Finite Group Databasebeta

← #29647005

Subgroups — #29647005

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459434C211yesmaximal, characteristic
2#21454944C211yesmaximal, characteristic, complemented
3#21444280C212yesmaximal, complemented
4#21444275C211yesmaximal, characteristic, complemented
5#21444267C212yesmaximal, complemented
6#21444259C211yesmaximal, characteristic, complemented
7#21444241C211yesmaximal, characteristic, complemented
8#21444233C212yesmaximal, complemented
9#21444202C211yesmaximal, characteristic, complemented
10C2×#16839494C212yesmaximal, complemented
11C4×#131×C61C211yesmaximal, characteristic, complemented
12#16840490C411yescharacteristic
13#16840485C411yescharacteristic
14#16840480—21no—
15#16840452C412yes—
16#16840445—22no—
17D4×Dic122C411yescharacteristic
18#16840421C412yes—
19#16840416C411yescharacteristic
20#16840411—21no—
21#16840319C2211yescharacteristic
22#16840316—21no—
23#16840296C2212yes—
24#16840292—22no—
25C4×#11758222C2211yescharacteristic
26C4×#11758224—21no—
27C2×C4×Dic122C2212yes—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2212yes—
32#16839815C2211yescharacteristic
33#16839786C2211yescharacteristic
34C2×C4⋊#92418C2212yes—
35#16839494C2211yescharacteristic
36#16839494C2212yes—
37#16839494C2212yes—
38#16839494C2212yes—
39#16839494C2211yescharacteristic
40#16839494C2212yes—
41#16839494C2212yes—
42#16839491C2211yescharacteristic
43#16839491C2211yescharacteristic
44#16839491C2212yes—
45C2×#131×C61—21no—
46C4×#41×C61C2211yescharacteristic
47C61×#405—21no—
48C61×#405—22no—
49C4×M16×C61C2212yes—
50C4×M16×C61C2211yescharacteristic
51C2×C4×C8×C61C2212yes—
52C2×C4×C8×C61C2211yescharacteristic
53C2×#11758222C2×C411yescharacteristic
54C2×#11758224—21no—
55C22×Dic122C2×C412yes—
56C2×#11758194—21no—
57C22×#92418—22no—
58#14519777C2×C411yescharacteristic
59#14519777C2×C411yescharacteristic
60#14519776—21no—
61#14519775—21no—
62#14519772—22no—
63#14519771—22no—
64#14519770—22no—
65#14519769—21no—
66#14519769C2×C412yes—
67#14519767—22no—
68#14519767—22no—
69#14519765C2×C412yes—
70#14519763—22no—
71#14519761—22no—
72#14519760—22no—
73#14519758—22no—
74#14519757—21no—
75#14519756—21no—
76#14519756C2×C412yes—
77#14519756C2×C412yes—
78#14519755—22no—
79#14519754—22no—
80#14519753—21no—
81#14519751C2×C412yes—
82#14519749—22no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85C4×Dic122—21no—
86C4×Dic122C2×C412yes—
87C4×Dic122C2×C412yes—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122C2311yescharacteristic
90C4×Dic122C2312yes—
91#14519742—21no—
92C4×D244C2×C411yescharacteristic
93C4×D244C2311yescharacteristic
94D4×Dic61—21no—
95C4×C22⋊D61—21no—
96C4×C22⋊D61—22no—
97C4×C22⋊D61—21no—
98#14519737—22no—
99#14519736—21no—
100#14519736C2×C411yescharacteristic
101#14519735—21no—
102#14519734—21no—
103#14519734—22no—
104#14519733—22no—
105C2×C4⋊Dic61C2×C412yes—
106C2×#11758196—22no—
107C2×C4×Dic61—22no—
108C42×D61—21no—
109C4⋊#92418—22no—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2312yes—
113C4⋊#92418C2311yescharacteristic
114C22⋊#92418—21no—
115C22⋊#92418—22no—
116#14519688—22no—
117C4×#92418—21no—
118C2×#41×C61D411yescharacteristic
119#131×C61—28no—
120C2×M16×C61—22no—
121C2×M16×C61—21no—
122C22×C8×C61—22no—
123C22×C8×C61—21no—
124C4×Q8×C61C2×C411yescharacteristic
125C4×Q8×C61D411yescharacteristic
126C4×D4×C61C2×C412yes—
127C4×D4×C61C2×C411yescharacteristic
128C4×D4×C61D411yescharacteristic
129C4×D4×C61D412yes—
130#117×C61D411yescharacteristic
131#117×C61D412yes—
132C2×C42×C61C2312yes—
133C2×C42×C61C2311yescharacteristic
134#97×C61—21no—
135#97×C61—22no—
136C4×C8×C61—21no—
137C4×C8×C61—22no—
138C4×C8×C61C2311yescharacteristic
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2312yes—
141#41×C61—24no—
142C2×Q8×C61C22⋊C411yescharacteristic
143C2×D4×C61C22⋊C412yes—
144C2×D4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22×C412yes—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—21no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155#11758224—44no—
156#11758222—22no—
157#11758222—41no—
158C2×D244C22×C411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×Dic122—24no—
163C2×Dic122—22no—
164C2×Dic122—21no—
165C2×Dic122—42no—
166C2×Dic122C22×C412yes—
167C2×Dic122C22×C411yescharacteristic
168M16×C61—28no—
169M16×C61—24no—
170C2×C8×C61—28no—
171C2×C8×C61—24no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—22no—
175C4⋊C4×C61C22⋊C412yes—
176C4⋊C4×C61C22⋊C411yescharacteristic
177C4⋊C4×C61C22⋊C411yescharacteristic
178C4⋊C4×C61C22⋊C412yes—
179C22⋊C4×C61C22⋊C412yes—
180C22⋊C4×C61C22⋊C412yes—
181C22⋊C4×C61C22⋊C412yes—
182C42×C61C22×C411yescharacteristic
183C42×C61C22×C412yes—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D412yes—
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D412yes—
188C42×C61C2411yescharacteristic
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—21no—
192#11758198—21no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C22×C411yescharacteristic
202C4⋊Dic61C22×C412yes—
203C4⋊Dic61C22×C411yescharacteristic
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—21no—
209#11758196—22no—
210#11758196—22no—
211#11758196—21no—
212#11758196—22no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—21no—
218C4×Dic61—22no—
219#11758194—81no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—22no—
223C2×#92418—82no—
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C2×C22⋊C412yes—
226C22×D61—21no—
227Q8×C61—22no—
228D4×C61—24no—
229D4×C61#10414yes—
230C2×C4×C61—24no—
231C2×C4×C61—22no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—21no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C22⋊D61—42no—
250C22⋊D61—44no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—44no—
258D244—21no—
259D244—21no—
260C4×D61—42no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—42no—
264Dic122—21no—
265Dic122—22no—
266Dic122—22no—
267Dic122—44no—
268Dic122—42no—
269Dic122#13112yescomplemented
270C8×C61—22no—
271C8×C61—24no—
272C8×C61—22no—
273#92418—41no—
274#92418—42no—
275#92418—41no—
276#17519—611nomaximal, Sylow
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61C2×#10414yes—
283C22×C61C22×C22⋊C411yescharacteristic
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C4×C61—22no—
288C4×C61—21no—
289C4×C61—22no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—22no—
293C4×C61C2×#10412yes—
294C4×C61C2×#13111yescharacteristic
295C4×C61#56811yescharacteristic
296C4×C61#57711yescharacteristic
297C4×C61#57811yescharacteristic
298Dic61—22no—
299Dic61—42no—
300Dic61—41no—
301Dic61—42no—
302Dic61—41no—
303#3482—611no—
304#2990—611no—
305C4×#131D6111yescharacteristic, complemented
306#1494—612no—
307#1493—611no—
308#1491—612no—
309#1489—611no—
310#1485—611no—
311#1483—612no—
312#1476—611no—
313C2×Q8⋊C8—612no—
314C2×C61—24no—
315C2×C61—41no—
316C2×C61#288011yescharacteristic
317C2×C61#289811yescharacteristic
318C2×C61C22×#10411yescharacteristic
319D61—81no—
320C2×#131—21no—
321#557—611no—
322#556—611no—
323#556—1221no—
324#552—612no—
325#552—1222no—
326D4×Q8—611no—
327#548—612no—
328#547—611no—
329#547—1221no—
330#533—611no—
331#533—1221no—
332#529—612no—
333#529—1222no—
334C4×#41—611no—
335C4×#41—1221no—
336C4×#41C2×D6111yescharacteristic
337C2×C4×Q8—612no—
338#446—1221no—
339#435—1222no—
340C8×D4—1221no—
341#432—612no—
342#431—611no—
343#423—611no—
344C2×C4⋊C8—612no—
345#405—22no—
346#405—21no—
347C4×M16C2×D6112yes—
348C4×M16C2×D6111yescharacteristic
349C2×C4×C8C2×D6112yes—
350C2×C4×C8C2×D6111yescharacteristic
351Q8⋊C8—611no—
352Q8⋊C8—612no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—611no—
356Q8⋊C8—612no—
357Q8⋊C8—612no—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8—612no—
361C61#1751911yescharacteristic, complemented, Sylow
362C2×#41—611no—
363C2×#41—1221no—
364C2×#41C22⋊D6111yescharacteristic
365C22×Q8—612no—
366#131—28no—
367C2×M16—22no—
368C2×M16—21no—
369C2×M16—1221no—
370C22×C8—22no—
371C22×C8—21no—
372C22×C8—1222no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8—1221no—
376C4⋊Q8—1221no—
377#126—1222no—
378#126—1222no—
379#126—1222no—
380#125—612no—
381#125—1222no—
382#125—1222no—
383#125—1221no—
384#124—612no—
385#124—1222no—
386#123—1222no—
387#123—1222no—
388#123—1222no—
389C22⋊Q8—612no—
390C22⋊Q8—612no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396#121—612no—
397#121—1222no—
398C4×Q8—612no—
399C4×Q8—612no—
400C4×Q8—611no—
401C4×Q8—611no—
402C4×Q8—612no—
403C4×Q8—1221no—
404C4×Q8—1222no—
405C4×Q8—1221no—
406C4×Q8—1222no—
407C4×Q8C4×D6111yescharacteristic
408C4×Q8D24411yescharacteristic
409C4×D4—611no—
410C4×D4—611no—
411C4×D4—1221no—
412C4×D4—1221no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4—1222no—
416C4×D4C4×D6111yescharacteristic
417C4×D4C4×D6112yes—
418C4×D4D24411yescharacteristic
419C4×D4D24412yes—
420#117—611no—
421#117—1221no—
422#117—1221no—
423#117—1222no—
424#117—1222no—
425#117—1221no—
426#117C22⋊D6112yes—
427#117C22⋊D6111yescharacteristic
428C2×C4⋊C4—612no—
429C2×C4⋊C4—1222no—
430C2×C42—1222no—
431C2×C42—1221no—
432C2×C42C22×D6112yes—
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—611no—
435C4⋊C8—612no—
436C4⋊C8—611no—
437C4⋊C8—1222no—
438C4⋊C8—1221no—
439C22⋊C8—1221no—
440C22⋊C8—1222no—
441#97—21no—
442#97—22no—
443#97—1222no—
444C4×C8—21no—
445C4×C8—22no—
446C4×C8—1221no—
447C4×C8C22×D6111yescharacteristic
448C4×C8C22×D6111yescharacteristic
449C4×C8C22×D6112yes—
450#41—24no—
451#41—1222no—
452#41—2441no—
453#41—2444no—
454C2×Q8—611no—
455C2×Q8—612no—
456C2×Q8—1224no—
457C2×Q8—1222no—
458C2×Q8—1221no—
459C2×Q8—2442no—
460C2×Q8#1175819811yescharacteristic
461C2×D4—611no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4#1175819812yes—
465C2×D4#1175819811yescharacteristic
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4C2×C4×D6112yes—
472C22×C4C2×C4×D6111yescharacteristic
473C22×C4C2×D24412yes—
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×C22⋊D6112yes—
476C22×C4C2×C22⋊D6111yescharacteristic
477M16—28no—
478M16—24no—
479M16—4881no—
480C2×C8—28no—
481C2×C8—24no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1222no—
488C2×C8—4882no—
489C4⋊C4—611no—
490C4⋊C4—612no—
491C4⋊C4—611no—
492C4⋊C4—1221no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4#1175819811yescharacteristic
506C4⋊C4#1175819812yes—
507C4⋊C4#1175819811yescharacteristic
508C4⋊C4#1175819812yes—
509C22⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1221no—
513C22⋊C4—1221no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4#1175819812yes—
518C22⋊C4#1175819812yes—
519C22⋊C4#1175819812yes—
520C42—1221no—
521C42—1222no—
522C42—1221no—
523C42—1222no—
524C42—1221no—
525C42—1222no—
526C42C2×C4×D6111yescharacteristic
527C42C2×C4×D6112yes—
528C42C2×D24411yescharacteristic
529C42C2×D24412yes—
530C42C2×C22⋊D6111yescharacteristic
531C42C2×C22⋊D6112yes—
532C42C23×D6111yescharacteristic
533C23—1221no—
534C23C2×#1175819811yescharacteristic
535C23C2×#1175819812yes—
536Q8—22no—
537Q8—612no—
538Q8—1221no—
539Q8—1222no—
540Q8—1222no—
541Q8—2444no—
542Q8—2442no—
543D4—24no—
544D4—1221no—
545D4—1221no—
546D4—2442no—
547D4—2444no—
548D4#1451970514yes—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—21no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2442no—
565C2×C4—2444no—
566C2×C4—2444no—
567C2×C4—2442no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4C2×#1175819812yes—
571C2×C4C2×#1175819811yescharacteristic
572C2×C4C2×#1175819812yes—
573C2×C4C2×#1175819811yescharacteristic
574C2×C4C2×#1175819812yes—
575C2×C4C2×#1175819811yescharacteristic
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C22×C22⋊D6111yescharacteristic
578C2×C4C22×D24411yescharacteristic
579C8—22no—
580C8—24no—
581C8—22no—
582C8—2441no—
583C8—2442no—
584C8—2441no—
585C22—22no—
586C22—22no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—2441no—
591C22—2441no—
592C22—2441no—
593C22C2×#1451970514yes—
594C22C22×#1175819811yescharacteristic
595C4—22no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—21no—
600C4—22no—
601C4—1222no—
602C4—2441no—
603C4—2441no—
604C4—2442no—
605C4—2442no—
606C4C2×#1451970512yes—
607C4#131×D6111yescharacteristic
608C4#1684052711yescharacteristic
609C4#1684057411yescharacteristic
610C4#1684057911yescharacteristic
611C2—24no—
612C2—41no—
613C2—4881no—
614C2#2145426611yescharacteristic
615C2#2145435311yescharacteristic
616C2C22×#1451970511yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC4×#131×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459434C2
cosocleC42×C61C24
hypercenterC4×#131D61
hypocenterC61#17519
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C4×#131

Complements

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

NH ≅ G/Ntype
2. #21454944C2semidirect
3. #21444280C2semidirect
4. #21444275C2semidirect
5. #21444267C2semidirect
6. #21444259C2semidirect
7. #21444241C2semidirect
8. #21444233C2semidirect
9. #21444202C2semidirect
10. C2×#16839494C2semidirect
11. C4×#131×C61C2semidirect
269. Dic122#131semidirect
305. C4×#131D61semidirect
361. C61#17519semidirect

Maximal subgroups

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

subgroupindex
#214594342
#214549442
#214442802
#214442752
#214442672
#214442592
#214442412
#214442332
#214442022
C2×#168394942
C4×#131×C612
#1751961