Finite Group Databasebeta

← #29646067

Subgroups — #29646067

Order 15616.

940 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
1C2×#16840378C211yesmaximal, characteristic
2#21454903C211yesmaximal, characteristic, complemented
3#21454899C211yesmaximal, characteristic, complemented
4#21444188C211yesmaximal, characteristic, complemented
5#21444187C212yesmaximal, complemented
6#21444172C211yesmaximal, characteristic, complemented
7#21444170C211yesmaximal, characteristic, complemented
8#21444169C212yesmaximal, complemented
9#21444138C212yesmaximal, complemented
10#21444132C212yesmaximal, complemented
11C61×#1473C211yesmaximal, characteristic, complemented
12#16840378—22no—
13#16840378—22no—
14#16840378—22no—
15#16840378—22no—
16C2×#14519776C411yescharacteristic
17C2×#14519772C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519763C411yescharacteristic
20C2×#14519763C2211yescharacteristic
21C2×#14519762C411yescharacteristic
22C2×#14519761C411yescharacteristic
23C2×#14519757C411yescharacteristic
24C2×#14519755C411yescharacteristic
25C2×#14519754C411yescharacteristic
26C2×#14519750C411yescharacteristic
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519737C2211yescharacteristic
29C2×#14519735C2211yescharacteristic
30#16839891—22no—
31#16839890—22no—
32C8×Dic122—22no—
33#16839886—22no—
34#16839819C2211yescharacteristic
35#16839816C2212yes—
36#16839815C2211yescharacteristic
37#16839812C2212yes—
38#16839812C2212yes—
39#16839812C2212yes—
40#16839488C2212yes—
41#16839486C2212yes—
42#16839485C2212yes—
43#16839485C2212yes—
44#16839485C2212yes—
45#16839485C2212yes—
46C2×#126×C61C2211yescharacteristic
47C61×#432C2211yescharacteristic
48C61×#431C2211yescharacteristic
49C61×#324C2212yes—
50C61×#323C2212yes—
51C22×Dic122C2×C411yescharacteristic
52C22×Dic122C2×C411yescharacteristic
53C22×Dic122C2×C411yescharacteristic
54C22×D244C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C22×C22⋊D61C2×C411yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C22×C4×D61C2×C411yescharacteristic
59C22×C4×D61C2×C411yescharacteristic
60#14519776—22no—
61#14519776—22no—
62#14519772—22no—
63#14519772—22no—
64#14519765—21no—
65#14519765—21no—
66#14519765—21no—
67#14519765—21no—
68#14519763—22no—
69#14519763—22no—
70#14519763—22no—
71#14519763—22no—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519761—22no—
76#14519757—22no—
77#14519757—22no—
78#14519755—22no—
79#14519755—22no—
80#14519754—22no—
81#14519754—22no—
82#14519750—22no—
83#14519750—22no—
84C4×Dic122D412yes—
85C4×Dic122D412yes—
86C4×Dic122D412yes—
87C4×Dic122D412yes—
88#14519737—22no—
89#14519737—22no—
90#14519735—22no—
91#14519735—22no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×#11758196C2×C411yescharacteristic
96C2×#11758196C2×C411yescharacteristic
97C2×#11758196C2×C411yescharacteristic
98C2×#11758196C2×C411yescharacteristic
99C2×C22⋊Dic61C2×C411yescharacteristic
100C2×#11758198C2×C411yescharacteristic
101C2×#11758198C2×C411yescharacteristic
102C2×#11758198C2×C411yescharacteristic
103C2×#11758198C2×C411yescharacteristic
104C2×#11758198C2×C411yescharacteristic
105C2×#11758198C2×C411yescharacteristic
106C2×#11758198C2311yescharacteristic
107C2×#11758198C2311yescharacteristic
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C4⋊#92418—21no—
112C4⋊#92418—21no—
113#14519706—22no—
114#14519706—22no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118C22⋊#92418C2312yes—
119#14519688—21no—
120#14519687—22no—
121C4×#92418—21no—
122C8×Dic61—22no—
123C8×Dic61—22no—
124C8×Dic61—22no—
125#126×C61—21no—
126#126×C61—21no—
127#126×C61—21no—
128#126×C61—21no—
129C2×C4⋊C4×C61C2×C411yescharacteristic
130C2×C4⋊C4×C61C2×C411yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C22⋊C4×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61—21no—
138C22⋊C8×C61C2312yes—
139C22⋊C8×C61C2312yes—
140#97×C61—21no—
141C4×C8×C61—21no—
142C24×C61C22×C411yescharacteristic
143C23×D61C22×C411yescharacteristic
144C23×D61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C2×C22⋊D61—22no—
152C2×C22⋊D61—22no—
153C2×C22⋊D61—24no—
154C2×C22⋊D61—81no—
155C22×Dic61—22no—
156C22×Dic61—81no—
157C22×Dic61C22×C411yescharacteristic
158C22×Dic61C22×C411yescharacteristic
159C22×Dic61C22×C411yescharacteristic
160C22×Dic61C22×C411yescharacteristic
161C22×Dic61C22×C411yescharacteristic
162C22×Dic61C22×C411yescharacteristic
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—81no—
166C2×C4×D61—22no—
167C2×C4×D61—24no—
168C2×C4×D61—22no—
169C2×C4×D61—81no—
170C2×Dic122—24no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—22no—
174C2×Dic122—81no—
175C2×Dic122—81no—
176C2×Dic122C22⋊C412yes—
177C2×Dic122C22⋊C412yes—
178C2×C8×C61—22no—
179C2×C8×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—21no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—21no—
191C22⋊C4×C61—21no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C22⋊Dic61—22no—
197#11758198—21no—
198#11758198—21no—
199#11758198—21no—
200#11758198—21no—
201#11758198—21no—
202#11758198—21no—
203#11758198—21no—
204#11758198—21no—
205#11758198—22no—
206#11758198—22no—
207#11758198—22no—
208#11758198—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61—22no—
211C4⋊Dic61C22⋊C412yes—
212C4⋊Dic61C22⋊C412yes—
213#11758196—22no—
214#11758196—22no—
215#11758196C22⋊C412yes—
216#11758196C22⋊C412yes—
217#11758196C22⋊C412yes—
218#11758196C22⋊C412yes—
219C4×Dic61—22no—
220C4×Dic61C2×D412yes—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C4×Dic61C2×D412yes—
224C2×#92418—22no—
225C2×#92418—22no—
226C23×C61—21no—
227C23×C61—22no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C23×C411yescharacteristic
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—22no—
234C22×D61—22no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C22⋊D61—44no—
255C22⋊D61—82no—
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—82no—
264C2×Dic61C2×C22⋊C412yes—
265C2×Dic61C2×C22⋊C412yes—
266C2×Dic61C2×C22⋊C412yes—
267C2×Dic61C2×C22⋊C412yes—
268C2×Dic61#13114yes—
269D244—81no—
270D244—81no—
271C4×D61—44no—
272C4×D61—82no—
273Dic122—24no—
274Dic122—24no—
275Dic122—82no—
276Dic122—81no—
277Dic122—81no—
278C8×C61—42no—
279C8×C61—42no—
280#92418—42no—
281#92418—42no—
282#17390—611nomaximal, Sylow
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61—41no—
288C22×C61—82no—
289C22×C61C22×C22⋊C411yescharacteristic
290C22×C61C2×#13112yes—
291C2×D61—41no—
292C2×D61—41no—
293C2×D61—41no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—82no—
298C2×D61—82no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—41no—
306C4×C61—41no—
307C4×C61—41no—
308Dic61—22no—
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61#40814yes—
314C2×#542—611no—
315#2983—611no—
316#2982—611no—
317#1473—612no—
318#1473—611no—
319#1473D6111yescharacteristic, complemented
320#1469—611no—
321#1469—611no—
322#1469—612no—
323#1460—612no—
324#1458—612no—
325C2×C61—22no—
326C2×C61—81no—
327C2×C61C2×#40811yescharacteristic
328C2×C61#288411yescharacteristic
329C2×C61#288811yescharacteristic
330D61—81no—
331D61—81no—
332#542—1222no—
333#542—1222no—
334#542—1222no—
335#542—1222no—
336C2×C4⋊Q8—611no—
337C2×#126—611no—
338C2×#126C2×D6111yescharacteristic
339C2×#124—611no—
340C2×#124—611no—
341C2×#124—611no—
342C2×#123—611no—
343C2×#123—611no—
344C2×C22⋊Q8—611no—
345C2×C22⋊Q8—611no—
346C2×C22⋊Q8—611no—
347C2×#121—611no—
348C2×C4×Q8—611no—
349C2×C4×D4—611no—
350C2×#117—611no—
351#446—1222no—
352#446—1222no—
353C8×Q8—1222no—
354C8×Q8—1222no—
355#432—611no—
356#432—612no—
357#432C2×D6111yescharacteristic
358#431—611no—
359#431—612no—
360#431—612no—
361#431—612no—
362#431C2×D6111yescharacteristic
363#324—612no—
364#324C2×D6112yes—
365#323—612no—
366#323—612no—
367#323—612no—
368#323—612no—
369#323—612no—
370#323C2×D6112yes—
371C61#1739011yescharacteristic, complemented, Sylow
372C22×Q8—611no—
373C22×Q8—611no—
374C22×Q8—611no—
375C22×D4—611no—
376C22×D4—611no—
377C22×D4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C4⋊Q8—1222no—
382C4⋊Q8—1222no—
383#126—21no—
384#126—21no—
385#126—21no—
386#126—21no—
387#126—1222no—
388#126—1222no—
389#124—1222no—
390#124—1222no—
391#124—1222no—
392#124—1222no—
393#124—1221no—
394#124—1221no—
395#124—1221no—
396#124—1221no—
397#123—1222no—
398#123—1222no—
399#123—1222no—
400#123—1222no—
401C22⋊Q8—1222no—
402C22⋊Q8—1222no—
403C22⋊Q8—1222no—
404C22⋊Q8—1222no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407#121—1222no—
408#121—1222no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—612no—
412C4×Q8—612no—
413C4×D4—1222no—
414C4×D4—1222no—
415#117—1222no—
416#117—1222no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4C4×D6111yescharacteristic
425C2×C4⋊C4C4×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—611no—
432C2×C22⋊C4—611no—
433C2×C22⋊C4—611no—
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4C4×D6111yescharacteristic
437C2×C22⋊C4C4×D6111yescharacteristic
438C2×C22⋊C4C22×D6111yescharacteristic
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42C22×D6111yescharacteristic
443C4⋊C8—21no—
444C4⋊C8—21no—
445C4⋊C8—1222no—
446C4⋊C8—1222no—
447C4⋊C8—1222no—
448C4⋊C8—1222no—
449C4⋊C8—1221no—
450C4⋊C8—1221no—
451C22⋊C8—612no—
452C22⋊C8—612no—
453C22⋊C8C22×D6112yes—
454C22⋊C8C22×D6112yes—
455#97—21no—
456#97—1222no—
457#97—1221no—
458C4×C8—21no—
459C4×C8—1222no—
460C4×C8—1222no—
461C4×C8—1222no—
462C4×C8—1221no—
463C24—611no—
464C24—611no—
465C24C2×C4×D6111yescharacteristic
466C2×Q8—612no—
467C2×Q8—612no—
468C2×Q8—1224no—
469C2×Q8—1222no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—4881no—
473C2×Q8—4881no—
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—1224no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—4881no—
480C2×D4—4881no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—1222no—
488C22×C4—1222no—
489C22×C4—1224no—
490C22×C4—1222no—
491C22×C4—4881no—
492C22×C4—4881no—
493C22×C4C2×C4×D6111yescharacteristic
494C22×C4C2×C4×D6111yescharacteristic
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4C2×C4×D6111yescharacteristic
497C22×C4C2×C4×D6111yescharacteristic
498C22×C4C23×D6111yescharacteristic
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—1222no—
502C2×C8—1222no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C22⋊C4—21no—
520C22⋊C4—21no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1221no—
531C22⋊C4—1221no—
532C22⋊C4—1221no—
533C22⋊C4—1221no—
534C22⋊C4—1221no—
535C22⋊C4—1221no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C42—612no—
539C42—612no—
540C42—612no—
541C42—612no—
542C42—1222no—
543C42D4×D6111yescharacteristic
544C42D4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42D4×D6111yescharacteristic
547C23—21no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—1222no—
552C23—1222no—
553C23—1221no—
554C23—1221no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23C22×C4×D6111yescharacteristic
560Q8—1224no—
561Q8—1224no—
562Q8—4881no—
563Q8—4881no—
564Q8—4882no—
565D4—4881no—
566D4—4881no—
567D4—4882no—
568D4—2444no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—612no—
579C2×C4—614no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—612no—
583C2×C4—1222no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—4882no—
587C2×C4—4882no—
588C2×C4—2444no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4C22⋊C4×D6111yescharacteristic
594C2×C4C22⋊C4×D6111yescharacteristic
595C2×C4C22⋊C4×D6111yescharacteristic
596C2×C4C22⋊C4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C2×D4×D6111yescharacteristic
599C8—42no—
600C8—42no—
601C8—2442no—
602C8—2442no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—82no—
609C22—4882no—
610C22—4882no—
611C22—2441no—
612C22—2441no—
613C22—2441no—
614C22—2441no—
615C22—2441no—
616C22—2441no—
617C22C2×C22⋊C4×D6111yescharacteristic
618C22#131×D6112yes—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—614no—
629C4—1222no—
630C4—1222no—
631C4—1222no—
632C4—2441no—
633C4—2441no—
634C2—22no—
635C2—81no—
636C2—4881no—
637C2—4881no—
638C2#408×D6111yescharacteristic
639C2#2145428111yescharacteristic
640C2#2145429611yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1473C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519750C4
cosocleC22×C4×C61C24
hypercenter#1473D61
hypocenterC61#17390
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1473

Complements

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

NH ≅ G/Ntype
2. #21454903C2semidirect
3. #21454899C2semidirect
4. #21444188C2semidirect
5. #21444187C2semidirect
6. #21444172C2semidirect
7. #21444170C2semidirect
8. #21444169C2semidirect
9. #21444138C2semidirect
10. #21444132C2semidirect
11. C61×#1473C2semidirect
319. #1473D61semidirect
371. C61#17390semidirect

Maximal subgroups

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

subgroupindex
C2×#168403782
#214549032
#214548992
#214441882
#214441872
#214441722
#214441702
#214441692
#214441382
#214441322
C61×#14732
#1739061