Finite Group Databasebeta

← #29647012

Subgroups — #29647012

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459445C211yesmaximal, characteristic
2#21454949C211yesmaximal, characteristic, complemented
3C4×#14519785C211yesmaximal, characteristic, complemented
4#21444269C211yesmaximal, characteristic, complemented
5#21444268C212yesmaximal, complemented
6#21444257C212yesmaximal, complemented
7#21444237C211yesmaximal, characteristic, complemented
8#21444234C212yesmaximal, complemented
9#21444201C211yesmaximal, characteristic, complemented
10#21444200C212yesmaximal, complemented
11C61×#1489C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13Q8×Dic122C411yescharacteristic
14#16840477—21no—
15#16840477C411yescharacteristic
16#16840475—22no—
17#16840475C412yes—
18#16840434C411yescharacteristic
19#16840433—21no—
20#16840433C411yescharacteristic
21#16840431C412yes—
22#16840312—21no—
23#16840312C2211yescharacteristic
24#16840309—22no—
25#16840309C2212yes—
26C4×#11758226C2212yes—
27C4×#11758222—21no—
28C4×#11758222C2211yescharacteristic
29#16839886—21no—
30#16839839—21no—
31#16839838—22no—
32#16839815C2211yescharacteristic
33#16839814C2212yes—
34#16839784C2212yes—
35#16839720—21no—
36#16839719—22no—
37C4×#11758189C2212yes—
38C2×C4×#92418C2211yescharacteristic
39C4×C8×D61C2212yes—
40Q8⋊#92418C2211yescharacteristic
41Q8⋊#92418C2211yescharacteristic
42#16839491C2212yes—
43#16839491C2212yes—
44D4⋊#92418C2211yescharacteristic
45D4⋊#92418C2211yescharacteristic
46#16839489C2212yes—
47#16839489C2212yes—
48C4×#41×C61C2211yescharacteristic
49C61×#423C2211yescharacteristic
50C4×M16×C61C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×D4⋊C8C2212yes—
53C2×#11758226C2×C412yes—
54C2×#11758222—21no—
55C2×#11758222—22no—
56C2×#11758222—21no—
57C2×#11758222D411yescharacteristic
58#14519785—28no—
59C2×#11758189—22no—
60C2×#11758189—22no—
61C22×#92418—21no—
62C2×C8×D61—22no—
63#14519777—21no—
64#14519777C2×C411yescharacteristic
65#14519776—22no—
66#14519776C2×C412yes—
67#14519774—21no—
68#14519774C2×C411yescharacteristic
69#14519773—22no—
70#14519773C2×C412yes—
71#14519765—21no—
72#14519765—22no—
73#14519763—22no—
74#14519763—22no—
75#14519763—22no—
76#14519752—22no—
77#14519752C2×C412yes—
78#14519751—21no—
79#14519751—21no—
80#14519751—22no—
81#14519751—21no—
82#14519751C2×C412yes—
83#14519750—22no—
84#14519750—22no—
85#14519750—22no—
86#14519748—22no—
87#14519748C2×C412yes—
88Q8×Dic61C2×C412yes—
89C4×Dic122—21no—
90C4×Dic122C2×C411yescharacteristic
91C4×Dic122D411yescharacteristic
92C4×D244—22no—
93C4×D244—21no—
94C4×D244—21no—
95C4×D244C2×C411yescharacteristic
96C4×D244D411yescharacteristic
97C4×D244C2312yes—
98#14519739—22no—
99#14519739C2×C412yes—
100C4×C22⋊D61—22no—
101C4×C22⋊D61—22no—
102C4×C22⋊D61C2×C412yes—
103C4×C22⋊D61D412yes—
104#14519736—21no—
105#14519736D411yescharacteristic
106#14519735—22no—
107#14519735C2×C412yes—
108#14519734—22no—
109#14519734D412yes—
110C42×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418C2311yescharacteristic
113C4⋊#92418C2311yescharacteristic
114#14519706—22no—
115C22⋊#92418—21no—
116#14519689—22no—
117#14519687—22no—
118C4×#92418—21no—
119C4×#92418—21no—
120C4×#92418C2311yescharacteristic
121C4×#92418C2311yescharacteristic
122C8×Dic61—22no—
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⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61C22⋊C412yes—
157#11758226—24no—
158#11758226—42no—
159#11758222—24no—
160#11758222—44no—
161#11758222—82no—
162#11758222—82no—
163#11758222—81no—
164#11758222—81no—
165C2×D244—22no—
166C2×D244—21no—
167C2×D244—22no—
168C2×D244—21no—
169C2×D244C22⋊C411yescharacteristic
170C2×D244C22×C412yes—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122C22⋊C411yescharacteristic
180M16×C61—24no—
181M16×C61—81no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C22×C411yescharacteristic
187C4⋊C4×C61C22×C411yescharacteristic
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C22×C411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2411yescharacteristic
198C22⋊Dic61—22no—
199C22⋊Dic61C22⋊C412yes—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198C22⋊C412yes—
206#11758198C22⋊C412yes—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61C22⋊C411yescharacteristic
210C4⋊Dic61C22⋊C411yescharacteristic
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196C22⋊C412yes—
214#11758196C22⋊C412yes—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—24no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225#11758189—28no—
226#11758189—82no—
227C8×D61—28no—
228C23×C61—21no—
229C23×C61C2×C22⋊C411yescharacteristic
230C22×D61—22no—
231C22×D61C2×C22⋊C412yes—
232Q8×C61—21no—
233Q8×C61#13112yescomplemented
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—42no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—42no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C23×C411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—24no—
256C22⋊D61—44no—
257C22⋊D61—82no—
258C22⋊D61—82no—
259C22⋊D61—82no—
260C2×Dic61—22no—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61C2×C22⋊C412yes—
263C2×Dic61C2×C22⋊C412yes—
264D244—22no—
265D244—22no—
266D244—22no—
267D244—44no—
268D244—42no—
269D244—81no—
270D244—82no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—24no—
276C4×D61—44no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—44no—
280Dic122—82no—
281Dic122—81no—
282Dic122#10414yes—
283C8×C61—24no—
284C8×C61—42no—
285#92418—22no—
286#92418—22no—
287#92418—41no—
288#92418—41no—
289#17520—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61C22×C22⋊C411yescharacteristic
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61C2×#10412yes—
310C4×C61C2×#13111yescharacteristic
311C4×C61#56811yescharacteristic
312C4×C61#57611yescharacteristic
313C4×C61#57711yescharacteristic
314Dic61—22no—
315Dic61—22no—
316Dic61—42no—
317Dic61C2×#10414yes—
318#3483—611no—
319#2991—611no—
320C4×#131—611no—
321#1492—611no—
322#1492—612no—
323#1489—612no—
324#1489D6111yescharacteristic, complemented
325#1484—611no—
326#1484—612no—
327#1476—612no—
328#1476—611no—
329C2×C61—41no—
330C2×C61—81no—
331C2×C61#288011yescharacteristic
332C2×C61#289711yescharacteristic
333C2×C61C22×#10411yescharacteristic
334D61—42no—
335D61—82no—
336C2×#131—1221no—
337Q82—611no—
338#555—611no—
339#555—612no—
340#555—1222no—
341#555—1221no—
342#550—611no—
343#550—612no—
344#550—611no—
345#550—1221no—
346#532—612no—
347#532—611no—
348#532—1221no—
349#532—1222no—
350C4×#41—611no—
351C4×#41—612no—
352C4×#41—1221no—
353C4×#41C2×D6111yescharacteristic
354C8×Q8—1221no—
355#436—1221no—
356#436—1222no—
357#431—611no—
358#431—612no—
359#423—612no—
360#423C2×D6111yescharacteristic
361#405—1222no—
362#405—1221no—
363C4×M16—612no—
364C4×M16C2×D6111yescharacteristic
365C2×C4×C8—611no—
366C2×C4×C8—612no—
367Q8⋊C8—611no—
368Q8⋊C8—611no—
369Q8⋊C8C2×D6112yes—
370D4⋊C8—611no—
371D4⋊C8—611no—
372D4⋊C8—612no—
373D4⋊C8—612no—
374D4⋊C8—612no—
375D4⋊C8—612no—
376D4⋊C8C2×D6112yes—
377C61#1752011yescharacteristic, complemented, Sylow
378C2×#41—611no—
379C2×#41—612no—
380C2×#41—1222no—
381C2×#41—1221no—
382C2×#41—1221no—
383C2×#41C4×D6111yescharacteristic
384#131—1228no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—1222no—
388C2×M16—1222no—
389C22×C8—1222no—
390C22×C8—1221no—
391C4⋊Q8—612no—
392C4⋊Q8—611no—
393C4⋊Q8—1221no—
394C4⋊Q8—1222no—
395#127—612no—
396#127—611no—
397#127—1221no—
398#127—1222no—
399#124—1222no—
400#124—1222no—
401#124—1222no—
402#124—1222no—
403#124—1221no—
404#121—612no—
405#121—612no—
406#121—612no—
407#121—1222no—
408#121—1222no—
409#121—1222no—
410#121—1222no—
411#121—1222no—
412#121—1222no—
413#121—1221no—
414#121—1221no—
415#121—1221no—
416C4×Q8—611no—
417C4×Q8—611no—
418C4×Q8—612no—
419C4×Q8—1221no—
420C4×Q8C4×D6111yescharacteristic
421C4×Q8C22×D6111yescharacteristic
422C4×D4—21no—
423C4×D4—21no—
424C4×D4—612no—
425C4×D4—612no—
426C4×D4—612no—
427C4×D4—611no—
428C4×D4—611no—
429C4×D4—612no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435C4×D4—1221no—
436C4×D4C4×D6111yescharacteristic
437C4×D4C22×D6111yescharacteristic
438#117—21no—
439#117—611no—
440#117—612no—
441#117—612no—
442#117—1222no—
443#117—1222no—
444#117—1221no—
445#117C4×D6111yescharacteristic
446C2×C42—21no—
447C2×C42—612no—
448C2×C42—1222no—
449C2×C42C22×D6111yescharacteristic
450C4⋊C8—21no—
451C4⋊C8—611no—
452C4⋊C8—611no—
453C4⋊C8—1222no—
454C4⋊C8C22×D6112yes—
455C22⋊C8—1222no—
456C22⋊C8—1221no—
457#97—21no—
458#97—1222no—
459C4×C8—611no—
460C4×C8—611no—
461C4×C8—1222no—
462C4×C8—1222no—
463C4×C8—1221no—
464C4×C8—1221no—
465C4×C8C22×D6112yes—
466#41—22no—
467#41—41no—
468#41—1224no—
469#41—1224no—
470#41—4881no—
471#41—4881no—
472#41—4882no—
473#41—4882no—
474#41—2444no—
475#41—2442no—
476C2×Q8—611no—
477C2×Q8—1222no—
478C2×Q8—1221no—
479C2×Q8C2×C4×D6111yescharacteristic
480C2×D4—21no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—611no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—1222no—
488C2×D4—1222no—
489C2×D4—1222no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4C2×C4×D6111yescharacteristic
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—612no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1222no—
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505M16—24no—
506M16—81no—
507M16—1228no—
508M16—4882no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—1228no—
512C2×C8—1224no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—611no—
522C4⋊C4—611no—
523C4⋊C4—612no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1221no—
527C4⋊C4C2×C4×D6111yescharacteristic
528C4⋊C4C2×C4×D6111yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—612no—
533C22⋊C4—612no—
534C22⋊C4—612no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—612no—
545C42—612no—
546C42—612no—
547C42—1222no—
548C42—1222no—
549C42C2×C4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C42C23×D6111yescharacteristic
553C23—21no—
554C23—612no—
555C23—1222no—
556C23C22⋊C4×D6111yescharacteristic
557Q8—21no—
558Q8—614no—
559Q8—4881no—
560Q8—4882no—
561Q8#1451978512yescomplemented
562D4—21no—
563D4—21no—
564D4—42no—
565D4—1224no—
566D4—1222no—
567D4—1222no—
568D4—1222no—
569D4—4881no—
570D4—4882no—
571D4—4882no—
572D4—4882no—
573D4—4882no—
574D4—2444no—
575D4—2444no—
576D4—2442no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1224no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22×C4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C8—24no—
609C8—42no—
610C8—1222no—
611C8—1222no—
612C8—2441no—
613C8—2441no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—2442no—
624C22—2442no—
625C22—2442no—
626C22C2×C22⋊C4×D6111yescharacteristic
627C4—21no—
628C4—21no—
629C4—21no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—614no—
634C4—1222no—
635C4—1222no—
636C4—2442no—
637C4#104×D6112yes—
638C4C2×#1451978511yescharacteristic
639C4#1684052811yescharacteristic
640C4#1684056611yescharacteristic
641C4#1684057111yescharacteristic
642C2—41no—
643C2—81no—
644C2—4882no—
645C2—2442no—
646C2#2145426911yescharacteristic
647C2#2145434311yescharacteristic
648C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1489C2
socleC22×C61C22×C22⋊C4
prime radical#21459445C2
cosocleC42×C61C24
hypercenter#1489D61
hypocenterC61#17520
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. #21454949C2semidirect
3. C4×#14519785C2semidirect
4. #21444269C2semidirect
5. #21444268C2semidirect
6. #21444257C2semidirect
7. #21444237C2semidirect
8. #21444234C2semidirect
9. #21444201C2semidirect
10. #21444200C2semidirect
11. C61×#1489C2semidirect
233. Q8×C61#131semidirect
324. #1489D61semidirect
377. C61#17520semidirect
561. Q8#14519785semidirect

Maximal subgroups

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

subgroupindex
#214594452
#214549492
C4×#145197852
#214442692
#214442682
#214442572
#214442372
#214442342
#214442012
#214442002
C61×#14892
#1752061