Finite Group Databasebeta

← #30623936

Subgroups — #30623936

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#21462429C211yesmaximal, characteristic, complemented
2#21462418C211yesmaximal, characteristic, complemented
3#21462081C212yesmaximal, complemented
4#21462074C214yesmaximal, complemented
5#21462070C212yesmaximal, complemented
6C2×#16840664C214yesmaximal, complemented
7#21459463C211yesmaximal, characteristic
8#16840840—21no—
9#16840829—21no—
10#16840749—22no—
11#16840748C2212yes—
12D61⋊#100—21no—
13C2×#14519837C2214yes—
14#16840742—21no—
15#16840741C2212yes—
16D61⋊#99—22no—
17C2×#14519835C2214yes—
18#16840664C2214yescomplemented
19#16840664C2218yescomplemented
20#16840662C2214yescomplemented
21C2×#14519822C2211yescharacteristic
22#16840411—21no—
23#16840411—21no—
24#16840411—22no—
25#16840400—22no—
26#16840400—21no—
27#16840400—21no—
28#16840336C411yescharacteristic, complemented
29#16840336C2211yescharacteristic
30#16840321C412yescomplemented
31#16840321C2212yes—
32#16840273C411yescharacteristic, complemented
33#16840273C2211yescharacteristic
34C2×#14519760C412yescomplemented
35C2×#14519760C2212yes—
36#16840237C411yescharacteristic, complemented
37C22×#41×C61C411yescharacteristic, complemented
38#14519864—44no—
39D61⋊M16—22no—
40C2×#11758217—24no—
41D4×#6923—44no—
42#14519856—22no—
43C2×C22⋊#6923—24no—
44#14519838—22no—
45#14519837—24no—
46#14519837C2314yes—
47#14519836—22no—
48#14519835—24no—
49#14519835C2314yes—
50#14519822—44no—
51C2×#11758222C2×C411yescharacteristic
52C2×#11758222C2311yescharacteristic
53C2×#11758224D411yescharacteristic
54C2×#11758224D411yescharacteristic
55C2×#11758224D412yes—
56C22×C22⋊D61C2×C412yes—
57C22×C22⋊D61C2312yes—
58C2×D4×D61D412yes—
59C2×D4×D61D411yescharacteristic
60C2×D4×D61D411yescharacteristic
61#14519775—21no—
62#14519775—21no—
63#14519766—21no—
64#14519766—21no—
65#14519760—22no—
66#14519760—22no—
67#14519760—22no—
68#14519760—22no—
69#14519760C2×C411yescharacteristic, complemented
70#14519760C2×C412yescomplemented
71#14519760C2×C412yescomplemented
72#14519760C2×C412yescomplemented
73#14519760C2×C412yescomplemented
74#14519760C2311yescharacteristic
75#14519760C2312yes—
76#14519758—22no—
77#14519758—22no—
78#14519756—21no—
79#14519756—21no—
80#14519753—22no—
81#14519753—21no—
82#14519753—21no—
83#14519751—21no—
84#14519751—21no—
85#14519749—22no—
86#14519749—22no—
87#14519749—22no—
88#14519749—22no—
89#14519746C2×C412yescomplemented
90#14519746C2×C411yescharacteristic, complemented
91#14519746C2311yescharacteristic
92#14519745—22no—
93#14519745—22no—
94D4×Dic61—22no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—21no—
99C4×C22⋊D61—21no—
100#14519736C2×C411yescharacteristic, complemented
101#14519736C2×C411yescharacteristic, complemented
102C2×#11758196—22no—
103C2×#11758196—22no—
104C2×C22⋊Dic61C2×C412yescomplemented
105C2×C22⋊Dic61C2×C412yescomplemented
106C2×#11758198—22no—
107C2×#11758198—22no—
108C2×#41×C61—21no—
109C2×#41×C61—22no—
110C2×#41×C61—21no—
111C2×#41×C61—44no—
112C22×Q8×C61C2×C411yescharacteristic, complemented
113C22×D4×C61C2×C411yescharacteristic
114C22×D4×C61C2×C412yescomplemented
115C23×C4×C61C2×C412yescomplemented
116C23×C4×C61C2×C411yescharacteristic, complemented
117C24×C61C22×C412yes—
118C24×C61C22×C411yescharacteristic
119C23×D61C22⋊C412yes—
120C23×D61C2×D412yes—
121C22×#6923—44no—
122#41×C61—48no—
123#41×C61—44no—
124#41×C61—44no—
125C2×Q8×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—42no—
128C2×D4×C61—24no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—44no—
132C2×D4×C61C22⋊C412yes—
133C2×D4×C61C22⋊C411yescharacteristic
134C2×D4×C61C22⋊C411yescharacteristic
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—42no—
140C22×C4×C61—44no—
141C22×C4×C61C22×C411yescharacteristic
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C22×C412yes—
144C2×C22⋊D61—22no—
145C2×C22⋊D61—21no—
146C2×C22⋊D61—21no—
147C2×C22⋊D61—44no—
148C2×C22⋊D61—44no—
149C2×C22⋊D61C22⋊C412yes—
150C2×C22⋊D61C22⋊C412yes—
151C2×C22⋊D61C22×C411yescharacteristic
152C2×C22⋊D61C22×C412yes—
153C2×C22⋊D61C2×D412yes—
154C2×C22⋊D61C2×D412yes—
155C2×C22⋊D61C2411yescharacteristic
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61C22⋊C412yes—
159C22×Dic61C2×D412yes—
160#11758224—24no—
161#11758224—42no—
162#11758224—44no—
163D4×D61—24no—
164D4×D61—42no—
165D4×D61—44no—
166#11758222—42no—
167#11758222—42no—
168C2×D244C22⋊C411yescharacteristic
169C2×D244C2×D411yescharacteristic
170C2×C4×D61C22⋊C412yes—
171C2×C4×D61C2×D411yescharacteristic
172C2×C4×D61C2×D411yescharacteristic
173C2×Dic122C22⋊C411yescharacteristic
174C2×Dic122C2×D411yescharacteristic
175C22⋊#6923—24no—
176C22⋊#6923—44no—
177#11758217—24no—
178#11758217—44no—
179C2×#92420—44no—
180C4⋊#6923—42no—
181C4×#6923—42no—
182#11758213—42no—
183D61⋊C8—42no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186C22⋊Dic61—22no—
187C22⋊Dic61—21no—
188C22⋊Dic61—21no—
189C22⋊Dic61C22×C412yes—
190C22⋊Dic61C22×C411yescharacteristic
191C22⋊Dic61C22×C411yescharacteristic
192#11758198—22no—
193#11758198—21no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—21no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201#11758196—22no—
202#11758196—21no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208C4×Dic61—21no—
209C4×Dic61—21no—
210C23×C61—24no—
211C23×C61—21no—
212C23×C61—41no—
213C23×C61—82no—
214C23×C61—81no—
215C23×C61C2×C22⋊C412yes—
216C23×C61C2×C22⋊C412yes—
217C23×C61C23×C411yescharacteristic
218C22×D61—24no—
219C22×D61—42no—
220C22×D61—82no—
221C22×D61—82no—
222C22×D61C2×C22⋊C412yes—
223C22×D61C2×C22⋊C411yescharacteristic
224C22×D61C22×D411yescharacteristic
225C2×#6923—44no—
226C2×#6923—164no—
227Q8×C61—42no—
228Q8×C61—42no—
229D4×C61—24no—
230D4×C61—22no—
231D4×C61—44no—
232D4×C61—44no—
233D4×C61#9914yes—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—22no—
237C2×C4×C61—42no—
238C2×C4×C61—44no—
239C2×C4×C61—44no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C22⋊D61—24no—
246C22⋊D61—24no—
247C22⋊D61—42no—
248C22⋊D61—41no—
249C22⋊D61—41no—
250C22⋊D61—42no—
251C22⋊D61—42no—
252C22⋊D61—82no—
253C22⋊D61—81no—
254C22⋊D61—81no—
255C2×Dic61—24no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—42no—
260C2×Dic61—82no—
261C2×Dic61—82no—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C22×D411yescharacteristic
265D244—22no—
266D244—41no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—42no—
270Dic122—22no—
271Dic122—41no—
272#92420—44no—
273#33350—611nomaximal, Sylow
274C22×C61—22no—
275C22×C61—21no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—41no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—84no—
283C22×C61—82no—
284C22×C61C2×#9914yes—
285C22×C61C22×C22⋊C411yescharacteristic
286C2×D61—21no—
287C2×D61—21no—
288C2×D61—22no—
289C2×D61—42no—
290C2×D61—41no—
291C2×D61—41no—
292C2×D61—84no—
293C2×D61—162no—
294#6923—84no—
295C4×C61—21no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61C2×#9912yes—
299Dic61—21no—
300Dic61—22no—
301Dic61—21no—
302Dic61—41no—
303Dic61—42no—
304Dic61—41no—
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—24no—
314C2×C61—42no—
315C2×C61—41no—
316C2×C61—81no—
317C2×C61C22×#9911yescharacteristic
318D61—41no—
319D61—42no—
320D61—41no—
321C2×H(5,𝔽2)—611no—
322C22×#41#692311yescharacteristic, complemented
323#568—1221no—
324#547—1221no—
325#547—1222no—
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—611no—
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—21no—
354C2×#41—22no—
355C2×#41—21no—
356C2×#41—44no—
357C2×#41—611no—
358C2×#41—612no—
359C2×#41—611no—
360C2×#41—611no—
361C2×#41—611no—
362C22×Q8C2×#692311yescharacteristic, complemented
363C22×D4—612no—
364C22×D4—612no—
365C22×D4—611no—
366C22×D4—612no—
367C22×D4—611no—
368C22×D4C2×#692311yescharacteristic
369C22×D4C2×#692312yescomplemented
370C23×C4C2×#692312yescomplemented
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—611no—
380#123—612no—
381#123—611no—
382#123—612no—
383#123—612no—
384#123—612no—
385#123—612no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389#123—1222no—
390#123—1222no—
391#123—1222no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396C22⋊Q8—1222no—
397#121—1221no—
398#121—1221no—
399#121—1222no—
400#121—1222no—
401#121—1222no—
402#121—1222no—
403#120—611no—
404#120—611no—
405#120—612no—
406#120—1222no—
407#120—1222no—
408C4×D4—1221no—
409C4×D4—1221no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—2444no—
415#117—611no—
416#117—611no—
417#117—1222no—
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—
433C24C22×#692312yes—
434C24C22×#692311yescharacteristic
435#41—48no—
436#41—44no—
437#41—44no—
438#41—1224no—
439#41—2442no—
440#41—2442no—
441#41—2444no—
442#41—2442no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—42no—
446C2×Q8—611no—
447C2×Q8—611no—
448C2×D4—24no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—44no—
452C2×D4—612no—
453C2×D4—612no—
454C2×D4—612no—
455C2×D4—612no—
456C2×D4—611no—
457C2×D4—611no—
458C2×D4—612no—
459C2×D4—611no—
460C2×D4—611no—
461C2×D4—1224no—
462C2×D4—1221no—
463C2×D4—1222no—
464C2×D4—1221no—
465C2×D4—2442no—
466C2×D4—2444no—
467C2×D4—2444no—
468C2×D4—2444no—
469C2×D4C22⋊#692312yes—
470C2×D4C22⋊#692311yescharacteristic
471C2×D4C22⋊#692311yescharacteristic
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—612no—
479C22×C4—612no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—612no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—2444no—
486C22×C4C22×#692311yescharacteristic
487C22×C4C22×#692311yescharacteristic
488C22×C4C22×#692312yes—
489M16—1224no—
490M16—2442no—
491M16—2444no—
492C2×C8—2442no—
493C2×C8—2444no—
494C4⋊C4—1221no—
495C4⋊C4—1222no—
496C4⋊C4—1221no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2442no—
504C22⋊C4—611no—
505C22⋊C4—611no—
506C22⋊C4—612no—
507C22⋊C4—1221no—
508C22⋊C4—1222no—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1221no—
515C22⋊C4—1222no—
516C22⋊C4—1221no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1224no—
520C22⋊C4—2444no—
521C42—1221no—
522C42—1221no—
523C42—2442no—
524C23—24no—
525C23—21no—
526C23—41no—
527C23—82no—
528C23—81no—
529C23—611no—
530C23—612no—
531C23—611no—
532C23—1224no—
533C23—4882no—
534C23—4882no—
535C23—2442no—
536C23C2×C22⋊#692312yes—
537C23C2×C22⋊#692312yes—
538C23C23×#692311yescharacteristic
539Q8—42no—
540Q8—42no—
541Q8—1222no—
542Q8—2441no—
543D4—24no—
544D4—22no—
545D4—44no—
546D4—44no—
547D4—1224no—
548D4—1224no—
549D4—1222no—
550D4—4881no—
551D4—4881no—
552D4—4882no—
553D4—2441no—
554D4—2441no—
555D4—2442no—
556D4—2441no—
557D4—2442no—
558D4—2442no—
559D4#1451983514yes—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—611no—
570C2×C4—612no—
571C2×C4—611no—
572C2×C4—1224no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—1221no—
576C2×C4—1222no—
577C2×C4—1221no—
578C2×C4—9764no—
579C2×C4—4882no—
580C2×C4—4882no—
581C2×C4—2442no—
582C2×C4—2442no—
583C2×C4—2444no—
584C2×C4C2×C22⋊#692311yescharacteristic
585C2×C4C2×C22⋊#692311yescharacteristic
586C8—2444no—
587C22—21no—
588C22—22no—
589C22—42no—
590C22—42no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—84no—
596C22—82no—
597C22—1221no—
598C22—1221no—
599C22—1222no—
600C22—9762no—
601C22—4884no—
602C22—2441no—
603C22—2441no—
604C22—2442no—
605C22C2×#1451983514yes—
606C22C22×C22⋊#692311yescharacteristic
607C4—21no—
608C4—41no—
609C4—42no—
610C4—1221no—
611C4—1221no—
612C4—1222no—
613C4—4884no—
614C4—2441no—
615C4—2442no—
616C4—2441no—
617C4C2×#1451983512yes—
618C2—21no—
619C2—24no—
620C2—41no—
621C2—42no—
622C2—81no—
623C2—2441no—
624C2—2441no—
625C2—2442no—
626C2C22×#1451983511yescharacteristic
subgroupG/H
centerC2C22×#14519835
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC22×#41×C61C4
socleC2×C61C22×#99
prime radical#21459463C2
cosocleC2×C22⋊D61C24
hypercenterC22×#41#6923
hypocenterC61#33350
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×D4 ◁ C22×D4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #21462429C2semidirect
2. #21462418C2semidirect
3. #21462081C2semidirect
4. #21462074C2semidirect
5. #21462070C2semidirect
6. C2×#16840664C2semidirect
18. #16840664C22semidirect
19. #16840664C22semidirect
20. #16840662C22semidirect
28. #16840336C4semidirect
30. #16840321C4semidirect
32. #16840273C4semidirect
34. C2×#14519760C4semidirect
36. #16840237C4semidirect
37. C22×#41×C61C4semidirect
69. #14519760C2×C4semidirect
70. #14519760C2×C4semidirect
71. #14519760C2×C4semidirect
72. #14519760C2×C4semidirect
73. #14519760C2×C4semidirect
89. #14519746C2×C4semidirect
90. #14519746C2×C4semidirect
100. #14519736C2×C4semidirect
101. #14519736C2×C4semidirect
104. C2×C22⋊Dic61C2×C4semidirect
105. C2×C22⋊Dic61C2×C4semidirect
112. C22×Q8×C61C2×C4semidirect
114. C22×D4×C61C2×C4semidirect
115. C23×C4×C61C2×C4semidirect
116. C23×C4×C61C2×C4semidirect
322. C22×#41#6923semidirect
351. C61#33350semidirect
362. C22×Q8C2×#6923semidirect
369. C22×D4C2×#6923semidirect
370. C23×C4C2×#6923semidirect
371. C23×C4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214624292
#214624182
#214620812
#214620742
#214620702
C2×#168406642
#214594632
#3335061