Finite Group Databasebeta

← #29646923

Subgroups — #29646923

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459377C211yesmaximal, characteristic
2#21454780C211yesmaximal, characteristic, complemented
3#21454388C211yesmaximal, characteristic, complemented
4#21444281C212yesmaximal, complemented
5#21444273C211yesmaximal, characteristic, complemented
6#21444255C212yesmaximal, complemented
7#21444229C212yesmaximal, complemented
8#21444223C211yesmaximal, characteristic, complemented
9#21444203C212yesmaximal, complemented
10C2×D4⋊#92418C211yesmaximal, characteristic, complemented
11C61×#1486C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840482C411yescharacteristic
14#16840477C411yescharacteristic
15#16840469C411yescharacteristic
16#16840458—21no—
17#16840452—21no—
18#16840447C412yes—
19#16840438—22no—
20#16840416C411yescharacteristic
21#16840406C412yes—
22#16840319C2211yescharacteristic
23#16840294C2212yes—
24#16840292—22no—
25#16840273—21no—
26C4×#11758222C2211yescharacteristic
27C4×#11758224C2212yes—
28C2×#14519736—21no—
29#16839889—21no—
30#16839839—21no—
31#16839830—22no—
32#16839813C2212yescomplemented
33#16839790—21no—
34#16839787—22no—
35#16839784C2212yescomplemented
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61C2212yescomplemented
38C4×#11758189C2212yescomplemented
39C2×C4×#92418C2211yescharacteristic
40#16839494C2212yescomplemented
41#16839494C2212yescomplemented
42#16839492C2212yescomplemented
43#16839492C2212yescomplemented
44D4⋊#92418C2211yescharacteristic
45D4⋊#92418C2211yescharacteristic
46D4⋊#92418C2211yescharacteristic
47D4⋊#92418C2211yescharacteristic
48C2×C4×D4×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C61×#423C2211yescharacteristic
51C61×D4⋊C8C2212yes—
52C61×D4⋊C8C2212yes—
53C2×#11758222D411yescharacteristic
54C2×#11758224C2×C412yes—
55#14519785—44no—
56C2×#11758189—22no—
57C2×#11758189—22no—
58C22×#92418—21no—
59C22×#92418—21no—
60C2×C8×D61—22no—
61#14519777C2×C411yescharacteristic
62#14519777C2×C411yescharacteristic
63#14519772—22no—
64#14519770—22no—
65#14519770—22no—
66#14519769—21no—
67#14519768C2×C412yes—
68#14519767—22no—
69#14519764C2×C412yes—
70#14519763—22no—
71#14519761—22no—
72#14519760—22no—
73#14519759—22no—
74#14519758—22no—
75#14519757C2×C412yes—
76#14519756—21no—
77#14519756—21no—
78#14519756—21no—
79#14519756C2×C412yes—
80#14519754—22no—
81#14519751—21no—
82#14519751—21no—
83#14519749—22no—
84#14519748C2×C412yes—
85#14519746—22no—
86C4×Dic122—21no—
87C4×Dic122C2×C411yescharacteristic
88C4×Dic122D411yescharacteristic
89C4×Dic122C2312yes—
90#14519742C2×C412yes—
91C4×D244C2×C411yescharacteristic
92C4×D244D411yescharacteristic
93D4×Dic61C2×C412yes—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96C4×C22⋊D61C2×C412yes—
97C4×C22⋊D61D412yes—
98#14519737—22no—
99#14519736—21no—
100#14519736—22no—
101#14519736—21no—
102#14519736—21no—
103#14519736—22no—
104#14519736D411yescharacteristic
105#14519735C2×C412yes—
106#14519734D412yes—
107#14519733—22no—
108C2×C4⋊Dic61—21no—
109C2×C4⋊Dic61—21no—
110C2×C22⋊Dic61—22no—
111C2×C4×Dic61—22no—
112C42×D61C2312yes—
113C4⋊#92418—21no—
114C4⋊#92418C2311yescharacteristic
115C4⋊#92418C2311yescharacteristic
116#14519706—22no—
117#14519706—22no—
118#14519706—22no—
119#14519689—22no—
120#14519687—22no—
121C4×#92418—21no—
122C4×#92418C2311yescharacteristic
123C4×#92418C2311yescharacteristic
124C22×D4×C61C2×C411yescharacteristic
125C23×C4×C61—21no—
126C2×M16×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61C2×C411yescharacteristic
130C4×D4×C61C2×C411yescharacteristic
131C4×D4×C61C2311yescharacteristic
132C4×D4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2312yes—
139C22⋊C8×C61—21no—
140C4×C8×C61C2312yes—
141C24×C61—21no—
142C2×D4×C61—22no—
143C2×D4×C61—21no—
144C2×D4×C61—41no—
145C2×D4×C61C22×C411yescharacteristic
146C2×D4×C61C22×C411yescharacteristic
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—42no—
153C22×C4×C61—42no—
154C22×C4×C61C22×C411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61C22⋊C412yes—
160C22×Dic61—22no—
161C22×Dic61—22no—
162#11758224—24no—
163#11758224—42no—
164#11758222—22no—
165#11758222—22no—
166C2×D244C22⋊C411yescharacteristic
167C2×C4×D61C22×C412yes—
168C2×C4×D61C2×D412yes—
169C2×C4×D61C2×D412yes—
170C2×Dic122—21no—
171C2×Dic122C22⋊C411yescharacteristic
172C2×Dic122C22×C412yes—
173M16×C61—42no—
174C2×C8×C61—22no—
175C2×C8×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61C22×C411yescharacteristic
178C4⋊C4×C61C22×C411yescharacteristic
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C42×C61C22×C411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61C22⋊C412yes—
193#11758198—22no—
194#11758198—22no—
195#11758198C22⋊C412yes—
196#11758198C22⋊C412yes—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22⋊C411yescharacteristic
203C4⋊Dic61C22⋊C411yescharacteristic
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196C22⋊C412yes—
210#11758196C22⋊C412yes—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61C22×C412yes—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D412yes—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—42no—
222C2×#92418—42no—
223#11758189—44no—
224#11758189—44no—
225C8×D61—44no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61—81no—
231C23×C61C2×C22⋊C411yescharacteristic
232C22×D61C2×C22⋊C412yes—
233D4×C61—21no—
234D4×C61—21no—
235D4×C61—21no—
236D4×C61—42no—
237D4×C61#13112yescomplemented
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—42no—
249C2×C4×C61—42no—
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61AGL(1,ℤ/8)12yes—
254C2×C4×C61#13712yes—
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—22no—
263C2×Dic61—44no—
264C2×Dic61—82no—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C412yes—
268D244—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—42no—
274Dic122—22no—
275Dic122—22no—
276Dic122—22no—
277Dic122—81no—
278C8×C61—42no—
279C8×C61—42no—
280#92418—22no—
281#92418—41no—
282#92418—41no—
283#92418—41no—
284#17512—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—22no—
288C22×C61—41no—
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—84no—
294C22×C61#41712yes—
295C22×C61C22×C22⋊C411yescharacteristic
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61#41712yes—
305C4×C61C2×#13111yescharacteristic
306C4×C61#56811yescharacteristic
307C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C4×C61C2×#13711yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—22no—
312Dic61—42no—
313Dic61—42no—
314#3475—611no—
315#2965—611no—
316#2906—611no—
317#1495—612no—
318#1493—611no—
319#1489—612no—
320#1486D6111yescharacteristic, complemented
321#1483—612no—
322#1481—611no—
323#1477—612no—
324C2×D4⋊C8—611no—
325C2×C61—22no—
326C2×C61—42no—
327C2×C61—81no—
328C2×C61#288011yescharacteristic
329C2×C61C2×#41711yescharacteristic
330C2×C61#290111yescharacteristic
331D61—42no—
332C2×#131—1221no—
333#556—611no—
334#555—611no—
335#554—611no—
336#553—1221no—
337#552—612no—
338#552—1221no—
339#551—1222no—
340#547—611no—
341#546—612no—
342#533—611no—
343#529—612no—
344#529—1222no—
345#525—1221no—
346C4×#41—611no—
347C4×#41—612no—
348C2×C4×D4C2×D6111yescharacteristic
349C2×#117—1221no—
350#446—1221no—
351#436—1221no—
352#435—1222no—
353#431—612no—
354#431C2×D6111yescharacteristic
355#424—1222no—
356#424—1221no—
357#423—612no—
358#423C2×D6111yescharacteristic
359C2×C4⋊C8—611no—
360C2×C4⋊C8—612no—
361C4×M16—612no—
362C2×C4×C8—611no—
363Q8⋊C8—612no—
364Q8⋊C8—612no—
365Q8⋊C8—612no—
366Q8⋊C8—612no—
367D4⋊C8—611no—
368D4⋊C8—611no—
369D4⋊C8—611no—
370D4⋊C8—611no—
371D4⋊C8C2×D6112yes—
372D4⋊C8C2×D6112yes—
373C61#1751211yescharacteristic, complemented, Sylow
374C2×#41—611no—
375C2×#41—612no—
376C22×D4C4×D6111yescharacteristic
377C23×C4—21no—
378#131—2444no—
379C2×M16—21no—
380C2×M16—1222no—
381C2×M16—1222no—
382C22×C8—1222no—
383C22×C8—1221no—
384C22×C8—1221no—
385C4⋊Q8—611no—
386C4⋊Q8—611no—
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1222no—
392#125—1221no—
393#124—612no—
394#124—1222no—
395#123—1222no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—612no—
401C22⋊Q8—1222no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405#121—612no—
406#121—1222no—
407#121—1221no—
408#121—1221no—
409#120—1222no—
410C4×Q8—611no—
411C4×Q8—611no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—1221no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—612no—
420C4×D4—611no—
421C4×D4—611no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4C4×D6111yescharacteristic
426C4×D4C4×D6111yescharacteristic
427C4×D4C22×D6111yescharacteristic
428C4×D4C22×D6111yescharacteristic
429#117—611no—
430#117—612no—
431#117—612no—
432#117—1222no—
433#117—1222no—
434#117—1222no—
435#117—1221no—
436#117—1221no—
437#117—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4C4×D6111yescharacteristic
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1222no—
443C2×C42—612no—
444C2×C42—1222no—
445C2×C42C22×D6111yescharacteristic
446C4⋊C8—21no—
447C4⋊C8—21no—
448C4⋊C8—611no—
449C4⋊C8—611no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453C4⋊C8—1221no—
454C4⋊C8C22×D6112yes—
455C22⋊C8—21no—
456C22⋊C8—1222no—
457#97—1222no—
458C4×C8—611no—
459C4×C8—611no—
460C4×C8—1221no—
461C4×C8C22×D6112yes—
462C24—21no—
463#41—1222no—
464#41—1222no—
465#41—1224no—
466#41—2442no—
467C2×Q8—611no—
468C2×Q8—612no—
469C2×Q8—1221no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—41no—
473C2×D4—612no—
474C2×D4—611no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4C2×C4×D6111yescharacteristic
478C2×D4C2×C4×D6111yescharacteristic
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—612no—
487C22×C4—612no—
488C22×C4—612no—
489C22×C4—1222no—
490C22×C4—1222no—
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4D4×D6111yescharacteristic
494M16—42no—
495M16—2444no—
496M16—2444no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—2444no—
504C2×C8—2442no—
505C2×C8—2442no—
506C4⋊C4—21no—
507C4⋊C4—611no—
508C4⋊C4—611no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4C2×C4×D6111yescharacteristic
522C4⋊C4C2×C4×D6111yescharacteristic
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—612no—
528C22⋊C4—612no—
529C22⋊C4—612no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C42—612no—
538C42—612no—
539C42—612no—
540C42—1222no—
541C42—1222no—
542C42—1222no—
543C42C2×C4×D6111yescharacteristic
544C42D4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C23—21no—
548C23—21no—
549C23—42no—
550C23—42no—
551C23—81no—
552C23—612no—
553C23C22⋊C4×D6111yescharacteristic
554Q8—1222no—
555Q8—1222no—
556Q8—1222no—
557Q8—4881no—
558D4—21no—
559D4—21no—
560D4—21no—
561D4—42no—
562D4—1224no—
563D4—1222no—
564D4—4882no—
565D4—2444no—
566D4#1451978512yescomplemented
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—612no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—4882no—
589C2×C4—2442no—
590C2×C4—2444no—
591C2×C4C22⋊C4×D6111yescharacteristic
592C2×C4C22⋊C4×D6111yescharacteristic
593C2×C4C22⋊C4×D6111yescharacteristic
594C2×C4#1451980112yes—
595C2×C4#1451980612yes—
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C8—42no—
600C8—42no—
601C8—1222no—
602C8—2441no—
603C8—2441no—
604C8—2441no—
605C22—21no—
606C22—21no—
607C22—22no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—84no—
614C22—1222no—
615C22—1222no—
616C22—1222no—
617C22#1683976012yes—
618C22C2×C22⋊C4×D6111yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—41no—
623C4—41no—
624C4—1222no—
625C4—1222no—
626C4—1222no—
627C4—2442no—
628C4—2442no—
629C4#1683976012yes—
630C4C2×#1451978511yescharacteristic
631C4#1684052811yescharacteristic
632C4C2×#1451980111yescharacteristic
633C4C2×#1451980611yescharacteristic
634C2—22no—
635C2—42no—
636C2—81no—
637C2—2442no—
638C2#2145426911yescharacteristic
639C2C2×#1683976011yescharacteristic
640C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1486C2
socleC22×C61C22×C22⋊C4
prime radical#21459377C2
cosocleC42×C61C24
hypercenter#1486D61
hypocenterC61#17512
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #21454780C2semidirect
3. #21454388C2semidirect
4. #21444281C2semidirect
5. #21444273C2semidirect
6. #21444255C2semidirect
7. #21444229C2semidirect
8. #21444223C2semidirect
9. #21444203C2semidirect
10. C2×D4⋊#92418C2semidirect
11. C61×#1486C2semidirect
32. #16839813C22semidirect
35. #16839784C22semidirect
37. C4⋊C8×D61C22semidirect
38. C4×#11758189C22semidirect
40. #16839494C22semidirect
41. #16839494C22semidirect
42. #16839492C22semidirect
43. #16839492C22semidirect
237. D4×C61#131semidirect
320. #1486D61semidirect
373. C61#17512semidirect
566. D4#14519785semidirect

Maximal subgroups

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

subgroupindex
#214593772
#214547802
#214543882
#214442812
#214442732
#214442552
#214442292
#214442232
#214442032
C2×D4⋊#924182
C61×#14862
#1751261