Finite Group Databasebeta

← #29646344

Subgroups — #29646344

Order 15616.

1016 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840494C211yesmaximal, characteristic
2#21454856C211yesmaximal, characteristic, complemented
3#21454853C211yesmaximal, characteristic, complemented
4#21444171C211yesmaximal, characteristic, complemented
5#21444164C212yesmaximal, complemented
6#21444164C212yesmaximal, complemented
7#21444162C211yesmaximal, characteristic, complemented
8#21444154C212yesmaximal, complemented
9#21444142C211yesmaximal, characteristic, complemented
10#21444134C212yesmaximal, complemented
11C61×#1469C211yesmaximal, characteristic, complemented
12#16840494—21no—
13#16840494—21no—
14#16840494—22no—
15#16840494—22no—
16#16840494—21no—
17#16840494—21no—
18C2×#14519773C411yescharacteristic
19C2×#14519766C411yescharacteristic
20C2×#14519763C412yes—
21C2×#14519761C412yes—
22C2×#14519759C2212yes—
23C2×Q8×Dic61C411yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×D4×Dic61C411yescharacteristic
27C2×#14519737C2212yes—
28#16839841—21no—
29#16839839—21no—
30#16839835—22no—
31#16839832—22no—
32C8×D244—21no—
33#16839827—21no—
34#16839819C2211yescharacteristic
35#16839815C2211yescharacteristic
36#16839724C2212yes—
37#16839724C2212yes—
38C22⋊C8×D61C2212yes—
39C22⋊C8×D61C2212yes—
40#16839487C2211yescharacteristic
41#16839487C2212yes—
42#16839487C2211yescharacteristic
43#16839487C2211yescharacteristic
44#16839487C2211yescharacteristic
45#16839487C2212yes—
46#16839486C2212yes—
47#16839486C2212yes—
48C2×#124×C61C2211yescharacteristic
49C61×#432C2211yescharacteristic
50C61×#431C2211yescharacteristic
51C61×#323C2212yes—
52C61×#323C2212yes—
53C22×Dic122C2×C411yescharacteristic
54C22×D244C2×C411yescharacteristic
55C22×C22⋊D61C2×C412yes—
56C22×C22⋊D61C2×C412yes—
57C23×Dic61C2×C412yes—
58C22×C4×D61C2312yes—
59C2×#11758189—22no—
60C2×#11758189—22no—
61C2×C8×D61—22no—
62C2×C8×D61—22no—
63#14519773—21no—
64#14519773—21no—
65#14519773—22no—
66#14519766—21no—
67#14519766—21no—
68#14519766—22no—
69#14519763—22no—
70#14519763—22no—
71#14519763—22no—
72#14519763—22no—
73#14519761—22no—
74#14519761—22no—
75#14519761—22no—
76#14519761—22no—
77#14519759—22no—
78#14519759—22no—
79#14519759—22no—
80#14519759—22no—
81Q8×Dic61—22no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×D244D411yescharacteristic
89C4×D244D411yescharacteristic
90C4×D244D412yes—
91C4×D244D412yes—
92C4×D244D411yescharacteristic
93C4×D244D411yescharacteristic
94D4×Dic61—22no—
95D4×Dic61—21no—
96D4×Dic61—21no—
97#14519737—22no—
98#14519737—22no—
99#14519737—22no—
100#14519737—22no—
101C2×C4⋊Dic61C2×C411yescharacteristic
102C2×C4⋊Dic61C2×C412yes—
103C2×C4⋊Dic61C2×C411yescharacteristic
104C2×#11758196C2311yescharacteristic
105C2×#11758196C2311yescharacteristic
106C2×C22⋊Dic61C2×C412yes—
107C2×#11758198C2×C412yes—
108C2×#11758198C2×C412yes—
109C2×#11758198C2×C412yes—
110C2×C4×Dic61C2×C412yes—
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61C2×C411yescharacteristic
113C4⋊#92418—21no—
114C4⋊#92418—21no—
115C22⋊#92418C2311yescharacteristic
116C22⋊#92418C2312yes—
117C22⋊#92418C2311yescharacteristic
118#14519689—22no—
119#14519689—22no—
120#14519689—22no—
121#14519689—22no—
122#14519688—21no—
123C4×#92418—21no—
124C22×Q8×C61C2×C411yescharacteristic
125C22×D4×C61C2×C411yescharacteristic
126#124×C61—21no—
127#124×C61—21no—
128#124×C61—21no—
129#124×C61—21no—
130C2×C22⋊C4×C61C2×C412yes—
131C2×C22⋊C4×C61C2312yes—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—21no—
134C4⋊C8×C61—21no—
135C22⋊C8×C61C2311yescharacteristic
136C22⋊C8×C61C2312yes—
137C22⋊C8×C61C2311yescharacteristic
138#97×C61—21no—
139C4×C8×C61—21no—
140C24×C61C22×C412yes—
141C23×D61C22×C412yes—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—42no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—42no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C22×C412yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—24no—
157C2×C22⋊D61—42no—
158C2×C22⋊D61—82no—
159C22×Dic61—22no—
160C22×Dic61—22no—
161C22×Dic61—82no—
162C22×Dic61C22×C411yescharacteristic
163C22×Dic61C22×C411yescharacteristic
164C22×Dic61C22×C411yescharacteristic
165C22×Dic61C22×C411yescharacteristic
166C22×Dic61C22×C412yes—
167C2×D244—42no—
168C2×D244C22⋊C411yescharacteristic
169C2×D244C22⋊C411yescharacteristic
170C2×D244C22⋊C412yes—
171C2×C4×D61—82no—
172C2×C4×D61C2×D412yes—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×D412yes—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—22no—
179C2×Dic122—41no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198C22⋊C412yes—
198#11758198C22⋊C412yes—
199#11758198C22⋊C412yes—
200#11758198C22⋊C412yes—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61C22⋊C412yes—
206C4⋊Dic61C22⋊C411yescharacteristic
207C4⋊Dic61C22⋊C411yescharacteristic
208#11758196—21no—
209#11758196—21no—
210#11758196—21no—
211#11758196—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—22no—
221#11758189—82no—
222#11758189—82no—
223C8×D61—82no—
224C8×D61—82no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61C23×C411yescharacteristic
231C22×D61—22no—
232C22×D61—22no—
233C22×D61—42no—
234C22×D61—42no—
235C22×D61C2×C22⋊C412yes—
236C22×D61C2×C22⋊C412yes—
237Q8×C61—42no—
238Q8×C61—42no—
239D4×C61—42no—
240D4×C61—42no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—41no—
246C2×C4×C61—42no—
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C22⋊D61—44no—
254C22⋊D61—44no—
255C22⋊D61—82no—
256C22⋊D61—82no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—42no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—82no—
268C2×Dic61—82no—
269C2×Dic61C2×C22⋊C412yes—
270C2×Dic61C2×C22⋊C412yes—
271C2×Dic61#13112yes—
272D244—22no—
273D244—22no—
274D244—42no—
275C4×D61—42no—
276C4×D61—42no—
277C4×D61—42no—
278C4×D61—42no—
279Dic122—42no—
280Dic122—42no—
281C8×C61—41no—
282C8×C61—42no—
283C8×C61—41no—
284#92418—41no—
285#92418—42no—
286#92418—41no—
287#17429—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61—82no—
294C22×C61—82no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61C2×#13111yescharacteristic
297C22×C61#56811yescharacteristic
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—82no—
305C2×D61—82no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—21no—
309C4×C61—21no—
310C4×C61—21no—
311C4×C61—42no—
312C4×C61#41312yes—
313Dic61—22no—
314Dic61—22no—
315Dic61—22no—
316Dic61—41no—
317Dic61—41no—
318Dic61—41no—
319C2×#559—611no—
320#2974—611no—
321#2974—611no—
322#1469—611no—
323#1469D6111yescharacteristic, complemented
324#1468—612no—
325#1468—612no—
326#1467—611no—
327#1465—612no—
328#1461—611no—
329#1459—612no—
330C2×C61—21no—
331C2×C61—21no—
332C2×C61—82no—
333C2×C61#288011yescharacteristic
334C2×C61#288311yescharacteristic
335C2×C61C2×#41311yescharacteristic
336D61—42no—
337D61—42no—
338#559—1221no—
339#559—1221no—
340#559—1222no—
341#559—1222no—
342#559—1221no—
343#559—1221no—
344C2×#127—611no—
345C2×#124—611no—
346C2×#124—612no—
347C2×#124C2×D6111yescharacteristic
348C2×#123—612no—
349C2×#123—612no—
350C2×C4×Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×D4—611no—
353C2×C4×D4—612no—
354C2×C4×D4—611no—
355#436—1221no—
356#436—1221no—
357#435—1222no—
358#435—1222no—
359C8×D4—1221no—
360C8×D4—1221no—
361#432—611no—
362#432C2×D6111yescharacteristic
363#431—611no—
364#431C2×D6111yescharacteristic
365#407—612no—
366#407—612no—
367C2×C22⋊C8—612no—
368C2×C22⋊C8—612no—
369#324—611no—
370#324—612no—
371#324—611no—
372#324—611no—
373#324—611no—
374#324—612no—
375#323—612no—
376#323—612no—
377#323C2×D6112yes—
378#323C2×D6112yes—
379C61#1742911yescharacteristic, complemented, Sylow
380C22×Q8—611no—
381C22×Q8C4×D6111yescharacteristic
382C22×D4—612no—
383C22×D4—612no—
384C22×D4—611no—
385C22×D4C4×D6111yescharacteristic
386C23×C4—612no—
387C23×C4—612no—
388C2×M16—1222no—
389C2×M16—1222no—
390C22×C8—1222no—
391C22×C8—1222no—
392#127—1221no—
393#127—1221no—
394#127—1222no—
395#124—21no—
396#124—21no—
397#124—21no—
398#124—21no—
399#124—1222no—
400#124—1222no—
401#124—1222no—
402#124—1222no—
403#124—1221no—
404#124—1221no—
405#124—1222no—
406#123—1222no—
407#123—1222no—
408#123—1222no—
409#123—1222no—
410#123—1222no—
411#123—1222no—
412#123—1222no—
413#123—1222no—
414C4×Q8—1222no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8—1221no—
420C4×Q8—1221no—
421C4×D4—611no—
422C4×D4—611no—
423C4×D4—612no—
424C4×D4—612no—
425C4×D4—611no—
426C4×D4—611no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—1222no—
431C4×D4—1222no—
432C4×D4—1221no—
433C4×D4—1221no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—612no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C22⋊C4—612no—
440C2×C22⋊C4—612no—
441C2×C22⋊C4—612no—
442C2×C22⋊C4—612no—
443C2×C22⋊C4C4×D6112yes—
444C2×C22⋊C4C22×D6112yes—
445C2×C42—612no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42C22×D6111yescharacteristic
449C4⋊C8—21no—
450C4⋊C8—21no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453C22⋊C8—611no—
454C22⋊C8—612no—
455C22⋊C8—611no—
456C22⋊C8—1222no—
457C22⋊C8—1222no—
458C22⋊C8—1222no—
459C22⋊C8—1222no—
460C22⋊C8C22×D6111yescharacteristic
461C22⋊C8C22×D6112yes—
462C22⋊C8C22×D6111yescharacteristic
463#97—21no—
464#97—1221no—
465C4×C8—21no—
466C4×C8—1221no—
467C24—612no—
468C24C2×C4×D6112yes—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—42no—
472C2×Q8—1222no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—2441no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—42no—
479C2×D4—611no—
480C2×D4—611no—
481C2×D4—612no—
482C2×D4—1224no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—1222no—
486C2×D4—1222no—
487C2×D4—4882no—
488C2×D4—2442no—
489C2×D4—2442no—
490C22×C4—611no—
491C22×C4—612no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—612no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—4882no—
502C22×C4—4882no—
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6112yes—
506C22×C4C23×D6111yescharacteristic
507M16—4882no—
508M16—4882no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—22no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1222no—
515C2×C8—4882no—
516C2×C8—4882no—
517C4⋊C4—612no—
518C4⋊C4—611no—
519C4⋊C4—611no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—612no—
533C22⋊C4—612no—
534C22⋊C4—612no—
535C22⋊C4—612no—
536C22⋊C4—1222no—
537C22⋊C4—1222no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C42—1222no—
543C42—1222no—
544C42—1221no—
545C42—1221no—
546C42—1221no—
547C42—1221no—
548C42D4×D6111yescharacteristic
549C42D4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C42D4×D6111yescharacteristic
552C23—22no—
553C23—22no—
554C23—22no—
555C23—42no—
556C23—42no—
557C23—612no—
558C23—612no—
559C23—1222no—
560C23—1222no—
561C23—2442no—
562C23—2442no—
563C23C22×C4×D6111yescharacteristic
564Q8—42no—
565Q8—42no—
566Q8—2442no—
567Q8—2442no—
568D4—42no—
569D4—42no—
570D4—1222no—
571D4—1222no—
572D4—4882no—
573D4—4882no—
574D4—2442no—
575D4—2444no—
576D4—2444no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—612no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—4882no—
594C2×C4—4882no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C8—41no—
609C8—42no—
610C8—41no—
611C8—2441no—
612C8—2442no—
613C8—2441no—
614C22—21no—
615C22—21no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—82no—
620C22—82no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—1222no—
626C22—1222no—
627C22—4882no—
628C22—4882no—
629C22C2×C22⋊C4×D6111yescharacteristic
630C22#131×D6111yescharacteristic
631C22#1684052711yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—42no—
638C4—1222no—
639C4—1222no—
640C4—1222no—
641C4—2441no—
642C4—2441no—
643C4—2441no—
644C4#1683974312yes—
645C2—21no—
646C2—21no—
647C2—82no—
648C2—2442no—
649C2—2442no—
650C2#2145426711yescharacteristic
651C2#2145427811yescharacteristic
652C2C2×#1683974311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1469C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519773C4
cosocleC22×C4×C61C24
hypercenter#1469D61
hypocenterC61#17429
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1469

Complements

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

NH ≅ G/Ntype
2. #21454856C2semidirect
3. #21454853C2semidirect
4. #21444171C2semidirect
5. #21444164C2semidirect
6. #21444164C2semidirect
7. #21444162C2semidirect
8. #21444154C2semidirect
9. #21444142C2semidirect
10. #21444134C2semidirect
11. C61×#1469C2semidirect
323. #1469D61semidirect
379. C61#17429semidirect

Maximal subgroups

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

subgroupindex
C2×#168404942
#214548562
#214548532
#214441712
#214441642
#214441622
#214441542
#214441422
#214441342
C61×#14692
#1742961