Finite Group Databasebeta

← #29646836

Subgroups — #29646836

Order 15616.

1112 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#423×D61C211yesmaximal, characteristic, complemented
3C2×C4×#11758189C211yesmaximal, characteristic, complemented
4#21444274C211yesmaximal, characteristic, complemented
5#21444271C211yesmaximal, characteristic, complemented
6#21444269C211yesmaximal, characteristic, complemented
7#21444257C214yesmaximal, complemented
8#21444231C214yesmaximal, complemented
9C61×#1493C211yesmaximal, characteristic, complemented
10C22×#11758189—21no—
11C2×M16×D61—21no—
12#16840317C411yescharacteristic
13#16840317C411yescharacteristic
14#16840313C411yescharacteristic
15#16840313C411yescharacteristic
16#16840312C2211yescharacteristic
17#16840299—21no—
18#16840299C2212yes—
19#16840298—21no—
20#16840298C412yes—
21#16840297—21no—
22#16840297—21no—
23#16840275—21no—
24#16840275C412yes—
25#16840274—21no—
26C4×#11758226—21no—
27C4×#11758226C2212yes—
28#16840230—21no—
29C2×C42×D61C2211yescharacteristic
30#16839786C2211yescharacteristic
31#16839785—21no—
32#16839784—21no—
33#16839784C2212yes—
34C4⋊C8×D61—21no—
35C4⋊C8×D61C2212yes—
36C4×#11758189—21no—
37C4×#11758189—21no—
38C4×#11758189C2212yes—
39C4×#11758189C2212yes—
40C2×#14519687—21no—
41C2×C4×#92418C2211yescharacteristic
42Q8⋊#92418C2214yescomplemented
43#16839491C2214yescomplemented
44#16839489C2214yescomplemented
45C61×#533C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C2×C4×C8×C61C2211yescharacteristic
48C61×Q8⋊C8C2214yescomplemented
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61D411yescharacteristic
55C2×#11758194—21no—
56C2×#11758189—28no—
57C2×#11758189—22no—
58C2×#11758189—22no—
59C2×#11758189—22no—
60M16×D61—44no—
61C22×#92418—21no—
62C2×C8×D61—22no—
63#14519777C2×C411yescharacteristic
64#14519774C2×C411yescharacteristic
65#14519772—22no—
66#14519772—22no—
67#14519768—21no—
68#14519768C2×C412yes—
69#14519766—21no—
70#14519766C2×C412yes—
71#14519765—21no—
72#14519762—22no—
73#14519762—22no—
74#14519761—22no—
75#14519757—21no—
76#14519757—21no—
77#14519753—21no—
78#14519753C2×C412yes—
79#14519752—21no—
80#14519752—21no—
81#14519752—21no—
82#14519752C2×C412yes—
83#14519751—21no—
84#14519751C2×C412yes—
85#14519747—22no—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yes—
88#14519742—21no—
89#14519742—21no—
90C4×D244—22no—
91C4×D244—22no—
92C4×D244—21no—
93C4×D244C2312yes—
94#14519739—22no—
95#14519739—22no—
96#14519739—21no—
97#14519739—21no—
98#14519739—21no—
99#14519739C2×C412yes—
100#14519737—22no—
101#14519737—22no—
102#14519735—21no—
103#14519735—22no—
104#14519735—22no—
105#14519735C2×C412yes—
106#14519734—22no—
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C42×D61—22no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊#92418—21no—
120C4⋊#92418C2312yes—
121#14519706—22no—
122#14519706—22no—
123#14519687—22no—
124#14519687—22no—
125C4×#92418—21no—
126C4×#92418C2312yes—
127C2×M16×C61—21no—
128C22×C8×C61—21no—
129C4⋊Q8×C61C2×C411yescharacteristic
130C4⋊Q8×C61C2×C411yescharacteristic
131#125×C61—21no—
132C22⋊Q8×C61—21no—
133C22⋊Q8×C61C2×C412yes—
134C4×Q8×C61—21no—
135C4×Q8×C61C2312yes—
136#117×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C4⋊C8×C61C2312yes—
140C4×C8×C61—21no—
141C4×C8×C61C2312yes—
142C23×D61C22⋊C411yescharacteristic
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C2×C22⋊D61—22no—
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151C22×Dic61C22⋊C411yescharacteristic
152#11758226—44no—
153#11758226—44no—
154#11758222—42no—
155#11758222—42no—
156C2×D244—22no—
157C2×D244—22no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244C22×C412yes—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—24no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—42no—
168C2×C4×D61C22⋊C411yescharacteristic
169C2×C4×D61C22⋊C411yescharacteristic
170C2×C4×D61C22⋊C411yescharacteristic
171C2×C4×D61C22⋊C411yescharacteristic
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22×C411yescharacteristic
175C2×C4×D61C22×C411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×C4×D61C2×D411yescharacteristic
178C2×C4×D61C2×D411yescharacteristic
179C2×C4×D61C2×D411yescharacteristic
180C2×Dic122—21no—
181M16×C61—42no—
182C2×C8×C61—24no—
183C2×C8×C61—22no—
184C2×C8×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61C22×C412yes—
191C22⋊C4×C61—22no—
192C42×C61—22no—
193C42×C61C22×C411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2411yescharacteristic
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202#11758198—22no—
203#11758198—22no—
204#11758198—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—22no—
207C4⋊Dic61—22no—
208C4⋊Dic61C22×C412yes—
209#11758196—22no—
210#11758196—22no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225#11758194—42no—
226C2×#92418—24no—
227C2×#92418—22no—
228C2×#92418—22no—
229#11758189—28no—
230#11758189—28no—
231#11758189—44no—
232C8×D61—44no—
233C23×C61C2×C22⋊C411yescharacteristic
234C22×D61—22no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—41no—
238C22×D61C2×C22⋊C411yescharacteristic
239C22×D61C2×C22⋊C411yescharacteristic
240Q8×C61—24no—
241Q8×C61—42no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61AGL(1,ℤ/8)12yes—
250C2×C4×C61#13712yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C22⋊D61—44no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—41no—
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265D244—24no—
266D244—44no—
267D244—44no—
268D244—42no—
269D244—42no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—21no—
274C4×D61—21no—
275C4×D61—21no—
276C4×D61—21no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—44no—
284C4×D61—42no—
285C4×D61—42no—
286C4×D61—44no—
287C4×D61#10414yes—
288Dic122—42no—
289C8×C61—24no—
290C8×C61—42no—
291#92418—24no—
292#92418—42no—
293#17504—611nomaximal, Sylow
294C22×C61—21no—
295C22×C61—21no—
296C22×C61#41712yes—
297C22×C61C22×C22⋊C411yescharacteristic
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—21no—
303C2×D61—44no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61C2×#10412yes—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61#41712yes—
313C4×C61C2×#10412yes—
314C4×C61#56812yes—
315C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
316C4×C61C2×#13711yescharacteristic
317Dic61—21no—
318Dic61—21no—
319Dic61—21no—
320Dic61—21no—
321Dic61—21no—
322Dic61—42no—
323Dic61C2×#10412yes—
324#3444—611no—
325C2×#423—611no—
326C2×C4×M16—611no—
327#1493—611no—
328#1493D6111yescharacteristic, complemented
329#1492—611no—
330#1492—611no—
331#1489—614no—
332#1483—614no—
333C2×C61—22no—
334C2×C61#287911yescharacteristic
335C2×C61C2×#41711yescharacteristic
336C2×C61C22×#10411yescharacteristic
337D61—22no—
338D61—41no—
339D61—82no—
340C22×M16—1221no—
341C22×M16—1221no—
342#533—611no—
343#533—611no—
344#533C2×D6111yescharacteristic
345#532—611no—
346#532—611no—
347#532—611no—
348#529—612no—
349#529—612no—
350#529—1221no—
351#529—1221no—
352#529—1221no—
353#529—1221no—
354#525—612no—
355#525—1221no—
356#525—1221no—
357C4×#41—612no—
358C4×#41—1221no—
359C4×#41—1221no—
360C22×C42—611no—
361#423—611no—
362#423—612no—
363#423—1221no—
364#423—1221no—
365#423C2×D6111yescharacteristic
366C2×C4⋊C8—612no—
367C2×C4⋊C8—1221no—
368C4×M16—612no—
369C4×M16—612no—
370C4×M16—1221no—
371C4×M16—1221no—
372C2×#97—1221no—
373C2×C4×C8—611no—
374C2×C4×C8C2×D6111yescharacteristic
375Q8⋊C8—614no—
376Q8⋊C8C2×D6114yescomplemented
377D4⋊C8—614no—
378D4⋊C8—614no—
379C61#1750411yescharacteristic, complemented, Sylow
380C2×#41—612no—
381C2×#41—1221no—
382C2×#41—1221no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C2×M16—21no—
387C2×M16—1222no—
388C2×M16—1222no—
389C2×M16—1228no—
390C2×M16—1222no—
391C2×M16—1221no—
392C2×M16—2444no—
393C22×C8—21no—
394C22×C8—1222no—
395C22×C8—1221no—
396C4⋊Q8—611no—
397C4⋊Q8C4×D6111yescharacteristic
398C4⋊Q8C4×D6111yescharacteristic
399#127—611no—
400#126—1222no—
401#126—1222no—
402#125—21no—
403#125—612no—
404#125—1221no—
405#124—612no—
406#124—1221no—
407#124—1221no—
408#123—1222no—
409#123—1222no—
410#123—1222no—
411C22⋊Q8—21no—
412C22⋊Q8—612no—
413C22⋊Q8—1221no—
414C22⋊Q8—1221no—
415C22⋊Q8—1221no—
416C22⋊Q8C4×D6112yes—
417#121—612no—
418#121—612no—
419#121—1221no—
420#121—1221no—
421#121—1221no—
422#121—1221no—
423#120—1222no—
424C4×Q8—21no—
425C4×Q8—612no—
426C4×Q8—1221no—
427C4×Q8—1221no—
428C4×Q8—1221no—
429C4×Q8C22×D6112yes—
430C4×D4—612no—
431C4×D4—612no—
432C4×D4—1222no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4—1221no—
436C4×D4—1221no—
437C4×D4—1222no—
438C4×D4—1222no—
439C4×D4—1222no—
440C4×D4—1222no—
441C4×D4—1221no—
442#117—21no—
443#117—612no—
444#117—1222no—
445#117—1222no—
446#117—1222no—
447#117—1221no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—611no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—1222no—
460C2×C42C22×D6111yescharacteristic
461C4⋊C8—21no—
462C4⋊C8—612no—
463C4⋊C8—1222no—
464C4⋊C8—1222no—
465C4⋊C8—1221no—
466C4⋊C8C22×D6112yes—
467#97—1222no—
468#97—1222no—
469C4×C8—21no—
470C4×C8—612no—
471C4×C8—1221no—
472C4×C8C22×D6112yes—
473C24—611no—
474#41—2444no—
475#41—2444no—
476#41—2442no—
477#41—2442no—
478C2×Q8—21no—
479C2×Q8—1221no—
480C2×Q8C2×C4×D6112yes—
481C2×D4—612no—
482C2×D4—1222no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—1224no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—2442no—
509C22×C4C2×C4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4D4×D6111yescharacteristic
512M16—42no—
513M16—1228no—
514M16—1228no—
515M16—2444no—
516M16—2442no—
517C2×C8—24no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—1224no—
521C2×C8—1222no—
522C2×C8—1222no—
523C2×C8—2444no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—22no—
529C4⋊C4—612no—
530C4⋊C4—1222no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4C2×C4×D6112yes—
536C22⋊C4—22no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—1222no—
544C22⋊C4—1222no—
545C42—22no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—611no—
550C42—611no—
551C42—611no—
552C42—611no—
553C42—611no—
554C42—611no—
555C42—611no—
556C42—611no—
557C42—611no—
558C42—1222no—
559C42—1222no—
560C42C2×C4×D6111yescharacteristic
561C42D4×D6111yescharacteristic
562C42D4×D6111yescharacteristic
563C42C23×D6111yescharacteristic
564C23—611no—
565C23—611no—
566C23—1222no—
567C23—1222no—
568C23—1222no—
569C23—2441no—
570C23C22⋊C4×D6111yescharacteristic
571Q8—24no—
572Q8—42no—
573Q8—2442no—
574D4—1224no—
575D4—2444no—
576D4—2444no—
577D4—2444no—
578D4—2442no—
579D4—2442no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—611no—
585C2×C4—614no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—611no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1222no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2444no—
608C2×C4—2444no—
609C2×C4—2441no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4C22⋊C4×D6111yescharacteristic
613C2×C4C22⋊C4×D6111yescharacteristic
614C2×C4C22⋊C4×D6111yescharacteristic
615C2×C4#1451980412yes—
616C2×C4#1451980912yes—
617C2×C4C22×C4×D6111yescharacteristic
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C8—24no—
621C8—42no—
622C8—1224no—
623C8—2442no—
624C22—21no—
625C22—21no—
626C22—612no—
627C22—1221no—
628C22—1221no—
629C22—1221no—
630C22—1221no—
631C22—1221no—
632C22—2444no—
633C22—2442no—
634C22—2442no—
635C22—2442no—
636C22#1683976212yes—
637C22C2×C22⋊C4×D6111yescharacteristic
638C4—21no—
639C4—21no—
640C4—42no—
641C4—42no—
642C4—612no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—1221no—
647C4—1221no—
648C4—2442no—
649C4#1683976212yes—
650C4#104×D6112yes—
651C4#1684052712yes—
652C4C2×#1451980411yescharacteristic
653C4C2×#1451980911yescharacteristic
654C2—22no—
655C2—1222no—
656C2—4882no—
657C2—2441no—
658C2#2145426111yescharacteristic
659C2C2×#1683976211yescharacteristic
660C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1493C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC42×C61C24
hypercenter#1493D61
hypocenterC61#17504
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1493

Complements

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

NH ≅ G/Ntype
2. #423×D61C2semidirect
3. C2×C4×#11758189C2semidirect
4. #21444274C2semidirect
5. #21444271C2semidirect
6. #21444269C2semidirect
7. #21444257C2semidirect
8. #21444231C2semidirect
9. C61×#1493C2semidirect
42. Q8⋊#92418C22semidirect
43. #16839491C22semidirect
44. #16839489C22semidirect
48. C61×Q8⋊C8C22semidirect
328. #1493D61semidirect
376. Q8⋊C8C2×D61semidirect
379. C61#17504semidirect

Maximal subgroups

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

subgroupindex
#214591162
#423×D612
C2×C4×#117581892
#214442742
#214442712
#214442692
#214442572
#214442312
C61×#14932
#1750461