Finite Group Databasebeta

← #30606279

Subgroups — #30606279

Order 15616.

768 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#410C211yesmaximal, characteristic
2#21462404C211yesmaximal, characteristic
3#21460906C214yesmaximal, complemented
4#410×D61C211yesmaximal, characteristic
5C22×#11758213C411yescharacteristic, complemented
6C2×D61⋊M16C411yescharacteristic, complemented
7C2×#14519856—21no—
8#16840743—21no—
9#16840742C2212yes—
10#16840741—21no—
11#16840741C412yescomplemented
12D61⋊#99—21no—
13D61⋊#99C412yescomplemented
14#16840737—21no—
15#16840736—21no—
16#16840735C412yescomplemented
17#16840735C2212yes—
18#16840733—21no—
19#16840733—21no—
20C2×#41×D61C2211yescharacteristic
21#117×D61—21no—
22#16839736C411yescharacteristic, complemented
23#16839735—21no—
24#16839734—21no—
25#16839734C2212yes—
26#99×D61—21no—
27#99×D61C412yescomplemented
28C61×#410C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×#11758213—21no—
31C2×#11758213—21no—
32C2×#11758213—41no—
33C2×#11758213—41no—
34D61⋊M16—41no—
35D61⋊M16—41no—
36C2×#11758217C2×C412yescomplemented
37C2×D61⋊C8—21no—
38C22×#92420C2×C412yescomplemented
39#14519856—22no—
40#14519856—22no—
41#14519856—41no—
42#14519856—41no—
43C2×C4⋊#6923—41no—
44C2×C22⋊#6923—22no—
45C2×C4×#6923—41no—
46#14519836—22no—
47#14519836—22no—
48#14519835—22no—
49#14519835C2×C414yescomplemented
50#14519834—22no—
51#14519834—22no—
52C22⋊#92420—22no—
53C22⋊#92420—22no—
54C2×#11758226D411yescharacteristic
55C2×#11758222C2×C411yescharacteristic
56C2×#11758222D411yescharacteristic
57C2×#11758222Q811yescharacteristic
58C2×#11758224C2×C411yescharacteristic
59C2×#11758224D411yescharacteristic
60C2×#11758224C2311yescharacteristic
61#41×D61—41no—
62#41×D61—41no—
63#41×D61—41no—
64#41×D61—41no—
65C2×Q8×D61Q811yescharacteristic
66C2×D4×D61C2×C411yescharacteristic
67C2×D4×D61C2×C411yescharacteristic
68C2×D4×D61D411yescharacteristic
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×C412yescomplemented
81#14519691—22no—
82#14519691—22no—
83C2×#41×C61C2×C411yescharacteristic
84#117×C61—21no—
85#99×C61—21no—
86#99×C61C2×C412yescomplemented
87C23×D61C4211yescharacteristic
88C23×D61C22⋊C411yescharacteristic
89C23×D61C22⋊C411yescharacteristic
90C22×#6923—42no—
91#41×C61—41no—
92#41×C61—41no—
93C2×Q8×C61C4⋊C411yescharacteristic
94C2×D4×C61C4211yescharacteristic
95C2×D4×C61C22⋊C411yescharacteristic
96C2×D4×C61C22×C411yescharacteristic
97C22×C4×C61C4211yescharacteristic
98C22×C4×C61C22⋊C411yescharacteristic
99C22×C4×C61C4⋊C411yescharacteristic
100C2×C22⋊D61C4211yescharacteristic
101C2×C22⋊D61C22⋊C411yescharacteristic
102C2×C22⋊D61C4⋊C411yescharacteristic
103C2×C22⋊D61C4⋊C411yescharacteristic
104C2×C22⋊D61C22×C411yescharacteristic
105C2×C22⋊D61C2×D411yescharacteristic
106C22×Dic61C22⋊C411yescharacteristic
107C22×Dic61C22×C411yescharacteristic
108C22×Dic61C2×D411yescharacteristic
109#11758226—41no—
110#11758226—41no—
111Q8×D61—41no—
112Q8×D61—41no—
113#11758224—41no—
114#11758224—41no—
115#11758224—41no—
116#11758224—41no—
117#11758224—41no—
118#11758224—41no—
119D4×D61—41no—
120D4×D61—41no—
121D4×D61—41no—
122D4×D61—41no—
123D4×D61—41no—
124D4×D61—41no—
125#11758222—41no—
126#11758222—41no—
127#11758222—41no—
128#11758222—41no—
129#11758222—41no—
130#11758222—41no—
131C2×D244C22⋊C411yescharacteristic
132C2×D244C22⋊C411yescharacteristic
133C2×D244C4⋊C411yescharacteristic
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—21no—
146C2×C4×D61—42no—
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C4⋊C411yescharacteristic
152C2×C4×D61C4⋊C411yescharacteristic
153C2×C4×D61C4⋊C411yescharacteristic
154C2×C4×D61C4⋊C411yescharacteristic
155C2×C4×D61C4⋊C411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122C4⋊C411yescharacteristic
158C2×Dic122C4⋊C411yescharacteristic
159C2×Dic122C2×Q811yescharacteristic
160C22⋊#6923—24no—
161C22⋊#6923—42no—
162#11758217—42no—
163C2×#92420—22no—
164C2×#92420—22no—
165C2×#92420—42no—
166C4⋊#6923—42no—
167C4⋊#6923—42no—
168C4×#6923—42no—
169C4×#6923—42no—
170#11758213—42no—
171#11758213—41no—
172#11758213—41no—
173#11758213—41no—
174#11758213—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×C4211yescharacteristic
187C23×C61C2×C22⋊C411yescharacteristic
188C22×D61—21no—
189C22×D61—21no—
190C22×D61—21no—
191C22×D61—21no—
192C22×D61—21no—
193C22×D61—21no—
194C22×D61—81no—
195C22×D61—81no—
196C22×D61—81no—
197C22×D61#9511yescharacteristic
198C22×D61#9511yescharacteristic
199C22×D61#9511yescharacteristic
200C22×D61C2×C22⋊C411yescharacteristic
201C2×#6923—44no—
202C2×#6923—84no—
203Q8×C61—41no—
204D4×C61—41no—
205D4×C61—41no—
206D4×C61—41no—
207C2×C4×C61—21no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—42no—
214C2×C4×C61#9511yescharacteristic
215C2×C4×C61#9511yescharacteristic
216C2×C4×C61#9511yescharacteristic
217C2×C4×C61C2×C4⋊C411yescharacteristic
218C22⋊D61—41no—
219C22⋊D61—41no—
220C22⋊D61—41no—
221C22⋊D61—41no—
222C22⋊D61—41no—
223C22⋊D61—41no—
224C2×Dic61—21no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—21no—
228C2×Dic61—21no—
229C2×Dic61—21no—
230C2×Dic61—42no—
231C2×Dic61#9511yescharacteristic
232C2×Dic61C2×C22⋊C411yescharacteristic
233C2×Dic61C2×C4⋊C411yescharacteristic
234C2×Dic61C2×C4⋊C411yescharacteristic
235D244—41no—
236D244—41no—
237D244—41no—
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—21no—
255C4×D61—21no—
256C4×D61—84no—
257C4×D61#9911yescharacteristic
258C4×D61#9911yescharacteristic
259C4×D61#9911yescharacteristic
260C4×D61#9911yescharacteristic
261Dic122—41no—
262Dic122—41no—
263Dic122—41no—
264#92420—42no—
265#92420—42no—
266#12577—611nomaximal, Sylow
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61C2×#9511yescharacteristic
274C2×D61—21no—
275C2×D61—21no—
276C2×D61—21no—
277C2×D61—41no—
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—41no—
281C2×D61—41no—
282C2×D61—41no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C2×D61—81no—
287C2×D61#34211yescharacteristic
288C2×D61#34211yescharacteristic
289#6923—84no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—82no—
294C4×C61#34211yescharacteristic
295C4×C61#34211yescharacteristic
296Dic61—21no—
297Dic61—21no—
298Dic61—21no—
299Dic61—82no—
300Dic61C2×#9911yescharacteristic
301Dic61C2×#9911yescharacteristic
302C2×#410—611no—
303C2×#410—611no—
304#2879—611no—
305#1391—614no—
306C2×C61—21no—
307C2×C61—41no—
308C2×C61—41no—
309C2×C61—41no—
310C2×C61C2×#34211yescharacteristic
311D61—21no—
312D61—21no—
313D61—41no—
314D61—41no—
315D61—41no—
316C22×#41—611no—
317C22×M16—611no—
318C22×M16—611no—
319C2×#117—1221no—
320C2×#117—1221no—
321#410—612no—
322#410—612no—
323#410—611no—
324#410—612no—
325#410—1221no—
326#410—1221no—
327#410—1221no—
328#410—1221no—
329#410#692311yescharacteristic, complemented
330C2×#99—612no—
331C2×#99—612no—
332C2×#99—1221no—
333C2×#99—1221no—
334#408—612no—
335#408—612no—
336#408—1221no—
337#408—1221no—
338#407—1221no—
339#407—1221no—
340C61#1257711yescharacteristic, complemented, Sylow
341C2×#41—611no—
342C2×#41—611no—
343C2×#41—611no—
344C2×#41—611no—
345C2×#41—611no—
346C2×#41—611no—
347C2×#41—611no—
348C2×#41—2441no—
349C2×#41—2441no—
350C2×#41—2441no—
351C2×#41—2441no—
352C2×#41C2×#692311yescharacteristic
353C22×Q8—611no—
354C22×D4—611no—
355C22×D4—611no—
356C22×D4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C2×M16—612no—
361C2×M16—1221no—
362C2×M16—1221no—
363C2×M16—1221no—
364C2×M16—2441no—
365C2×M16—2441no—
366C2×M16—2441no—
367C2×M16—2441no—
368C22×C8—612no—
369C22×C8—1221no—
370#117—21no—
371#117—1222no—
372#117—1222no—
373#117—1222no—
374#117—1221no—
375#117—2441no—
376#117—2441no—
377#117—2441no—
378#117—2441no—
379C2×C4⋊C4—2441no—
380C2×C4⋊C4—2441no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1222no—
383C2×C42—2441no—
384C2×C42—2441no—
385#99—21no—
386#99—614no—
387#99—612no—
388#99—1222no—
389#99—1222no—
390#99—1222no—
391#99—1222no—
392#99—1222no—
393#99—1221no—
394#99C2×#692312yescomplemented
395C22⋊C8—1222no—
396C22⋊C8—1222no—
397C22⋊C8—1222no—
398C22⋊C8—1222no—
399C24—611no—
400C24—611no—
401C24—611no—
402#41—41no—
403#41—41no—
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—
416#41—2441no—
417#41—2441no—
418C2×Q8—611no—
419C2×Q8—611no—
420C2×Q8—611no—
421C2×Q8—2441no—
422C2×Q8—2441no—
423C2×Q8C4⋊#692311yescharacteristic
424C2×D4—611no—
425C2×D4—611no—
426C2×D4—611no—
427C2×D4—611no—
428C2×D4—611no—
429C2×D4—611no—
430C2×D4—611no—
431C2×D4—611no—
432C2×D4—611no—
433C2×D4—2441no—
434C2×D4—2441no—
435C2×D4—2441no—
436C2×D4—2441no—
437C2×D4—2441no—
438C2×D4—2441no—
439C2×D4C4×#692311yescharacteristic
440C2×D4C22⋊#692311yescharacteristic
441C2×D4C22×#692311yescharacteristic
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—611no—
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—1221no—
466C22×C4—1221no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4C4×#692311yescharacteristic
470C22×C4C4⋊#692311yescharacteristic
471C22×C4C22⋊#692311yescharacteristic
472M16—2441no—
473M16—2441no—
474M16—2441no—
475M16—2441no—
476M16—2442no—
477M16—2442no—
478C2×C8—1222no—
479C2×C8—1222no—
480C2×C8—2442no—
481C2×C8—2442no—
482C4⋊C4—41no—
483C4⋊C4—2441no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C22⋊C4—22no—
488C22⋊C4—1224no—
489C22⋊C4—1222no—
490C22⋊C4—2442no—
491C22⋊C4—2442no—
492C42—41no—
493C42—2441no—
494C42—2442no—
495C42—2442no—
496C42—2442no—
497C23—611no—
498C23—611no—
499C23—611no—
500C23—611no—
501C23—1221no—
502C23—1221no—
503C23—1221no—
504C23—1221no—
505C23—1221no—
506C23—1221no—
507C23—4881no—
508C23—4881no—
509C23—4881no—
510C23(C2×#6923)⋊C411yescharacteristic
511C23C2×C4×#692311yescharacteristic
512C23C2×C22⋊#692311yescharacteristic
513Q8—41no—
514Q8—2441no—
515Q8—2441no—
516Q8—2441no—
517D4—41no—
518D4—41no—
519D4—41no—
520D4—2441no—
521D4—2441no—
522D4—2441no—
523D4—2441no—
524D4—2441no—
525D4—2441no—
526D4—2441no—
527D4—2441no—
528D4—2441no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—42no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—611no—
543C2×C4—611no—
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—1221no—
567C2×C4—1221no—
568C2×C4—4884no—
569C2×C4—4884no—
570C2×C4—2442no—
571C2×C4—2444no—
572C2×C4(C2×#6923)⋊C411yescharacteristic
573C2×C4(C2×#6923)⋊C411yescharacteristic
574C2×C4(C2×#6923)⋊C411yescharacteristic
575C2×C4C2×C4⋊#692311yescharacteristic
576C8—2442no—
577C8—2442no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—611no—
585C22—611no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
589C22—4881no—
590C22—4881no—
591C22—4881no—
592C22—4881no—
593C22—2441no—
594C22—2441no—
595C22—2441no—
596C22—2441no—
597C22—2442no—
598C22—2442no—
599C22C2×(C2×#6923)⋊C411yescharacteristic
600C4—21no—
601C4—21no—
602C4—21no—
603C4—82no—
604C4—611no—
605C4—611no—
606C4—1221no—
607C4—1221no—
608C4—1221no—
609C4—4882no—
610C4—4884no—
611C4#1684063911yescharacteristic
612C4#1684063911yescharacteristic
613C2—21no—
614C2—41no—
615C2—41no—
616C2—41no—
617C2—1221no—
618C2—1221no—
619C2—2441no—
620C2—2441no—
621C2—2441no—
622C2C2×#1684063911yescharacteristic
subgroupG/H
centerC2C2×#16840639
commutatorC23×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×#410C4
socleC2×C61C2×#342
prime radicalC2×#41×D61C22
cosocleC2×#11758224C23
hypercenter#410#6923
hypocenterC61#12577
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #21460906C2semidirect
5. C22×#11758213C4semidirect
6. C2×D61⋊M16C4semidirect
11. #16840741C4semidirect
13. D61⋊#99C4semidirect
16. #16840735C4semidirect
22. #16839736C4semidirect
27. #99×D61C4semidirect
28. C61×#410C4semidirect
36. C2×#11758217C2×C4semidirect
38. C22×#92420C2×C4semidirect
49. #14519835C2×C4semidirect
80. #14519692C2×C4semidirect
86. #99×C61C2×C4semidirect
329. #410#6923semidirect
340. C61#12577semidirect
394. #99C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4102
#214624042
#214609062
#410×D612
#1257761