Finite Group Databasebeta

← #29646986

Subgroups — #29646986

Order 15616.

1096 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459429C211yesmaximal, characteristic
2#21454945C211yesmaximal, characteristic, complemented
3C4×#14519785C211yesmaximal, characteristic, complemented
4#21444278C212yesmaximal, complemented
5#21444269C211yesmaximal, characteristic, complemented
6#21444260C212yesmaximal, complemented
7#21444237C211yesmaximal, characteristic, complemented
8#21444231C212yesmaximal, complemented
9#21444201C211yesmaximal, characteristic, complemented
10D4⋊C8×D61C212yesmaximal, complemented
11C61×#1489C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840497C411yescharacteristic
14#16840472C411yescharacteristic
15#16840469C411yescharacteristic
16#16840458—21no—
17#16840456C412yes—
18#16840436—22no—
19#16840408—21no—
20#16840400C412yes—
21D4×D244C411yescharacteristic
22#16840312C2211yescharacteristic
23#16840299C2212yes—
24#16840296—21no—
25#16840271—22no—
26C4×#11758222C2211yescharacteristic
27C2×C4×D244—21no—
28C4×D4×D61C2212yes—
29#16839889—21no—
30#16839831—22no—
31#16839827—21no—
32#16839818C2212yescomplemented
33#16839815C2211yescharacteristic, complemented
34C4⋊C8×D61C2212yescomplemented
35#16839720—21no—
36#16839719—22no—
37C4×#11758189C2212yescomplemented
38C2×C4×#92418C2211yescharacteristic, complemented
39C4×C8×D61C2212yescomplemented
40Q8⋊#92418C2211yescharacteristic, complemented
41Q8⋊#92418C2211yescharacteristic, complemented
42#16839491C2212yescomplemented
43#16839491C2212yescomplemented
44D4⋊#92418C2211yescharacteristic
45D4⋊#92418C2211yescharacteristic
46#16839489C2212yescomplemented
47#16839489C2212yescomplemented
48C4×#41×C61C2211yescharacteristic
49C61×#423C2211yescharacteristic, complemented
50C4×M16×C61C2211yescharacteristic, complemented
51C61×Q8⋊C8C2212yescomplemented
52C61×D4⋊C8C2212yes—
53C2×#11758222—21no—
54C2×#11758222D411yescharacteristic
55C22×D244—21no—
56C2×D4×D61C2×C412yes—
57C22×C4×D61—22no—
58#14519785—28no—
59C2×#11758189—22no—
60C22×#92418—21no—
61C2×C8×D61—22no—
62C2×C8×D61—22no—
63#14519777C2×C411yescharacteristic
64#14519774C2×C411yescharacteristic
65#14519772—22no—
66#14519771—22no—
67#14519769—21no—
68#14519768C2×C412yes—
69#14519766C2×C412yes—
70#14519765—21no—
71#14519765—21no—
72#14519762—22no—
73#14519762—22no—
74#14519760—22no—
75#14519759—22no—
76#14519756—21no—
77#14519755—22no—
78#14519754—22no—
79#14519753C2×C412yes—
80#14519752C2×C412yes—
81#14519751—21no—
82#14519751—21no—
83#14519751C2×C412yes—
84#14519750—22no—
85#14519749—22no—
86#14519747—22no—
87#14519745—22no—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122D411yescharacteristic
90C4×D244—21no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244—22no—
94C4×D244—21no—
95C4×D244C2×C411yescharacteristic
96C4×D244D411yescharacteristic
97C4×D244C2312yes—
98D4×Dic61C2×C412yes—
99#14519739C2×C412yes—
100C4×C22⋊D61—22no—
101C4×C22⋊D61C2×C412yes—
102C4×C22⋊D61D412yes—
103#14519737—22no—
104#14519736D411yescharacteristic
105#14519734—22no—
106#14519734D412yes—
107C2×C4⋊Dic61—21no—
108C4⋊C4×D61C2×C412yes—
109C2×#11758198—22no—
110C22⋊C4×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—
141C23×D61—22no—
142#41×C61—22no—
143#41×C61—41no—
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61—21no—
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61C22⋊C412yes—
156C22×Dic61—22no—
157D4×D61—24no—
158D4×D61—42no—
159#11758222—24no—
160#11758222—81no—
161#11758222—81no—
162C2×D244—22no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—44no—
166C2×D244C22⋊C411yescharacteristic
167C2×D244C22×C412yes—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—22no—
171C2×C4×D61—82no—
172C2×C4×D61—82no—
173C2×C4×D61C22×C412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×D412yes—
176C2×Dic122—21no—
177C2×Dic122C22⋊C411yescharacteristic
178M16×C61—24no—
179M16×C61—81no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C22×C411yescharacteristic
185C4⋊C4×C61C22×C411yescharacteristic
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61C22×C411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2411yescharacteristic
196C22⋊Dic61—22no—
197C22⋊Dic61C22⋊C412yes—
198#11758198—22no—
199#11758198—22no—
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#11758196—22no—
214#11758196—22no—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
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—
226C8×D61—28no—
227C8×D61—82no—
228C23×C61—21no—
229C23×C61C2×C22⋊C411yescharacteristic
230C22×D61—22no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61—82no—
234C22×D61C2×C22⋊C412yes—
235Q8×C61—21no—
236Q8×C61—21no—
237D4×C61—21no—
238D4×C61—42no—
239D4×C61#13112yescomplemented
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—42no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C23×C411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C22⋊D61—24no—
259C22⋊D61—44no—
260C22⋊D61—82no—
261C2×Dic61—22no—
262C2×Dic61—82no—
263C2×Dic61C2×C22⋊C412yes—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266D244—22no—
267D244—22no—
268D244—44no—
269D244—42no—
270D244—81no—
271D244#10414yes—
272C4×D61—24no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—42no—
278C4×D61—42no—
279C4×D61—82no—
280C4×D61—82no—
281Dic122—22no—
282Dic122—81no—
283C8×C61—24no—
284C8×C61—42no—
285#92418—22no—
286#92418—22no—
287#92418—41no—
288#92418—41no—
289#17517—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61C22×C22⋊C411yescharacteristic
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—42no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—88no—
303C2×D61C2×#10414yes—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—41no—
308C4×C61—41no—
309C4×C61—41no—
310C4×C61C2×#10412yes—
311C4×C61C2×#13111yescharacteristic
312C4×C61#56811yescharacteristic
313C4×C61#57611yescharacteristic
314C4×C61#57711yescharacteristic
315Dic61—22no—
316Dic61—22no—
317Dic61—22no—
318Dic61—42no—
319#3481—611no—
320#2990—611no—
321C4×#131—611no—
322#1494—612no—
323#1492—611no—
324#1490—612no—
325#1489D6111yescharacteristic, complemented
326#1484—611no—
327#1483—612no—
328#1476—611no—
329C2×D4⋊C8—612no—
330C2×C61—42no—
331C2×C61—41no—
332C2×C61#288011yescharacteristic
333C2×C61#289711yescharacteristic
334C2×C61C22×#10411yescharacteristic
335D61—24no—
336D61—82no—
337C2×#131—1221no—
338#559—611no—
339#554—611no—
340#554—611no—
341#553—612no—
342#553—1221no—
343#551—1222no—
344#546—612no—
345#546—1221no—
346D42—611no—
347#532—611no—
348#529—612no—
349#529—1221no—
350#525—1222no—
351C4×#41—611no—
352C4×#41C2×D6111yescharacteristic
353C2×C4×D4—612no—
354C2×C4×D4—1221no—
355#446—1221no—
356#435—1222no—
357C8×D4—1221no—
358#432—612no—
359#431—611no—
360#423C2×D6111yescharacteristic, complemented
361C2×C4⋊C8—612no—
362#405—1222no—
363#405—1221no—
364C4×M16—612no—
365C4×M16C2×D6111yescharacteristic, complemented
366C2×C4×C8—611no—
367C2×C4×C8—612no—
368Q8⋊C8—611no—
369Q8⋊C8—611no—
370Q8⋊C8C2×D6112yescomplemented
371D4⋊C8—611no—
372D4⋊C8—611no—
373D4⋊C8—612no—
374D4⋊C8—612no—
375D4⋊C8—612no—
376D4⋊C8—612no—
377D4⋊C8C2×D6112yes—
378C61#1751711yescharacteristic, complemented, Sylow
379C2×#41—611no—
380C2×#41—1221no—
381C2×#41C4×D6111yescharacteristic
382C22×D4—612no—
383C22×D4—1221no—
384C23×C4—1222no—
385#131—1228no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—1222no—
389C22×C8—1222no—
390C22×C8—1222no—
391C22×C8—1221no—
392C4⋊Q8—611no—
393#127—611no—
394#126—1222no—
395#126—1222no—
396#125—612no—
397#125—1221no—
398#124—612no—
399#124—1221no—
400#124—1221no—
401#123—1222no—
402#123—1222no—
403#123—1222no—
404#123—1222no—
405C22⋊Q8—612no—
406C22⋊Q8—1222no—
407C22⋊Q8—1222no—
408C22⋊Q8—1221no—
409#121—612no—
410#121—612no—
411#121—1222no—
412#121—1222no—
413#121—1221no—
414#121—1221no—
415#120—1222no—
416#120—1222no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8C4×D6111yescharacteristic
420C4×Q8C22×D6111yescharacteristic
421C4×D4—21no—
422C4×D4—21no—
423C4×D4—612no—
424C4×D4—612no—
425C4×D4—612no—
426C4×D4—612no—
427C4×D4—611no—
428C4×D4—611no—
429C4×D4—612no—
430C4×D4—1222no—
431C4×D4—1221no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1222no—
435C4×D4—1222no—
436C4×D4—1221no—
437C4×D4C4×D6111yescharacteristic
438C4×D4C22×D6111yescharacteristic
439#117—21no—
440#117—611no—
441#117—612no—
442#117—1222no—
443#117C4×D6111yescharacteristic
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1221no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1222no—
448C2×C42—21no—
449C2×C42—612no—
450C2×C42C22×D6111yescharacteristic
451C4⋊C8—21no—
452C4⋊C8—611no—
453C4⋊C8—611no—
454C4⋊C8—1222no—
455C4⋊C8C22×D6112yes—
456C22⋊C8—1222no—
457C22⋊C8—1221no—
458#97—21no—
459#97—1222no—
460#97—1222no—
461C4×C8—611no—
462C4×C8—611no—
463C4×C8—1222no—
464C4×C8—1221no—
465C4×C8—1221no—
466C4×C8C22×D6112yes—
467C24—1222no—
468#41—22no—
469#41—41no—
470#41—1224no—
471#41—4881no—
472#41—4881no—
473C2×Q8—611no—
474C2×Q8—1221no—
475C2×Q8C2×C4×D6111yescharacteristic
476C2×D4—21no—
477C2×D4—612no—
478C2×D4—612no—
479C2×D4—611no—
480C2×D4—1222no—
481C2×D4—1222no—
482C2×D4—1224no—
483C2×D4—1222no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—2444no—
487C2×D4—2442no—
488C2×D4C2×C4×D6111yescharacteristic
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—4882no—
500C22×C4—4882no—
501C22×C4C2×C4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4D4×D6111yescharacteristic
504M16—24no—
505M16—81no—
506M16—1228no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—1228no—
510C2×C8—1224no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—4882no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—611no—
519C4⋊C4—612no—
520C4⋊C4—612no—
521C4⋊C4—611no—
522C4⋊C4—612no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4C2×C4×D6111yescharacteristic
529C4⋊C4C2×C4×D6111yescharacteristic
530C22⋊C4—21no—
531C22⋊C4—21no—
532C22⋊C4—21no—
533C22⋊C4—612no—
534C22⋊C4—612no—
535C22⋊C4—612no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—612no—
548C42—612no—
549C42—612no—
550C42C2×C4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C42D4×D6111yescharacteristic
553C42C23×D6111yescharacteristic
554C23—21no—
555C23—612no—
556C23—1222no—
557C23—4882no—
558C23—2444no—
559C23—2444no—
560C23C22⋊C4×D6111yescharacteristic
561Q8—21no—
562Q8—21no—
563Q8—1222no—
564Q8—4881no—
565D4—21no—
566D4—42no—
567D4—614no—
568D4—1224no—
569D4—1222no—
570D4—1222no—
571D4—4881no—
572D4—4882no—
573D4—2444no—
574D4—2444no—
575D4—2442no—
576D4#1451978512yescomplemented
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
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—4882no—
599C2×C4—4882no—
600C2×C4—4882no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C2×C4C2×D4×D6111yescharacteristic
609C8—24no—
610C8—42no—
611C8—1222no—
612C8—1222no—
613C8—2441no—
614C8—2441no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—614no—
622C22—1222no—
623C22—1222no—
624C22—4888no—
625C22—2442no—
626C22—2442no—
627C22—2442no—
628C22C2×C22⋊C4×D6111yescharacteristic
629C4—21no—
630C4—21no—
631C4—21no—
632C4—41no—
633C4—41no—
634C4—41no—
635C4—1222no—
636C4—1222no—
637C4—1222no—
638C4—2442no—
639C4#104×D6112yes—
640C4C2×#1451978511yescharacteristic
641C4#1684052811yescharacteristic
642C4#1684056611yescharacteristic
643C4#1684057111yescharacteristic
644C2—41no—
645C2—42no—
646C2—1224no—
647C2—4882no—
648C2#2145426911yescharacteristic
649C2#2145434311yescharacteristic
650C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1489C2
socleC22×C61C22×C22⋊C4
prime radical#21459429C2
cosocleC42×C61C24
hypercenter#1489D61
hypocenterC61#17517
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. #21444278C2semidirect
5. #21444269C2semidirect
6. #21444260C2semidirect
7. #21444237C2semidirect
8. #21444231C2semidirect
9. #21444201C2semidirect
10. D4⋊C8×D61C2semidirect
11. C61×#1489C2semidirect
32. #16839818C22semidirect
33. #16839815C22semidirect
34. C4⋊C8×D61C22semidirect
37. C4×#11758189C22semidirect
38. C2×C4×#92418C22semidirect
39. C4×C8×D61C22semidirect
40. Q8⋊#92418C22semidirect
41. Q8⋊#92418C22semidirect
42. #16839491C22semidirect
43. #16839491C22semidirect
46. #16839489C22semidirect
47. #16839489C22semidirect
49. C61×#423C22semidirect
50. C4×M16×C61C22semidirect
51. C61×Q8⋊C8C22semidirect
239. D4×C61#131semidirect
325. #1489D61semidirect
360. #423C2×D61semidirect
365. C4×M16C2×D61semidirect
370. Q8⋊C8C2×D61semidirect
378. C61#17517semidirect
576. D4#14519785semidirect

Maximal subgroups

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

subgroupindex
#214594292
#214549452
C4×#145197852
#214442782
#214442692
#214442602
#214442372
#214442312
#214442012
D4⋊C8×D612
C61×#14892
#1751761