Finite Group Databasebeta

← #29646987

Subgroups — #29646987

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#21454944C211yesmaximal, characteristic, complemented
3#21444280C212yesmaximal, complemented
4#21444271C211yesmaximal, characteristic, complemented
5#21444263C212yesmaximal, complemented
6#21444259C211yesmaximal, characteristic, complemented
7#21444237C211yesmaximal, characteristic, complemented
8#21444233C212yesmaximal, complemented
9#21444202C211yesmaximal, characteristic, complemented
10C2×#16839491C212yesmaximal, complemented
11C4×#131×C61C211yesmaximal, characteristic, complemented
12#16840497C411yescharacteristic
13#16840472C411yescharacteristic
14#16840469C411yescharacteristic
15#16840458C412yes—
16#16840456—21no—
17#16840436—22no—
18#16840408C412yes—
19#16840400—21no—
20D4×D244C411yescharacteristic
21#16840312C2211yescharacteristic
22#16840299—21no—
23#16840296C2212yes—
24#16840271—22no—
25C4×#11758222C2211yescharacteristic
26C2×C4×D244C2212yes—
27C4×D4×D61—21no—
28#16839893—21no—
29#16839834—22no—
30D4×#92418—21no—
31#16839819C2212yescomplemented
32#16839815C2211yescharacteristic, complemented
33#16839786C2211yescharacteristic, complemented
34C2×C4⋊#92418C2212yescomplemented
35#16839494C2211yescharacteristic, complemented
36#16839494C2211yescharacteristic, complemented
37#16839494C2212yescomplemented
38#16839491C2211yescharacteristic
39#16839491C2212yes—
40#16839491C2211yescharacteristic
41#16839491C2212yescomplemented
42#16839491C2212yescomplemented
43#16839491C2212yescomplemented
44#16839491C2212yescomplemented
45C2×#131×C61—21no—
46C4×#41×C61C2211yescharacteristic
47C61×#405—21no—
48C61×#405—22no—
49C4×M16×C61C2212yescomplemented
50C4×M16×C61C2211yescharacteristic, complemented
51C2×C4×C8×C61C2212yescomplemented
52C2×C4×C8×C61C2211yescharacteristic, complemented
53C2×#11758222—21no—
54C2×#11758222C2×C411yescharacteristic
55C22×D244C2×C412yes—
56C2×D4×D61—21no—
57C22×C4×D61—22no—
58C2×#11758194—21no—
59C22×#92418—22no—
60#14519777C2×C411yescharacteristic
61#14519774C2×C411yescharacteristic
62#14519772—22no—
63#14519771—22no—
64#14519769C2×C412yes—
65#14519768—21no—
66#14519766—21no—
67#14519765—21no—
68#14519765C2×C412yes—
69#14519762—22no—
70#14519762—22no—
71#14519760—22no—
72#14519759—22no—
73#14519756C2×C412yes—
74#14519755—22no—
75#14519754—22no—
76#14519753—21no—
77#14519752—21no—
78#14519751—21no—
79#14519751C2×C412yes—
80#14519751C2×C412yes—
81#14519750—22no—
82#14519749—22no—
83#14519747—22no—
84#14519745—22no—
85C4×Dic122C2×C411yescharacteristic
86C4×Dic122C2311yescharacteristic
87C4×D244—22no—
88C4×D244—21no—
89C4×D244—22no—
90C4×D244C2×C412yes—
91C4×D244C2×C412yes—
92C4×D244C2×C411yescharacteristic
93C4×D244C2311yescharacteristic
94C4×D244C2312yes—
95D4×Dic61—21no—
96#14519739—21no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99C4×C22⋊D61—21no—
100#14519737—22no—
101#14519736C2×C411yescharacteristic
102#14519734—21no—
103#14519734—22no—
104C2×C4⋊Dic61C2×C412yes—
105C4⋊C4×D61—21no—
106C2×#11758198—22no—
107C22⋊C4×D61—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×C61C2312yes—
140C4×C8×C61C2311yescharacteristic
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—28no—
179M16×C61—24no—
180C2×C8×C61—28no—
181C2×C8×C61—24no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—22no—
185C4⋊C4×C61C22⋊C411yescharacteristic
186C4⋊C4×C61C22⋊C412yes—
187C4⋊C4×C61C22⋊C411yescharacteristic
188C4⋊C4×C61C22⋊C412yes—
189C22⋊C4×C61C22⋊C412yes—
190C22⋊C4×C61C22⋊C412yes—
191C22⋊C4×C61C22⋊C412yes—
192C42×C61C22×C412yes—
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D412yes—
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D412yes—
198C42×C61C2411yescharacteristic
199C22⋊Dic61—22no—
200C22⋊Dic61—21no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—21no—
206#11758198—22no—
207#11758198—22no—
208#11758198—21no—
209#11758198—22no—
210C4⋊Dic61—21no—
211C4⋊Dic61—22no—
212C4⋊Dic61C22×C411yescharacteristic
213C4⋊Dic61C22×C412yes—
214C4⋊Dic61C22×C411yescharacteristic
215#11758196—22no—
216#11758196—21no—
217#11758196—22no—
218#11758196—22no—
219#11758196—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223#11758194—81no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—22no—
227C2×#92418—82no—
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—22no—
284C8×C61—24no—
285C8×C61—22no—
286#92418—41no—
287#92418—42no—
288#92418—41no—
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#2990—611no—
321C4×#131D6111yescharacteristic, complemented
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—21no—
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#423—611no—
361C2×C4⋊C8—612no—
362#405—22no—
363#405—21no—
364C4×M16C2×D6112yescomplemented
365C4×M16C2×D6111yescharacteristic, complemented
366C2×C4×C8C2×D6112yescomplemented
367C2×C4×C8C2×D6111yescharacteristic, complemented
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×#41C22⋊D6111yescharacteristic
382C22×D4—612no—
383C22×D4—1221no—
384C23×C4—1222no—
385#131—28no—
386C2×M16—22no—
387C2×M16—21no—
388C2×M16—1221no—
389C22×C8—22no—
390C22×C8—21no—
391C22×C8—1222no—
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×Q8D24411yescharacteristic
421C4×D4—612no—
422C4×D4—612no—
423C4×D4—611no—
424C4×D4—611no—
425C4×D4—612no—
426C4×D4—1221no—
427C4×D4—1222no—
428C4×D4—1221no—
429C4×D4—1221no—
430C4×D4—1221no—
431C4×D4—1222no—
432C4×D4—1222no—
433C4×D4—1221no—
434C4×D4—1222no—
435C4×D4C4×D6111yescharacteristic
436C4×D4C4×D6112yes—
437C4×D4D24411yescharacteristic
438C4×D4D24412yes—
439#117—611no—
440#117—1222no—
441#117—1221no—
442#117C22⋊D6112yes—
443#117C22⋊D6111yescharacteristic
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—611no—
452C4⋊C8—612no—
453C4⋊C8—611no—
454C4⋊C8—1222no—
455C4⋊C8—1221no—
456C22⋊C8—1221no—
457C22⋊C8—1222no—
458#97—21no—
459#97—22no—
460#97—1222no—
461C4×C8—21no—
462C4×C8—22no—
463C4×C8—1221no—
464C4×C8C22×D6111yescharacteristic
465C4×C8C22×D6112yes—
466C4×C8C22×D6111yescharacteristic
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—28no—
505M16—24no—
506M16—4881no—
507C2×C8—28no—
508C2×C8—24no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1222no—
515C2×C8—4882no—
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#1451970514yes—
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×#1175819812yes—
603C2×C4C2×#1175819811yescharacteristic
604C2×C4C2×#1175819812yes—
605C2×C4C2×#1175819811yescharacteristic
606C2×C4C22×C4×D6111yescharacteristic
607C2×C4C22×C22⋊D6111yescharacteristic
608C2×C4C22×D24411yescharacteristic
609C8—22no—
610C8—24no—
611C8—22no—
612C8—2441no—
613C8—2442no—
614C8—2441no—
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×#1451970514yes—
628C22C22×#1175819811yescharacteristic
629C4—22no—
630C4—21no—
631C4—21no—
632C4—22no—
633C4—21no—
634C4—22no—
635C4—2441no—
636C4—2442no—
637C4—2441no—
638C4—2441no—
639C4C2×#1451970512yes—
640C4#131×D6111yescharacteristic
641C4#1684052711yescharacteristic
642C4#1684056711yescharacteristic
643C4#1684057411yescharacteristic
644C2—24no—
645C2—41no—
646C2—4882no—
647C2—2442no—
648C2#2145426611yescharacteristic
649C2#2145434611yescharacteristic
650C2C22×#1451970511yescharacteristic
subgroupG/H
centerC2×C4C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC4×#131×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459429C2
cosocleC42×C61C24
hypercenterC4×#131D61
hypocenterC61#17517
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. #21444271C2semidirect
5. #21444263C2semidirect
6. #21444259C2semidirect
7. #21444237C2semidirect
8. #21444233C2semidirect
9. #21444202C2semidirect
10. C2×#16839491C2semidirect
11. C4×#131×C61C2semidirect
31. #16839819C22semidirect
32. #16839815C22semidirect
33. #16839786C22semidirect
34. C2×C4⋊#92418C22semidirect
35. #16839494C22semidirect
36. #16839494C22semidirect
37. #16839494C22semidirect
41. #16839491C22semidirect
42. #16839491C22semidirect
43. #16839491C22semidirect
44. #16839491C22semidirect
49. C4×M16×C61C22semidirect
50. C4×M16×C61C22semidirect
51. C2×C4×C8×C61C22semidirect
52. C2×C4×C8×C61C22semidirect
271. D244#131semidirect
321. C4×#131D61semidirect
364. C4×M16C2×D61semidirect
365. C4×M16C2×D61semidirect
366. C2×C4×C8C2×D61semidirect
367. 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
#214549442
#214442802
#214442712
#214442632
#214442592
#214442372
#214442332
#214442022
C2×#168394912
C4×#131×C612
#1751761