Finite Group Databasebeta

← #29646981

Subgroups — #29646981

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
2C4×#14519784C211yesmaximal, characteristic, complemented
3#21444277C212yesmaximal, complemented
4#21444269C211yesmaximal, characteristic, complemented
5#21444261C212yesmaximal, complemented
6#21444256C211yesmaximal, characteristic, complemented
7#21444235C211yesmaximal, characteristic, complemented
8#21444230C212yesmaximal, complemented
9#21444199C211yesmaximal, characteristic, complemented
10C2×#16839489C212yesmaximal, complemented
11C61×#2990C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13#16840497C411yescharacteristic
14#16840472C411yescharacteristic
15#16840469C411yescharacteristic
16#16840458C412yes—
17#16840456—21no—
18#16840436—22no—
19#16840408C412yes—
20#16840400—21no—
21D4×D244C411yescharacteristic
22#16840312C2211yescharacteristic
23#16840299—21no—
24#16840296C2212yes—
25#16840271—22no—
26C4×#11758222C2211yescharacteristic
27C2×C4×D244C2212yes—
28C4×D4×D61—21no—
29#16839718—21no—
30#16839718—22no—
31C4×#11758194C2212yescomplemented
32C4×#11758194C2211yescharacteristic, complemented
33C2×C4×#92418C2212yescomplemented
34C2×C4×#92418C2211yescharacteristic, complemented
35#16839492C2211yescharacteristic, complemented
36#16839492C2211yescharacteristic, complemented
37#16839492C2212yescomplemented
38#16839489C2211yescharacteristic
39#16839489C2212yes—
40#16839489C2211yescharacteristic
41#16839489C2212yescomplemented
42#16839489C2212yescomplemented
43#16839489C2212yescomplemented
44#16839489C2212yescomplemented
45C4×#41×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#435—22no—
48C8×D4×C61—21no—
49C61×#432C2212yescomplemented
50C61×#431C2211yescharacteristic, complemented
51C61×#423C2211yescharacteristic, complemented
52C2×C4⋊C8×C61C2212yescomplemented
53C2×#11758222—21no—
54C2×#11758222C2×C411yescharacteristic
55C22×D244C2×C412yes—
56C2×D4×D61—21no—
57C22×C4×D61—22no—
58#14519784—28no—
59C2×#11758194—22no—
60C2×#11758194—21no—
61C22×#92418—22no—
62C22×#92418—21no—
63#14519777C2×C411yescharacteristic
64#14519774C2×C411yescharacteristic
65#14519772—22no—
66#14519771—22no—
67#14519769C2×C412yes—
68#14519768—21no—
69#14519766—21no—
70#14519765—21no—
71#14519765C2×C412yes—
72#14519762—22no—
73#14519762—22no—
74#14519760—22no—
75#14519759—22no—
76#14519756C2×C412yes—
77#14519755—22no—
78#14519754—22no—
79#14519753—21no—
80#14519752—21no—
81#14519751—21no—
82#14519751C2×C412yes—
83#14519751C2×C412yes—
84#14519750—22no—
85#14519749—22no—
86#14519747—22no—
87#14519745—22no—
88C4×Dic122C2×C411yescharacteristic
89C4×Dic122C2311yescharacteristic
90C4×D244—22no—
91C4×D244—21no—
92C4×D244—22no—
93C4×D244C2×C412yes—
94C4×D244C2×C412yes—
95C4×D244C2×C411yescharacteristic
96C4×D244C2311yescharacteristic
97C4×D244C2312yes—
98D4×Dic61—21no—
99#14519739—21no—
100C4×C22⋊D61—21no—
101C4×C22⋊D61—22no—
102C4×C22⋊D61—21no—
103#14519737—22no—
104#14519736C2×C411yescharacteristic
105#14519734—21no—
106#14519734—22no—
107C2×C4⋊Dic61C2×C412yes—
108C4⋊C4×D61—21no—
109C2×#11758198—22no—
110C22⋊C4×D61—22no—
111C42×D61—21no—
112#14519688—21no—
113#14519688—22no—
114C4×#92418—21no—
115C4×#92418—22no—
116C4×#92418C2311yescharacteristic
117C4×#92418C2312yes—
118C4×#92418C2311yescharacteristic
119C2×#41×C61D411yescharacteristic
120C2×M16×C61—21no—
121C22×C8×C61—22no—
122C4×Q8×C61C2×C411yescharacteristic
123C4×Q8×C61D411yescharacteristic
124C4×D4×C61C2×C412yes—
125C4×D4×C61C2×C411yescharacteristic
126C4×D4×C61D411yescharacteristic
127C4×D4×C61D412yes—
128#117×C61D411yescharacteristic
129#117×C61D412yes—
130C2×C42×C61C2312yes—
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—22no—
133C4⋊C8×C61—21no—
134C4⋊C8×C61C2311yescharacteristic
135C4⋊C8×C61C2312yes—
136C4⋊C8×C61C2311yescharacteristic
137C22⋊C8×C61—21no—
138C22⋊C8×C61—22no—
139#97×C61—22no—
140C4×C8×C61—21no—
141C23×D61—22no—
142#41×C61—24no—
143C2×Q8×C61C22⋊C411yescharacteristic
144C2×D4×C61C22⋊C412yes—
145C2×D4×C61C22⋊C411yescharacteristic
146C22×C4×C61C22×C412yes—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—22no—
155C22×Dic61—22no—
156D4×D61—44no—
157#11758222—22no—
158#11758222—41no—
159#11758222—81no—
160#11758222—81no—
161C2×D244—24no—
162C2×D244—21no—
163C2×D244—22no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244C22×C412yes—
167C2×D244C22×C411yescharacteristic
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—22no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—82no—
175C2×C4×D61—82no—
176C2×Dic122—21no—
177C2×Dic122C22×C411yescharacteristic
178M16×C61—81no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—22no—
182C2×C8×C61—82no—
183C4⋊C4×C61C22⋊C411yescharacteristic
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22⋊C412yes—
187C22⋊C4×C61C22⋊C412yes—
188C22⋊C4×C61C22⋊C412yes—
189C22⋊C4×C61C22⋊C412yes—
190C42×C61C22×C412yes—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D412yes—
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D412yes—
196C42×C61C2411yescharacteristic
197C22⋊Dic61—22no—
198C22⋊Dic61—21no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—21no—
204#11758198—22no—
205#11758198—22no—
206#11758198—21no—
207#11758198—22no—
208C4⋊Dic61—21no—
209C4⋊Dic61—22no—
210C4⋊Dic61C22×C411yescharacteristic
211C4⋊Dic61C22×C412yes—
212C4⋊Dic61C22×C411yescharacteristic
213#11758196—22no—
214#11758196—21no—
215#11758196—22no—
216#11758196—22no—
217#11758196—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221#11758194—28no—
222#11758194—24no—
223C2×#92418—28no—
224C2×#92418—24no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—22no—
228C23×C61C2×C22⋊C411yescharacteristic
229C23×C61C2×C22⋊C412yes—
230C22×D61—21no—
231C22×D61—22no—
232C22×D61—44no—
233C22×D61—44no—
234C22×D61—82no—
235Q8×C61—22no—
236D4×C61—24no—
237D4×C61#10414yes—
238C2×C4×C61—24no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×D411yescharacteristic
257C22⋊D61—42no—
258C22⋊D61—44no—
259C22⋊D61—82no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—82no—
265D244—22no—
266D244—21no—
267D244—22no—
268D244—44no—
269D244—42no—
270D244—81no—
271D244#13112yescomplemented
272C4×D61—42no—
273C4×D61—41no—
274C4×D61—42no—
275C4×D61—41no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—82no—
279C4×D61—82no—
280Dic122—21no—
281Dic122—21no—
282Dic122—81no—
283C8×C61—41no—
284C8×C61—42no—
285C8×C61—41no—
286#92418—22no—
287#92418—24no—
288#92418—22no—
289#17517—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61C2×#10414yes—
296C22×C61C22×C22⋊C411yescharacteristic
297C2×D61—22no—
298C2×D61—41no—
299C2×D61—41no—
300C2×D61—42no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—88no—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61—21no—
307C4×C61—22no—
308C4×C61—21no—
309C4×C61—22no—
310C4×C61C2×#10412yes—
311C4×C61C2×#13111yescharacteristic
312C4×C61#56811yescharacteristic
313C4×C61#57611yescharacteristic
314C4×C61#57711yescharacteristic
315Dic61—41no—
316Dic61—42no—
317Dic61—41no—
318Dic61—41no—
319#3481—611no—
320#2990D6111yescharacteristic, complemented
321C4×#131—611no—
322#1494—612no—
323#1492—611no—
324#1490—612no—
325#1489—611no—
326#1484—611no—
327#1483—612no—
328#1476—611no—
329C2×D4⋊C8—612no—
330C2×C61—24no—
331C2×C61—41no—
332C2×C61#288011yescharacteristic
333C2×C61#289711yescharacteristic
334C2×C61C22×#10411yescharacteristic
335D61—42no—
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—21no—
356#435—22no—
357C8×D4—21no—
358#432C2×D6112yescomplemented
359#431C2×D6111yescharacteristic, complemented
360#423C2×D6111yescharacteristic, complemented
361C2×C4⋊C8C2×D6112yescomplemented
362#405—1222no—
363#405—1221no—
364C4×M16—612no—
365C4×M16—611no—
366C2×C4×C8—612no—
367C2×C4×C8—611no—
368Q8⋊C8—611no—
369Q8⋊C8—611no—
370Q8⋊C8—612no—
371D4⋊C8—611no—
372D4⋊C8—612no—
373D4⋊C8—611no—
374D4⋊C8—612no—
375D4⋊C8—612no—
376D4⋊C8—612no—
377D4⋊C8—612no—
378C61#1751711yescharacteristic, complemented, Sylow
379C2×#41—611no—
380C2×#41—1221no—
381C2×#41D24411yescharacteristic
382C22×D4—612no—
383C22×D4—1221no—
384C23×C4—1222no—
385#131—1228no—
386C2×M16—21no—
387C2×M16—1222no—
388C2×M16—1221no—
389C22×C8—22no—
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—1221no—
407C22⋊Q8—1222no—
408C22⋊Q8—1222no—
409#121—612no—
410#121—612no—
411#121—1222no—
412#121—1221no—
413#121—1221no—
414#121—1222no—
415#120—1222no—
416#120—1222no—
417C4×Q8—611no—
418C4×Q8—611no—
419C4×Q8C4×D6111yescharacteristic
420C4×Q8C22⋊D6111yescharacteristic
421C4×D4—612no—
422C4×D4—612no—
423C4×D4—611no—
424C4×D4—611no—
425C4×D4—612no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4—1222no—
429C4×D4—1221no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4C4×D6111yescharacteristic
436C4×D4C4×D6112yes—
437C4×D4C22⋊D6111yescharacteristic
438C4×D4C22⋊D6112yes—
439#117—611no—
440#117—1221no—
441#117—1222no—
442#117D24412yes—
443#117D24411yescharacteristic
444C2×C4⋊C4—612no—
445C2×C4⋊C4—1221no—
446C2×C22⋊C4—1222no—
447C2×C22⋊C4—1222no—
448C2×C42—1221no—
449C2×C42C22×D6112yes—
450C2×C42C22×D6111yescharacteristic
451C4⋊C8—22no—
452C4⋊C8—21no—
453C4⋊C8C22×D6111yescharacteristic
454C4⋊C8C22×D6112yes—
455C4⋊C8C22×D6111yescharacteristic
456C22⋊C8—21no—
457C22⋊C8—22no—
458#97—22no—
459#97—1221no—
460#97—1222no—
461C4×C8—21no—
462C4×C8—611no—
463C4×C8—612no—
464C4×C8—611no—
465C4×C8—1221no—
466C4×C8—1222no—
467C24—1222no—
468#41—24no—
469#41—1222no—
470#41—4881no—
471#41—4881no—
472#41—2441no—
473C2×Q8—611no—
474C2×Q8—1221no—
475C2×Q8#1175819811yescharacteristic
476C2×D4—611no—
477C2×D4—612no—
478C2×D4—1224no—
479C2×D4—1222no—
480C2×D4—1221no—
481C2×D4—1222no—
482C2×D4—1221no—
483C2×D4—1222no—
484C2×D4—1221no—
485C2×D4—2442no—
486C2×D4—2444no—
487C2×D4#1175819812yes—
488C2×D4#1175819811yescharacteristic
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—4882no—
497C22×C4—4882no—
498C22×C4C2×C4×D6112yes—
499C22×C4C2×C4×D6111yescharacteristic
500C22×C4C2×D24412yes—
501C22×C4C2×D24411yescharacteristic
502C22×C4C2×C22⋊D6112yes—
503C22×C4C2×C22⋊D6111yescharacteristic
504M16—81no—
505M16—1228no—
506M16—1224no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—22no—
510C2×C8—82no—
511C2×C8—1228no—
512C2×C8—1224no—
513C2×C8—1221no—
514C2×C8—1222no—
515C2×C8—1221no—
516C4⋊C4—611no—
517C4⋊C4—612no—
518C4⋊C4—611no—
519C4⋊C4—1221no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1222no—
525C4⋊C4—1222no—
526C4⋊C4#1175819811yescharacteristic
527C4⋊C4#1175819811yescharacteristic
528C4⋊C4#1175819812yes—
529C4⋊C4#1175819812yes—
530C22⋊C4—1222no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1221no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4#1175819812yes—
542C22⋊C4#1175819812yes—
543C22⋊C4#1175819812yes—
544C42—1221no—
545C42—1221no—
546C42—1221no—
547C42C2×C4×D6112yes—
548C42C2×C4×D6111yescharacteristic
549C42C2×D24411yescharacteristic
550C42C2×D24412yes—
551C42C2×C22⋊D6111yescharacteristic
552C42C2×C22⋊D6112yes—
553C42C23×D6111yescharacteristic
554C23—1221no—
555C23—1222no—
556C23—4882no—
557C23—2444no—
558C23—2444no—
559C23C2×#1175819811yescharacteristic
560C23C2×#1175819812yes—
561Q8—22no—
562Q8—1221no—
563Q8—1221no—
564Q8—4881no—
565D4—24no—
566D4—612no—
567D4—1222no—
568D4—1221no—
569D4—1222no—
570D4—4881no—
571D4—4882no—
572D4—2444no—
573D4—2442no—
574D4—2444no—
575D4—2442no—
576D4#1451970314yes—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—4882no—
592C2×C4—4882no—
593C2×C4—4882no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2441no—
597C2×C4—2442no—
598C2×C4—2441no—
599C2×C4—2442no—
600C2×C4C2×#1175819811yescharacteristic
601C2×C4C2×#1175819812yes—
602C2×C4C2×#1175819811yescharacteristic
603C2×C4C2×#1175819812yes—
604C2×C4C2×#1175819812yes—
605C2×C4C2×#1175819811yescharacteristic
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4C22×D24411yescharacteristic
609C8—41no—
610C8—42no—
611C8—41no—
612C8—1222no—
613C8—1224no—
614C8—1222no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—21no—
620C22—1222no—
621C22—4888no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22—2441no—
626C22—2441no—
627C22C2×#1451970314yes—
628C22C22×#1175819811yescharacteristic
629C4—22no—
630C4—21no—
631C4—21no—
632C4—22no—
633C4—21no—
634C4—22no—
635C4—2441no—
636C4—2442no—
637C4—2441no—
638C4—2441no—
639C4C2×#1451970312yes—
640C4#131×D6111yescharacteristic
641C4#1684052711yescharacteristic
642C4#1684056511yescharacteristic
643C4#1684057011yescharacteristic
644C2—24no—
645C2—41no—
646C2—4882no—
647C2—2442no—
648C2#2145426611yescharacteristic
649C2#2145434211yescharacteristic
650C2C22×#1451970311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#2990C2
socleC22×C61C22×C22⋊C4
prime radical#21459429C2
cosocleC42×C61C24
hypercenter#2990D61
hypocenterC61#17517
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2990

Complements

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

NH ≅ G/Ntype
2. C4×#14519784C2semidirect
3. #21444277C2semidirect
4. #21444269C2semidirect
5. #21444261C2semidirect
6. #21444256C2semidirect
7. #21444235C2semidirect
8. #21444230C2semidirect
9. #21444199C2semidirect
10. C2×#16839489C2semidirect
11. C61×#2990C2semidirect
31. C4×#11758194C22semidirect
32. C4×#11758194C22semidirect
33. C2×C4×#92418C22semidirect
34. C2×C4×#92418C22semidirect
35. #16839492C22semidirect
36. #16839492C22semidirect
37. #16839492C22semidirect
41. #16839489C22semidirect
42. #16839489C22semidirect
43. #16839489C22semidirect
44. #16839489C22semidirect
49. C61×#432C22semidirect
50. C61×#431C22semidirect
51. C61×#423C22semidirect
52. C2×C4⋊C8×C61C22semidirect
271. D244#131semidirect
320. #2990D61semidirect
358. #432C2×D61semidirect
359. #431C2×D61semidirect
360. #423C2×D61semidirect
361. C2×C4⋊C8C2×D61semidirect
378. C61#17517semidirect

Maximal subgroups

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

subgroupindex
#214594292
C4×#145197842
#214442772
#214442692
#214442612
#214442562
#214442352
#214442302
#214441992
C2×#168394892
C61×#29902
#1751761