—2441no—
420C2×Q8—2441no—
421C2×Q8C4×#692311yescharacteristic
422C2×D4—611no—
423C2×D4—611no—
424C2×D4—611no—
425C2×D4—611no—
426C2×D4—611no—
427C2×D4—611no—
428C2×D4—611no—
429C2×D4—611no—
430C2×D4—611no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4—2441no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2441no—
437C2×D4C4⋊#692311yescharacteristic
438C2×D4C22⋊#692311yescharacteristic
439C2×D4C22×#692311yescharacteristic
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—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—2442no—
466C22×C4C4×#692311yescharacteristic
467C22×C4C4⋊#692311yescharacteristic
468C22×C4C22⋊#692311yescharacteristic
469M16—1224no—
470M16—2442no—
471M16—2442no—
472M16—2442no—
473M16—2442no—
474M16—2442no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—2441no—
478C2×C8—2441no—
479C2×C8—2441no—
480C2×C8—2441no—
481C2×C8—2442no—
482C2×C8—2442no—
483C2×C8—2442no—
484C2×C8—2442no—
485C2×C8—2442no—
486C4⋊C4—41no—
487C4⋊C4—2441no—
488C4⋊C4—2442no—
489C22⋊C4—22no—
490C22⋊C4—1222no—
491C22⋊C4—2442no—
492C42—41no—
493C42—2441no—
494C42—2442no—
495C23—611no—
496C23—611no—
497C23—611no—
498C23—611no—
499C23—1221no—
500C23—1221no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23(C2×#6923)⋊C411yescharacteristic
509C23C2×C22⋊#692311yescharacteristic
510C23C2×C4⋊#692311yescharacteristic
511Q8—41no—
512Q8—2441no—
513Q8—2441no—
514Q8—2441no—
515D4—41no—
516D4—41no—
517D4—41no—
518D4—2441no—
519D4—2441no—
520D4—2441no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—42no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—4884no—
567C2×C4—2442no—
568C2×C4(C2×#6923)⋊C411yescharacteristic
569C2×C4(C2×#6923)⋊C411yescharacteristic
570C2×C4(C2×#6923)⋊C411yescharacteristic
571C2×C4C2×C4×#692311yescharacteristic
572C8—2442no—
573C8—2442no—
574C8—2444no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—611no—
582C22—611no—
583C22—1222no—
584C22—1221no—
585C22—1221no—
586C22—1221no—
587C22—4881no—
588C22—4881no—
589C22—2441no—
590C22—2441no—
591C22—2441no—
592C22—2442no—
593C22—2442no—
594C22—2442no—
595C22—2442no—
596C22C2×(C2×#6923)⋊C411yescharacteristic
597C4—21no—
598C4—21no—
599C4—21no—
600C4—82no—
601C4—611no—
602C4—611no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—4882no—
607C4#1684063911yescharacteristic
608C4#1684063911yescharacteristic
609C2—21no—
610C2—41no—
611C2—41no—
612C2—41no—
613C2—612no—
614C2—1221no—
615C2—2441no—
616C2—2441no—
617C2—2441no—
618C2C2×#1684063911yescharacteristic
subgroupG/H
centerC4#16840639
commutatorC2×C4×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×#410C4
socleC2×C61C2×#342
prime radicalC2×#41×D61C22
cosocleC2×#11758224C23
hypercenter#410#6923
hypocenterC61C2×#1395
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #21460918C2semidirect
6. C2×D61⋊M16C4semidirect
7. C22×D61⋊C8C4semidirect
11. #16840748C4semidirect
12. D61⋊#101C4semidirect
14. #16840737C4semidirect
30. C2×#11758213C2×C4semidirect
44. C2×D61⋊C8C2×C4semidirect
47. #14519838C2×C4semidirect
313. D61#1395semidirect
338. C61C2×#1395semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4132
D61⋊#4082
#214609182
#410×D612
C2×#139561