Finite Group Databasebeta

← #30623938

Subgroups — #30623938

Order 15616.

1240 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462430C211yesmaximal, characteristic, complemented
2#21462419C211yesmaximal, characteristic, complemented
3#21462083C212yesmaximal, complemented
4#21462075C214yesmaximal, complemented
5#21462070C212yesmaximal, complemented
6D61⋊#352C214yesmaximal, complemented
7#21459461C211yesmaximal, characteristic
8#16840842—21no—
9#16840829—21no—
10#16840748—22no—
11#16840748C2212yes—
12D61⋊#100—21no—
13D61⋊#100C2214yes—
14#16840743—21no—
15#16840741C2212yes—
16C2×#14519836—22no—
17D61⋊#99C2214yes—
18#16840665C2218yescomplemented
19#16840664C2214yescomplemented
20#16840662C2214yescomplemented
21C2×#14519823C2211yescharacteristic
22C22×#11758222C411yescharacteristic, complemented
23#16840417—22no—
24#16840414—21no—
25#16840414—21no—
26#16840409—21no—
27#16840409—21no—
28#16840408—22no—
29#16840338C2211yescharacteristic
30#16840333C411yescharacteristic, complemented
31#16840322C412yescomplemented
32#16840322C2212yes—
33#16840273C2211yescharacteristic
34#123×D61C412yescomplemented
35#123×D61C2212yes—
36#16840236C411yescharacteristic, complemented
37C61×#525C411yescharacteristic, complemented
38#14519865—44no—
39D61⋊M16—22no—
40D61⋊M16—24no—
41D4×#6923—44no—
42#14519856—22no—
43C2×C22⋊#6923—24no—
44#14519838—22no—
45#14519837—24no—
46#14519837C2314yes—
47#14519836—24no—
48#14519836—22no—
49#14519835C2314yes—
50#14519823—44no—
51C2×#11758226D411yescharacteristic
52C2×#11758226D411yescharacteristic
53C2×#11758222—22no—
54C2×#11758222—21no—
55C2×#11758222—21no—
56C2×#11758222—44no—
57C2×#11758222D412yes—
58C2×#11758222C2311yescharacteristic
59C22×Dic122C2×C411yescharacteristic, complemented
60C22×D244C2×C411yescharacteristic
61C22×D244D412yes—
62C22×C22⋊D61C2×C412yescomplemented
63C2×D4×D61C2×C412yes—
64C2×D4×D61D411yescharacteristic
65C2×D4×D61D411yescharacteristic
66C2×D4×D61C2312yes—
67C22×C4×D61C2×C412yescomplemented
68#14519777—21no—
69#14519775—21no—
70#14519765—21no—
71#14519764—21no—
72#14519762—22no—
73#14519762—22no—
74#14519761C2×C412yescomplemented
75#14519761C2312yes—
76#14519760C2×C411yescharacteristic, complemented
77#14519760C2311yescharacteristic
78#14519759—22no—
79#14519759—22no—
80#14519759—22no—
81#14519759—22no—
82#14519758C2×C412yescomplemented
83#14519758C2×C412yescomplemented
84#14519757—21no—
85#14519757—21no—
86#14519756—22no—
87#14519756—21no—
88#14519751—21no—
89#14519750—22no—
90#14519750—22no—
91#14519748—22no—
92#14519748—22no—
93#14519746C2×C411yescharacteristic, complemented
94#14519746C2311yescharacteristic
95#14519745—22no—
96#14519745—22no—
97C4×D244—22no—
98#14519739—21no—
99#14519739—21no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—21no—
102#14519736C2×C411yescharacteristic, complemented
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C22⋊C4×D61—22no—
106C22⋊C4×D61—22no—
107C22⋊C4×D61C2×C412yescomplemented
108C22⋊C4×D61C2×C412yescomplemented
109C2×#41×C61C2×C411yescharacteristic
110C23×C4×C61C2×C411yescharacteristic, complemented
111#123×C61C2×C412yescomplemented
112C22⋊Q8×C61—21no—
113#121×C61—21no—
114#120×C61C2×C412yescomplemented
115C4×D4×C61—22no—
116#117×C61C2×C411yescharacteristic, complemented
117C24×C61C22×C411yescharacteristic
118C23×D61C22⋊C412yes—
119C23×D61C22×C412yes—
120C23×D61C2×D412yes—
121C22×#6923—44no—
122#41×C61—42no—
123C2×Q8×C61C22⋊C411yescharacteristic
124C2×D4×C61—22no—
125C2×D4×C61C22⋊C411yescharacteristic
126C2×D4×C61C22×C412yes—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—42no—
130C22×C4×C61C22⋊C412yes—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C2×C22⋊D61—22no—
134C2×C22⋊D61—22no—
135C2×C22⋊D61—21no—
136C2×C22⋊D61—21no—
137C2×C22⋊D61—44no—
138C2×C22⋊D61C22⋊C412yes—
139C2×C22⋊D61C22×C411yescharacteristic
140C2×C22⋊D61C2×D412yes—
141C2×C22⋊D61C2411yescharacteristic
142C22×Dic61C22×C412yes—
143#11758226—24no—
144#11758226—42no—
145D4×D61—24no—
146D4×D61—42no—
147D4×D61—44no—
148D4×D61—44no—
149#11758222—42no—
150#11758222—44no—
151#11758222—48no—
152#11758222—44no—
153#11758222—44no—
154C2×D244—24no—
155C2×D244—44no—
156C2×D244C22⋊C412yes—
157C2×D244C22⋊C411yescharacteristic
158C2×D244C22⋊C412yes—
159C2×D244C22⋊C411yescharacteristic
160C2×D244C2×D412yes—
161C2×D244C2×D411yescharacteristic
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—44no—
167C2×C4×D61C22⋊C412yes—
168C2×C4×D61C2×D411yescharacteristic
169C2×C4×D61C2×D411yescharacteristic
170C2×C4×D61C2×D412yes—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—42no—
174C2×Dic122C2×D411yescharacteristic
175C22⋊#6923—24no—
176C22⋊#6923—44no—
177#11758217—24no—
178C4⋊#6923—42no—
179C4×#6923—42no—
180#11758213—44no—
181#11758213—42no—
182D61⋊C8—44no—
183D61⋊C8—42no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61C22×C412yes—
189C42×C61—21no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—21no—
193C22⋊Dic61—21no—
194C22⋊Dic61C22×C411yescharacteristic
195C22⋊Dic61C22×C411yescharacteristic
196#11758198—21no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—21no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—21no—
208#11758196—21no—
209C4×Dic61—21no—
210C23×C61—21no—
211C23×C61—41no—
212C23×C61—81no—
213C23×C61C2×C22⋊C412yes—
214C23×C61C23×C411yescharacteristic
215C22×D61—24no—
216C22×D61—24no—
217C22×D61—42no—
218C22×D61—82no—
219C22×D61—82no—
220C22×D61—82no—
221C22×D61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C412yes—
223C22×D61C2×C22⋊C412yes—
224C22×D61C22×D411yescharacteristic
225C2×#6923—44no—
226C2×#6923—164no—
227Q8×C61—22no—
228D4×C61—22no—
229D4×C61—42no—
230D4×C61—82no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61—41no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C22⋊D61—24no—
242C22⋊D61—41no—
243C22⋊D61—42no—
244C22⋊D61—41no—
245C22⋊D61—44no—
246C22⋊D61—44no—
247C22⋊D61—81no—
248C22⋊D61—81no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—42no—
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C22×D411yescharacteristic
255D244—24no—
256D244—24no—
257D244—22no—
258D244—41no—
259D244—42no—
260D244#9914yes—
261C4×D61—24no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—42no—
265C4×D61—44no—
266C4×D61—44no—
267C4×D61—82no—
268C4×D61—82no—
269Dic122—41no—
270Dic122—42no—
271Dic122—42no—
272#92420—44no—
273#33350—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—22no—
276C22×C61—42no—
277C22×C61—41no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—82no—
282C22×C61C22×C22⋊C411yescharacteristic
283C2×D61—21no—
284C2×D61—21no—
285C2×D61—22no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—41no—
289C2×D61—41no—
290C2×D61—84no—
291C2×D61—84no—
292C2×D61—162no—
293C2×D61C2×#9914yes—
294#6923—84no—
295C4×C61—21no—
296C4×C61—22no—
297C4×C61—42no—
298C4×C61—41no—
299C4×C61C2×#9912yes—
300Dic61—21no—
301Dic61—21no—
302Dic61—41no—
303Dic61—41no—
304Dic61—42no—
305#3485—611no—
306#2888—611no—
307#2883—611no—
308#2123—612no—
309#2121—614no—
310#2120—612no—
311C2×#352—614no—
312C2×C61—21no—
313C2×C61—41no—
314C2×C61—42no—
315C2×C61—81no—
316C2×C61C22×#9911yescharacteristic
317D61—24no—
318D61—41no—
319D61—42no—
320D61—41no—
321C2×H(5,𝔽2)—611no—
322C22×#41—611no—
323#568—1221no—
324#547—1222no—
325#547—1221no—
326#547—1221no—
327#546—1222no—
328#546—1221no—
329#546—1221no—
330#536—611no—
331#536—611no—
332#534—612no—
333#534—612no—
334#525—611no—
335#525#692311yescharacteristic, complemented
336C2×#123—612no—
337C2×#123—612no—
338#518—611no—
339#518—1221no—
340#413—612no—
341#413—1222no—
342C2×#100—614no—
343C2×#100—1221no—
344#410—612no—
345#410—1221no—
346C2×#99—614no—
347C2×#99—1222no—
348#352—614no—
349#352—618no—
350AΣL(1,𝔽16)—614no—
351C61#3335011yescharacteristic, complemented, Sylow
352H(5,𝔽2)—2444no—
353C2×#41—611no—
354C2×#41—611no—
355C2×#41—612no—
356C2×#41—611no—
357C2×#41—1222no—
358C2×#41—1221no—
359C2×#41—1221no—
360C2×#41—2444no—
361C2×#41C2×#692311yescharacteristic
362C22×Q8—611no—
363C22×D4—611no—
364C22×D4—611no—
365C22×D4—612no—
366C22×D4—612no—
367C22×D4—612no—
368C22×D4—612no—
369C22×D4—611no—
370C23×C4—612no—
371C23×C4C2×#692311yescharacteristic, complemented
372#131—2444no—
373C2×M16—1224no—
374C2×M16—1222no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377#124—1221no—
378#124—1221no—
379#123—612no—
380#123—612no—
381#123—611no—
382#123—611no—
383#123—612no—
384#123—612no—
385#123—1222no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123C2×#692312yescomplemented
392C22⋊Q8—21no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—1222no—
397#121—21no—
398#121—1222no—
399#121—1222no—
400#121—1221no—
401#121—1222no—
402#121—1222no—
403#120—611no—
404#120—611no—
405#120—1222no—
406#120—1222no—
407#120C2×#692312yescomplemented
408C4×D4—22no—
409C4×D4—1222no—
410C4×D4—1221no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—2444no—
415#117—611no—
416#117—1222no—
417#117C2×#692311yescharacteristic, complemented
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1222no—
420C2×C22⋊C4—612no—
421C2×C22⋊C4—612no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C22⋊C4—1224no—
425#101—1222no—
426#100—614no—
427#100—1224no—
428#99—614no—
429#99—1224no—
430#99—1222no—
431C24—612no—
432C24—612no—
433C24—612no—
434C24C22×#692311yescharacteristic
435#41—42no—
436#41—1224no—
437#41—2442no—
438#41—2444no—
439#41—2444no—
440#41—2448no—
441#41—2444no—
442#41—2442no—
443C2×Q8—611no—
444C2×Q8—1221no—
445C2×Q8—1221no—
446C2×Q8—2442no—
447C2×Q8C22⋊#692311yescharacteristic
448C2×D4—22no—
449C2×D4—612no—
450C2×D4—612no—
451C2×D4—611no—
452C2×D4—611no—
453C2×D4—612no—
454C2×D4—612no—
455C2×D4—612no—
456C2×D4—611no—
457C2×D4—611no—
458C2×D4—611no—
459C2×D4—1224no—
460C2×D4—1224no—
461C2×D4—1221no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1222no—
465C2×D4—2444no—
466C2×D4—2442no—
467C2×D4—2444no—
468C2×D4—2444no—
469C2×D4—2444no—
470C2×D4C22⋊#692311yescharacteristic
471C2×D4C22×#692312yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—42no—
475C22×C4—612no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—1222no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—2444no—
485C22×C4—2444no—
486C22×C4C22⋊#692312yes—
487C22×C4C22×#692311yescharacteristic
488C22×C4C22×#692311yescharacteristic
489M16—1224no—
490M16—2444no—
491M16—2442no—
492C2×C8—2444no—
493C2×C8—2442no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—2442no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—611no—
507C22⋊C4—611no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1221no—
511C22⋊C4—1221no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1221no—
515C22⋊C4—1221no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1224no—
519C22⋊C4—2444no—
520C22⋊C4C22×#692312yes—
521C42—21no—
522C42—1221no—
523C42—2442no—
524C23—21no—
525C23—41no—
526C23—81no—
527C23—612no—
528C23—611no—
529C23—612no—
530C23—611no—
531C23—1224no—
532C23—1224no—
533C23—4882no—
534C23—4882no—
535C23—4882no—
536C23—2442no—
537C23C2×C22⋊#692312yes—
538C23C23×#692311yescharacteristic
539Q8—22no—
540Q8—2441no—
541Q8—2442no—
542Q8—2442no—
543D4—22no—
544D4—42no—
545D4—82no—
546D4—614no—
547D4—1224no—
548D4—1224no—
549D4—1224no—
550D4—1222no—
551D4—4881no—
552D4—4881no—
553D4—2441no—
554D4—2442no—
555D4—2441no—
556D4—2442no—
557D4—2441no—
558D4—2444no—
559D4—2444no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—1224no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—9764no—
576C2×C4—4882no—
577C2×C4—4882no—
578C2×C4—2442no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2442no—
583C2×C4C2×C22⋊#692311yescharacteristic
584C2×C4C2×C22⋊#692312yes—
585C2×C4C2×C22⋊#692311yescharacteristic
586C8—2444no—
587C22—21no—
588C22—22no—
589C22—42no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—82no—
595C22—614no—
596C22—1221no—
597C22—1221no—
598C22—1222no—
599C22—9762no—
600C22—4884no—
601C22—4884no—
602C22—2442no—
603C22—2442no—
604C22—2441no—
605C22—2441no—
606C22C22×C22⋊#692311yescharacteristic
607C4—21no—
608C4—22no—
609C4—42no—
610C4—41no—
611C4—1221no—
612C4—1221no—
613C4—4884no—
614C4—2441no—
615C4—2441no—
616C4—2442no—
617C4D61⋊#9912yes—
618C2—21no—
619C2—41no—
620C2—42no—
621C2—81no—
622C2—1224no—
623C2—2441no—
624C2—2441no—
625C2—2442no—
626C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC61×#525C4
socleC2×C61C22×#99
prime radical#21459461C2
cosocleC2×C22⋊D61C24
hypercenter#525#6923
hypocenterC61#33350
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #21462430C2semidirect
2. #21462419C2semidirect
3. #21462083C2semidirect
4. #21462075C2semidirect
5. #21462070C2semidirect
6. D61⋊#352C2semidirect
18. #16840665C22semidirect
19. #16840664C22semidirect
20. #16840662C22semidirect
22. C22×#11758222C4semidirect
30. #16840333C4semidirect
31. #16840322C4semidirect
34. #123×D61C4semidirect
36. #16840236C4semidirect
37. C61×#525C4semidirect
59. C22×Dic122C2×C4semidirect
62. C22×C22⋊D61C2×C4semidirect
67. C22×C4×D61C2×C4semidirect
74. #14519761C2×C4semidirect
76. #14519760C2×C4semidirect
82. #14519758C2×C4semidirect
83. #14519758C2×C4semidirect
93. #14519746C2×C4semidirect
102. #14519736C2×C4semidirect
107. C22⋊C4×D61C2×C4semidirect
108. C22⋊C4×D61C2×C4semidirect
110. C23×C4×C61C2×C4semidirect
111. #123×C61C2×C4semidirect
114. #120×C61C2×C4semidirect
116. #117×C61C2×C4semidirect
335. #525#6923semidirect
351. C61#33350semidirect
371. C23×C4C2×#6923semidirect
391. #123C2×#6923semidirect
407. #120C2×#6923semidirect
417. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624302
#214624192
#214620832
#214620752
#214620702
D61⋊#3522
#214594612
#3335061