Finite Group Databasebeta

← #29685045

Subgroups — #29685045

Order 15616.

1164 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2C4×M16×D61C211yesmaximal, characteristic, complemented
3#21448215C211yesmaximal, characteristic, complemented
4#21446763C211yesmaximal, characteristic, complemented
5#21446762C211yesmaximal, characteristic, complemented
6#21446739C211yesmaximal, characteristic, complemented
7#21446118C214yesmaximal, complemented
8#21445631C214yesmaximal, complemented
9C61×#1980C211yesmaximal, characteristic, complemented
10C2×M16×D61C2212yes—
11#16840319C2211yescharacteristic
12#16840317C411yescharacteristic, complemented
13#16840317C411yescharacteristic, complemented
14#16840310C411yescharacteristic, complemented
15#16840310C411yescharacteristic, complemented
16#16840294—21no—
17#16840294—21no—
18#16840293—21no—
19#16840293—21no—
20#16840291C412yescomplemented
21#16840291C412yescomplemented
22#16840272—21no—
23#16840272—21no—
24#16840269C2212yes—
25#41×Dic61C2212yes—
26C4×#11758224—21no—
27C4×#11758224—21no—
28C2×C42×D61C2211yescharacteristic
29C2×#14519704C2214yes—
30C4×#11758194C2211yescharacteristic
31#16839714—21no—
32M16×Dic61C2212yes—
33C4×#11758189—21no—
34#97×D61—21no—
35C4×C8×D61—21no—
36#16839629—21no—
37#16839627—21no—
38#16839624—21no—
39#16839619—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41C4×#11758198C2212yes—
42#16839608C2212yes—
43#16839523C2214yes—
44#16839522C2214yes—
45C61×#532C2211yescharacteristic
46C2×#104×C61C2214yes—
47C4×M16×C61C2211yescharacteristic
48C4×C4⋊C4×C61C2211yescharacteristic
49C2×#11758224—21no—
50C2×#11758224—21no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758194C2312yes—
55C2×#11758189—22no—
56M16×D61—24no—
57M16×D61—24no—
58C2×C8×D61—22no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519770—22no—
62#14519770—22no—
63#14519769C2×C412yes—
64#14519768—21no—
65#14519768—21no—
66#14519764—21no—
67#14519764—21no—
68#14519759—22no—
69#14519759—22no—
70#14519758—22no—
71#14519757—21no—
72#14519757—21no—
73#14519756—21no—
74#14519756—21no—
75#14519753C2×C412yes—
76#14519753C2×C412yes—
77#14519748—21no—
78#14519748—21no—
79#14519748C2×C412yes—
80#14519748C2×C412yes—
81#14519745—22no—
82Q8×Dic61C2×C412yes—
83Q8×Dic61C2×C412yes—
84C4×Dic122—21no—
85C4×Dic122—21no—
86#14519742—21no—
87#14519742—21no—
88D4×Dic61—22no—
89D4×Dic61—22no—
90D4×Dic61—21no—
91D4×Dic61—21no—
92D4×Dic61C2×C412yes—
93D4×Dic61C2×C412yes—
94C4×C22⋊D61—22no—
95C4×C22⋊D61—22no—
96#14519737—22no—
97#14519737—22no—
98#14519736—22no—
99#14519736C2312yes—
100#14519735—21no—
101#14519735—21no—
102#14519733—22no—
103#14519733—22no—
104C2×C4⋊Dic61—21no—
105C2×#11758198—22no—
106C2×C4×Dic61—22no—
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61C2312yes—
110C2×C4×Dic61C2311yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119#14519704—28no—
120#14519688—21no—
121#14519687—22no—
122C4×#92418—21no—
123C8×Dic61—22no—
124#14519684—21no—
125#14519684—22no—
126#14519684—22no—
127#14519684—22no—
128C2×#41×C61C2312yes—
129C2×M16×C61C2312yes—
130C4⋊Q8×C61C2×C411yescharacteristic
131#127×C61C2×C411yescharacteristic
132#124×C61C2×C412yes—
133#121×C61—21no—
134#121×C61—21no—
135C4×D4×C61—21no—
136C4×D4×C61—21no—
137C2×C4⋊C4×C61—21no—
138C2×C42×C61C2312yes—
139C2×C42×C61C2311yescharacteristic
140#104×C61—28no—
141#97×C61—21no—
142C4×C8×C61—21no—
143#95×C61—21no—
144C23×D61C2×D411yescharacteristic
145#41×C61—24no—
146#41×C61—24no—
147C2×Q8×C61C22×C412yes—
148C2×D4×C61—22no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C22×C412yes—
152C22×C4×C61—22no—
153C22×C4×C61—22no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165#11758224—44no—
166#11758224—44no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22×C411yescharacteristic
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×Dic122—21no—
184C2×Dic122—21no—
185M16×C61—22no—
186M16×C61#4114yescomplemented
187C2×C8×C61—22no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—42no—
191C22⋊C4×C61—22no—
192C22⋊C4×C61—22no—
193C42×C61—24no—
194C42×C61C22×C411yescharacteristic
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203#11758198—22no—
204#11758198—22no—
205#11758198—44no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61—42no—
211C4⋊Dic61C22×C412yes—
212C4⋊Dic61C22×C412yes—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—22no—
220C4×Dic61—24no—
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22×C411yescharacteristic
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233#11758194—22no—
234#11758194#4114yescomplemented
235C2×#92418—22no—
236#11758189—28no—
237C8×D61—28no—
238C23×C61—22no—
239C23×C61C22×D411yescharacteristic
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—22no—
243C22×D61—41no—
244C22×D61C2×C22⋊C411yescharacteristic
245C22×D61C2×C22⋊C411yescharacteristic
246Q8×C61—24no—
247D4×C61—24no—
248D4×C61—42no—
249D4×C61—42no—
250D4×C61—42no—
251D4×C61—42no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—41no—
255C2×C4×C61—44no—
256C2×C4×C61—42no—
257C2×C4×C61—42no—
258C2×C4×C61—42no—
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61C2×C22⋊C411yescharacteristic
261C2×C4×C61C2×C22⋊C411yescharacteristic
262C2×C4×C61C2×C22⋊C411yescharacteristic
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4112yes—
266C2×C4×C61C2×#4112yes—
267C22⋊D61—44no—
268C22⋊D61—44no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—44no—
276C2×Dic61—44no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61C2×C22⋊C411yescharacteristic
285C2×Dic61C2×C22⋊C411yescharacteristic
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—41no—
295C4×D61—41no—
296C4×D61—41no—
297C4×D61—41no—
298Dic122—42no—
299Dic122—42no—
300C8×C61—24no—
301#92418—24no—
302#21383—611nomaximal, Sylow
303C22×C61—21no—
304C22×C61—21no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—42no—
308C22×C61C22×C22⋊C411yescharacteristic
309C22×C61#53211yescharacteristic
310C22×C61#53211yescharacteristic
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—21no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61#42112yes—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—21no—
322C4×C61—42no—
323C4×C61—82no—
324C4×C61#42112yes—
325C4×C61C4×#4112yes—
326C4×C61#52512yes—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—21no—
331Dic61—21no—
332Dic61—21no—
333Dic61—42no—
334Dic61—42no—
335Dic61—82no—
336#3444—611no—
337C2×C4×M16—611no—
338#2283—611no—
339#1986—611no—
340#1986—611no—
341#1980—611no—
342#1980D6111yescharacteristic, complemented
343#1871—614no—
344#1759—614no—
345C2×C61—21no—
346C2×C61—21no—
347C2×C61—82no—
348C2×C61#229711yescharacteristic
349C2×C61C2×#42111yescharacteristic
350C2×C61C2×#42111yescharacteristic
351D61—22no—
352D61—41no—
353C22×M16—612no—
354#533—611no—
355#533—611no—
356#533—611no—
357#532—611no—
358#532—611no—
359#532C2×D6111yescharacteristic
360#529—612no—
361#529—612no—
362#529—1221no—
363#529—1221no—
364#529—1221no—
365#529—1221no—
366#525—612no—
367#525—1221no—
368#525—1221no—
369C4×#41—612no—
370C4×#41—1221no—
371C4×#41—1221no—
372C22×C42—611no—
373C2×#104—614no—
374C2×#104C2×D6114yes—
375C4×M16—612no—
376C4×M16—611no—
377C4×M16—1221no—
378C4×M16—1221no—
379C4×M16C2×D6111yescharacteristic
380C2×#97—1221no—
381C2×C4×C8—1221no—
382#383—1221no—
383#382—1221no—
384#381—1221no—
385#380—1221no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4C2×D6111yescharacteristic
388C4×C22⋊C4—612no—
389#376—612no—
390#339—614no—
391#339—614no—
392C61#2138311yescharacteristic, complemented, Sylow
393C2×#41—1221no—
394C2×#41—1221no—
395C2×#41C22×D6112yes—
396C23×C4—611no—
397C23×C4—611no—
398C23×C4—611no—
399C2×M16—612no—
400C2×M16—1222no—
401C2×M16—1224no—
402C2×M16—1224no—
403C2×M16C22×D6112yes—
404C22×C8—1222no—
405C4⋊Q8—611no—
406C4⋊Q8—611no—
407C4⋊Q8C4×D6111yescharacteristic
408#127C4×D6111yescharacteristic
409#126—1222no—
410#126—1222no—
411#125—612no—
412#125—1221no—
413#125—1221no—
414#124—1221no—
415#124—1221no—
416#124C4×D6112yes—
417#123—1222no—
418#123—1222no—
419#123—1222no—
420C22⋊Q8—612no—
421C22⋊Q8—612no—
422C22⋊Q8—1221no—
423C22⋊Q8—1221no—
424C22⋊Q8—1221no—
425C22⋊Q8—1221no—
426#121—21no—
427#121—21no—
428#121—612no—
429#121—612no—
430#121—1221no—
431#121—1221no—
432#120—1222no—
433C4×Q8—612no—
434C4×Q8—612no—
435C4×Q8—1221no—
436C4×Q8—1221no—
437C4×Q8—1221no—
438C4×Q8—1221no—
439C4×D4—21no—
440C4×D4—21no—
441C4×D4—612no—
442C4×D4—612no—
443C4×D4—1221no—
444C4×D4—1221no—
445C4×D4—1222no—
446C4×D4—1222no—
447C4×D4—1222no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4—1222no—
451#117—612no—
452#117—1222no—
453#117—1222no—
454#117—1222no—
455#117—1221no—
456#117—1221no—
457C2×C4⋊C4—21no—
458C2×C4⋊C4—1221no—
459C2×C22⋊C4—1222no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—612no—
472C2×C42—1222no—
473C2×C42C22×D6112yes—
474C2×C42C22×D6111yescharacteristic
475#104—28no—
476#104—1228no—
477#97—21no—
478#97—1222no—
479#97—1221no—
480C4×C8—21no—
481C4×C8—1222no—
482C4×C8—1221no—
483#95—21no—
484#95—1222no—
485#95—1222no—
486#95—1222no—
487#95—1221no—
488C24—611no—
489#41—24no—
490#41—24no—
491#41—2444no—
492#41—2444no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8C2×C4×D6112yes—
496C2×D4—22no—
497C2×D4—21no—
498C2×D4—21no—
499C2×D4—1222no—
500C2×D4—1222no—
501C2×D4C2×C4×D6112yes—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1222no—
523C22×C4—1222no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4D4×D6111yescharacteristic
527C22×C4D4×D6111yescharacteristic
528C22×C4C23×D6111yescharacteristic
529M16—22no—
530M16—614no—
531M16—1228no—
532M16—1222no—
533M16#1175822414yescomplemented
534C2×C8—22no—
535C2×C8—1228no—
536C2×C8—1222no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—42no—
540C4⋊C4—612no—
541C4⋊C4—612no—
542C4⋊C4—1222no—
543C4⋊C4—1222no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—2442no—
551C22⋊C4—22no—
552C22⋊C4—22no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C22⋊C4—1222no—
557C22⋊C4—1222no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—2444no—
561C42—24no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—611no—
569C42—611no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—1222no—
575C42—1222no—
576C42—1222no—
577C42—1224no—
578C42C2×C4×D6111yescharacteristic
579C42C2×C4×D6111yescharacteristic
580C42D4×D6111yescharacteristic
581C42D4×D6111yescharacteristic
582C23—22no—
583C23—611no—
584C23—611no—
585C23—1222no—
586C23—1221no—
587C23—1221no—
588C23—2441no—
589C23C2×D4×D6111yescharacteristic
590Q8—24no—
591Q8—2442no—
592Q8—2442no—
593D4—24no—
594D4—42no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—2444no—
599D4—2444no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—611no—
610C2×C4—611no—
611C2×C4—611no—
612C2×C4—611no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2444no—
632C2×C4—2444no—
633C2×C4—2442no—
634C2×C4—2442no—
635C2×C4—2442no—
636C2×C4C22⋊C4×D6111yescharacteristic
637C2×C4C22⋊C4×D6111yescharacteristic
638C2×C4C22⋊C4×D6111yescharacteristic
639C2×C4C22⋊C4×D6111yescharacteristic
640C2×C4C22×C4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4#41×D6112yes—
643C2×C4C2×#1175822412yes—
644C8—24no—
645C8—1224no—
646C22—21no—
647C22—21no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—612no—
652C22—1221no—
653C22—1221no—
654C22—1221no—
655C22—1221no—
656C22—1221no—
657C22—2442no—
658C22—2442no—
659C22C2×C22⋊C4×D6111yescharacteristic
660C22#1684031011yescharacteristic
661C22#1684031111yescharacteristic
662C4—21no—
663C4—21no—
664C4—21no—
665C4—42no—
666C4—82no—
667C4—1221no—
668C4—1221no—
669C4—1221no—
670C4—1221no—
671C4—1221no—
672C4—1221no—
673C4—4882no—
674C4—2442no—
675C4—2442no—
676C4#1683977712yes—
677C4C4×#1175822412yes—
678C4#1684027212yes—
679C2—21no—
680C2—21no—
681C2—82no—
682C2—1222no—
683C2—2441no—
684C2#2144830611yescharacteristic
685C2#421×D6111yescharacteristic
686C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1980C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1980D61
hypocenterC61#21383
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. C4×M16×D61C2semidirect
3. #21448215C2semidirect
4. #21446763C2semidirect
5. #21446762C2semidirect
6. #21446739C2semidirect
7. #21446118C2semidirect
8. #21445631C2semidirect
9. C61×#1980C2semidirect
12. #16840317C4semidirect
13. #16840317C4semidirect
14. #16840310C4semidirect
15. #16840310C4semidirect
20. #16840291C4semidirect
21. #16840291C4semidirect
186. M16×C61#41semidirect
234. #11758194#41semidirect
342. #1980D61semidirect
392. C61#21383semidirect
533. M16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214591142
C4×M16×D612
#214482152
#214467632
#214467622
#214467392
#214461182
#214456312
C61×#19802
#2138361