Finite Group Databasebeta

← #29647004

Subgroups — #29647004

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#21459441C211yesmaximal, characteristic
2#21454945C211yesmaximal, characteristic, complemented
3C4×#14519785C211yesmaximal, characteristic, complemented
4#21444276C212yesmaximal, complemented
5#21444273C211yesmaximal, characteristic, complemented
6#21444264C212yesmaximal, complemented
7#21444241C211yesmaximal, characteristic, complemented
8#21444229C212yesmaximal, complemented
9#21444201C211yesmaximal, characteristic, complemented
10Q8⋊C8×D61C212yesmaximal, complemented
11C61×#1489C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840491C411yescharacteristic
14#16840482—21no—
15#16840482C411yescharacteristic
16#16840451—22no—
17#16840451C412yes—
18Q8×D244C411yescharacteristic
19#16840419C412yes—
20#16840417—21no—
21#16840417C411yescharacteristic
22#16840319—21no—
23#16840319C2211yescharacteristic
24#16840294—22no—
25#16840294C2212yes—
26C4×#11758222—21no—
27C4×#11758222C2211yescharacteristic
28C4×Q8×D61C2212yes—
29#16839889—21no—
30#16839831—22no—
31#16839827—21no—
32#16839818C2212yes—
33#16839815C2211yescharacteristic
34C4⋊C8×D61C2212yes—
35#16839720—21no—
36#16839719—22no—
37C4×#11758189C2212yes—
38C2×C4×#92418C2211yescharacteristic
39C4×C8×D61C2212yes—
40#16839494C2212yes—
41#16839494C2212yes—
42Q8⋊#92418C2211yescharacteristic
43Q8⋊#92418C2211yescharacteristic
44#16839492C2212yes—
45#16839492C2212yes—
46D4⋊#92418C2211yescharacteristic
47D4⋊#92418C2211yescharacteristic
48C4×#41×C61C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×D4⋊C8C2212yes—
53C2×#11758222—21no—
54C2×#11758222D411yescharacteristic
55C2×Q8×D61C2×C412yes—
56#14519785—28no—
57C2×#11758189—22no—
58C22×#92418—21no—
59C2×C8×D61—22no—
60C2×C8×D61—22no—
61#14519777—21no—
62#14519777—21no—
63#14519777C2×C411yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519770—22no—
66#14519770—22no—
67#14519770—22no—
68#14519769—21no—
69#14519769—22no—
70#14519768—22no—
71#14519768C2×C412yes—
72#14519764—22no—
73#14519764C2×C412yes—
74#14519759—22no—
75#14519759—22no—
76#14519759—22no—
77#14519757—22no—
78#14519757C2×C412yes—
79#14519756—21no—
80#14519756—21no—
81#14519756—21no—
82#14519756—22no—
83#14519756C2×C412yes—
84#14519748—22no—
85#14519748C2×C412yes—
86Q8×Dic61C2×C412yes—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122C2×C411yescharacteristic
91C4×Dic122D411yescharacteristic
92C4×Dic122C2312yes—
93#14519742—22no—
94#14519742C2×C412yes—
95C4×D244—21no—
96C4×D244C2×C411yescharacteristic
97C4×D244D411yescharacteristic
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61C2×C412yes—
101C4×C22⋊D61D412yes—
102#14519736—21no—
103#14519736—21no—
104#14519736—22no—
105#14519736D411yescharacteristic
106#14519734—22no—
107#14519734D412yes—
108C4⋊C4×D61—22no—
109C4⋊C4×D61C2×C412yes—
110C42×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—22no—
115C22⋊#92418—21no—
116#14519689—22no—
117#14519687—22no—
118#14519687—22no—
119C4×#92418—21no—
120C4×#92418—21no—
121C4×#92418C2311yescharacteristic
122C4×#92418C2311yescharacteristic
123C8×Dic61—22no—
124C2×#41×C61C2×C411yescharacteristic
125C2×M16×C61—21no—
126C2×M16×C61—21no—
127C4×Q8×C61C2×C411yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61C2×C411yescharacteristic
132C4×D4×C61C2311yescharacteristic
133#117×C61—21no—
134#117×C61C2×C411yescharacteristic
135C2×C42×C61—21no—
136C2×C42×C61C2311yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139#97×C61—21no—
140C4×C8×C61C2312yes—
141#41×C61—22no—
142#41×C61—41no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×D4×C61—21no—
145C2×D4×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61C22⋊C412yes—
154Q8×D61—24no—
155Q8×D61—42no—
156#11758222—24no—
157#11758222—44no—
158C2×D244C22⋊C411yescharacteristic
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C2×D412yes—
163C2×C4×D61C2×D412yes—
164C2×Dic122—21no—
165C2×Dic122—22no—
166C2×Dic122C22⋊C411yescharacteristic
167C2×Dic122C22×C412yes—
168M16×C61—24no—
169M16×C61—81no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61C22×C411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2411yescharacteristic
186C22⋊Dic61—22no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61C22⋊C412yes—
191#11758198—22no—
192#11758198C22⋊C412yes—
193#11758198C22⋊C412yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22⋊C411yescharacteristic
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196C22⋊C412yes—
209#11758196C22⋊C412yes—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22×C412yes—
214C4×Dic61C2×D412yes—
215C4×Dic61C2×D412yes—
216C2×#92418—24no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221#11758189—28no—
222C8×D61—28no—
223C8×D61—82no—
224C23×C61—21no—
225C23×C61C2×C22⋊C411yescharacteristic
226C22×D61C2×C22⋊C412yes—
227Q8×C61—21no—
228Q8×C61#13112yescomplemented
229D4×C61—21no—
230D4×C61—21no—
231D4×C61—42no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—42no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C22⋊D61—24no—
251C22⋊D61—44no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61C2×C22⋊C412yes—
255C2×Dic61C2×C22⋊C412yes—
256C2×Dic61C2×C22⋊C412yes—
257D244#10414yes—
258C4×D61—24no—
259C4×D61—22no—
260C4×D61—22no—
261C4×D61—22no—
262C4×D61—22no—
263C4×D61—44no—
264C4×D61—44no—
265Dic122—22no—
266Dic122—22no—
267Dic122—22no—
268Dic122—44no—
269Dic122—42no—
270C8×C61—24no—
271C8×C61—42no—
272#92418—22no—
273#92418—22no—
274#92418—41no—
275#92418—41no—
276#17519—611nomaximal, Sylow
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61C22×C22⋊C411yescharacteristic
284C2×D61—22no—
285C2×D61—22no—
286C2×D61C2×#10414yes—
287C4×C61—22no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61C2×#10412yes—
294C4×C61C2×#13111yescharacteristic
295C4×C61#56811yescharacteristic
296C4×C61#57711yescharacteristic
297C4×C61#57811yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—22no—
301Dic61—42no—
302Dic61—42no—
303#3482—611no—
304#2990—611no—
305C4×#131—611no—
306#1494—612no—
307#1493—611no—
308#1491—612no—
309#1489D6111yescharacteristic, complemented
310#1485—611no—
311#1483—612no—
312#1476—611no—
313C2×Q8⋊C8—612no—
314C2×C61—41no—
315C2×C61—81no—
316C2×C61#288011yescharacteristic
317C2×C61#289811yescharacteristic
318C2×C61C22×#10411yescharacteristic
319D61—24no—
320C2×#131—1221no—
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#423C2×D6111yescharacteristic
344C2×C4⋊C8—612no—
345#405—1222no—
346#405—1221no—
347C4×M16—612no—
348C4×M16C2×D6111yescharacteristic
349C2×C4×C8—611no—
350C2×C4×C8—612no—
351Q8⋊C8—611no—
352Q8⋊C8—611no—
353Q8⋊C8—612no—
354Q8⋊C8—612no—
355Q8⋊C8—612no—
356Q8⋊C8—612no—
357Q8⋊C8C2×D6112yes—
358D4⋊C8—611no—
359D4⋊C8—611no—
360D4⋊C8C2×D6112yes—
361C61#1751911yescharacteristic, complemented, Sylow
362C2×#41—611no—
363C2×#41—1221no—
364C2×#41C4×D6111yescharacteristic
365C22×Q8—612no—
366#131—1228no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—1222no—
370C22×C8—1222no—
371C22×C8—1222no—
372C22×C8—1221no—
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—1222no—
392C22⋊Q8—1221no—
393C22⋊Q8—1222no—
394C22⋊Q8—1221no—
395C22⋊Q8—1221no—
396#121—612no—
397#121—1222no—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—612no—
401C4×Q8—612no—
402C4×Q8—612no—
403C4×Q8—1222no—
404C4×Q8—1221no—
405C4×Q8—1222no—
406C4×Q8—1221no—
407C4×Q8C4×D6111yescharacteristic
408C4×Q8C22×D6111yescharacteristic
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—612no—
412C4×D4—612no—
413C4×D4—611no—
414C4×D4—611no—
415C4×D4—1221no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4C4×D6111yescharacteristic
419C4×D4C22×D6111yescharacteristic
420#117—21no—
421#117—611no—
422#117—612no—
423#117—1222no—
424#117—1222no—
425#117—1221no—
426#117—1221no—
427#117C4×D6111yescharacteristic
428C2×C4⋊C4—612no—
429C2×C4⋊C4—1222no—
430C2×C42—21no—
431C2×C42—612no—
432C2×C42—1222no—
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—21no—
435C4⋊C8—611no—
436C4⋊C8—611no—
437C4⋊C8—1222no—
438C4⋊C8C22×D6112yes—
439C22⋊C8—1222no—
440C22⋊C8—1221no—
441#97—21no—
442#97—1222no—
443#97—1222no—
444C4×C8—611no—
445C4×C8—611no—
446C4×C8—1222no—
447C4×C8—1221no—
448C4×C8—1221no—
449C4×C8C22×D6112yes—
450#41—22no—
451#41—41no—
452#41—1224no—
453#41—2444no—
454C2×Q8—611no—
455C2×Q8—612no—
456C2×Q8—1222no—
457C2×Q8—1221no—
458C2×Q8—1224no—
459C2×Q8—2442no—
460C2×Q8C2×C4×D6111yescharacteristic
461C2×D4—21no—
462C2×D4—612no—
463C2×D4—611no—
464C2×D4—1222no—
465C2×D4C2×C4×D6111yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—612no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4C2×C4×D6111yescharacteristic
475C22×C4D4×D6111yescharacteristic
476C22×C4D4×D6111yescharacteristic
477M16—24no—
478M16—81no—
479M16—1228no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—1228no—
483C2×C8—1224no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—4882no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—612no—
492C4⋊C4—612no—
493C4⋊C4—611no—
494C4⋊C4—611no—
495C4⋊C4—612no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4C2×C4×D6111yescharacteristic
508C4⋊C4C2×C4×D6111yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—612no—
513C22⋊C4—612no—
514C22⋊C4—612no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—612no—
524C42—612no—
525C42—612no—
526C42—1222no—
527C42—1222no—
528C42—1222no—
529C42C2×C4×D6111yescharacteristic
530C42D4×D6111yescharacteristic
531C42D4×D6111yescharacteristic
532C42C23×D6111yescharacteristic
533C23—21no—
534C23—612no—
535C23C22⋊C4×D6111yescharacteristic
536Q8—21no—
537Q8—1222no—
538Q8—1222no—
539Q8—1222no—
540Q8—2444no—
541Q8—2442no—
542Q8#1451978512yescomplemented
543D4—21no—
544D4—21no—
545D4—42no—
546D4—614no—
547D4—1224no—
548D4—2444no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—612no—
562C2×C4—612no—
563C2×C4—612no—
564C2×C4—1224no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—2444no—
572C2×C4—2444no—
573C2×C4C22⋊C4×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C22×C4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C2×C4C2×D4×D6111yescharacteristic
579C8—24no—
580C8—42no—
581C8—1222no—
582C8—1222no—
583C8—2441no—
584C8—2441no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—614no—
592C22—1222no—
593C22—1222no—
594C22C2×C22⋊C4×D6111yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—41no—
600C4—41no—
601C4—1222no—
602C4—1222no—
603C4—1222no—
604C4—2442no—
605C4—2442no—
606C4#104×D6112yes—
607C4C2×#1451978511yescharacteristic
608C4#1684052811yescharacteristic
609C4#1684057211yescharacteristic
610C4#1684057811yescharacteristic
611C2—41no—
612C2—81no—
613C2—1224no—
614C2#2145426911yescharacteristic
615C2#2145435011yescharacteristic
616C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1489C2
socleC22×C61C22×C22⋊C4
prime radical#21459441C2
cosocleC42×C61C24
hypercenter#1489D61
hypocenterC61#17519
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1489

Complements

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

NH ≅ G/Ntype
2. #21454945C2semidirect
3. C4×#14519785C2semidirect
4. #21444276C2semidirect
5. #21444273C2semidirect
6. #21444264C2semidirect
7. #21444241C2semidirect
8. #21444229C2semidirect
9. #21444201C2semidirect
10. Q8⋊C8×D61C2semidirect
11. C61×#1489C2semidirect
228. Q8×C61#131semidirect
309. #1489D61semidirect
361. C61#17519semidirect
542. Q8#14519785semidirect

Maximal subgroups

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

subgroupindex
#214594412
#214549452
C4×#145197852
#214442762
#214442732
#214442642
#214442412
#214442292
#214442012
Q8⋊C8×D612
C61×#14892
#1751961