Finite Group Databasebeta

← #29647284

Subgroups — #29647284

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459116C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3#21448216C211yesmaximal, characteristic, complemented
4#21444370C211yesmaximal, characteristic, complemented
5#21444367C211yesmaximal, characteristic, complemented
6#21444365C211yesmaximal, characteristic, complemented
7#21444361C214yesmaximal, complemented
8#21444304C214yesmaximal, complemented
9C61×#1515C211yesmaximal, characteristic, complemented
10#16840317C411yescharacteristic, complemented
11#16840317C411yescharacteristic, complemented
12#16840313C411yescharacteristic, complemented
13#16840313C411yescharacteristic, complemented
14#16840312C2211yescharacteristic
15#16840299—21no—
16#16840299C412yescomplemented
17#16840298—21no—
18#16840298C2212yes—
19#16840297—21no—
20#16840297—21no—
21#16840275—21no—
22#16840275C2212yes—
23#16840274—21no—
24C4×#11758226—21no—
25C4×#11758226C412yescomplemented
26#16840230—21no—
27C2×C42×D61C2211yescharacteristic
28#16839819C2211yescharacteristic
29#16839817—21no—
30#16839814—21no—
31#16839812C2212yes—
32#16839784—21no—
33#16839724C2212yes—
34#16839714—21no—
35#16839630C2211yescharacteristic
36#16839626—21no—
37#16839624—21no—
38#16839621C2212yes—
39#16839613—21no—
40#16839609—21no—
41#16839607C2212yes—
42#16839499C2214yes—
43#16839497C2214yes—
44#16839495C2214yes—
45C61×#533C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#383C2211yescharacteristic
48C61×#328C2214yes—
49C2×#11758226—21no—
50C2×#11758226C2×C412yes—
51C2×#11758222—21no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758189—22no—
56#14519777C2×C411yescharacteristic
57#14519774C2×C411yescharacteristic
58#14519772—22no—
59#14519772—22no—
60#14519768—21no—
61#14519768C2×C412yes—
62#14519766—21no—
63#14519766C2×C412yes—
64#14519765—21no—
65#14519762—22no—
66#14519762—22no—
67#14519761—22no—
68#14519757—21no—
69#14519757—21no—
70#14519753—21no—
71#14519753C2×C412yes—
72#14519752—21no—
73#14519752—21no—
74#14519752—21no—
75#14519752C2×C412yes—
76#14519751—21no—
77#14519751C2312yes—
78#14519747—22no—
79Q8×Dic61—21no—
80Q8×Dic61C2×C412yes—
81#14519742—21no—
82#14519742—21no—
83C4×D244—22no—
84C4×D244—22no—
85C4×D244—21no—
86C4×D244C2×C412yes—
87#14519739—22no—
88#14519739—22no—
89#14519739—21no—
90#14519739—21no—
91#14519739—21no—
92#14519739C2×C412yes—
93#14519737—22no—
94#14519737—22no—
95#14519735—21no—
96#14519735—22no—
97#14519735—22no—
98#14519735C2×C412yes—
99#14519734—22no—
100C2×#11758196—22no—
101C2×#11758198—22no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C42×D61—22no—
107C42×D61C2×C411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61C2×C411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C4⋊#92418—21no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118#14519689—22no—
119#14519688—21no—
120C8×Dic61—22no—
121#14519684—21no—
122#14519684—21no—
123#14519684—22no—
124#14519684C2312yes—
125C4⋊Q8×C61C2×C411yescharacteristic
126C4⋊Q8×C61C2×C411yescharacteristic
127#125×C61—21no—
128C22⋊Q8×C61—21no—
129C22⋊Q8×C61C2312yes—
130C4×Q8×C61—21no—
131C4×Q8×C61C2×C412yes—
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C22⋊C8×C61C2312yes—
136#97×C61—21no—
137#95×C61—21no—
138#95×C61—21no—
139#95×C61C2312yes—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61C22×C412yes—
143C22×C4×C61—22no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C2×C22⋊D61—22no—
148C22×Dic61—22no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152#11758226—44no—
153#11758226—44no—
154#11758222—44no—
155C2×D244—22no—
156C2×D244—22no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C22×C412yes—
160C2×C4×D61—22no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61C22⋊C411yescharacteristic
167C2×C4×D61C22⋊C411yescharacteristic
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×Dic122—21no—
179C2×C8×C61—22no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C42×C61—22no—
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—44no—
201C4⋊Dic61—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207#11758196—44no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—44no—
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
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
223C2×#92418—22no—
224#11758189—44no—
225C23×C61C22×D411yescharacteristic
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61C2×C22⋊C411yescharacteristic
231C22×D61C2×C22⋊C411yescharacteristic
232Q8×C61—42no—
233Q8×C61—42no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—42no—
240C2×C4×C61—42no—
241C2×C4×C61—42no—
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C23×C411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61—44no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61C2×C22⋊C411yescharacteristic
258C2×Dic61C2×C22⋊C411yescharacteristic
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262D244—42no—
263D244—42no—
264D244—42no—
265D244—44no—
266D244—44no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—22no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61—41no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—44no—
280C4×D61—44no—
281C4×D61—44no—
282Dic122—42no—
283C8×C61—42no—
284#92418—42no—
285#17558—611nomaximal, Sylow
286C22×C61—21no—
287C22×C61—21no—
288C22×C61C22×C22⋊C411yescharacteristic
289C22×C61#57512yes—
290C2×D61—21no—
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—44no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—21no—
306C4×C61—42no—
307C4×C61—42no—
308C4×C61—82no—
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—21no—
313Dic61—42no—
314Dic61—82no—
315Dic61#41012yes—
316Dic61#42112yes—
317#3444—611no—
318#2920—611no—
319#2283—611no—
320#1515—611no—
321#1515D6111yescharacteristic, complemented
322#1514—611no—
323#1514—611no—
324#1513—614no—
325#1500—614no—
326C2×C61—22no—
327C2×C61C2×#41011yescharacteristic
328C2×C61#289611yescharacteristic
329C2×C61C2×#42111yescharacteristic
330D61—41no—
331D61—41no—
332D61—82no—
333#533—611no—
334#533—611no—
335#533C2×D6111yescharacteristic
336#532—611no—
337#532—611no—
338#532—611no—
339#529—612no—
340#529—612no—
341#529—1221no—
342#529—1221no—
343#529—1221no—
344#529—1221no—
345#525—612no—
346#525—1221no—
347#525—1221no—
348C4×#41—612no—
349C4×#41—1221no—
350C4×#41—1221no—
351C22×C42—611no—
352#432—611no—
353#432—1221no—
354#432C2×D6111yescharacteristic
355#431—612no—
356#431—1221no—
357#423—1221no—
358#407—612no—
359C4×M16—1221no—
360#383—611no—
361#383C2×D6111yescharacteristic
362#382—1221no—
363#381—1221no—
364#380—612no—
365C4×C4⋊C4—1221no—
366C4×C22⋊C4—1221no—
367#376—612no—
368#328—614no—
369#328C2×D6114yes—
370#327—614no—
371#327—614no—
372C61#1755811yescharacteristic, complemented, Sylow
373C2×#41—612no—
374C2×#41—1221no—
375C2×#41—1221no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C2×M16—1222no—
380C4⋊Q8—611no—
381C4⋊Q8C4×D6111yescharacteristic
382C4⋊Q8C4×D6111yescharacteristic
383#127—611no—
384#126—1222no—
385#126—1222no—
386#125—21no—
387#125—612no—
388#125—1221no—
389#124—612no—
390#124—1221no—
391#124—1221no—
392#123—1222no—
393#123—1222no—
394#123—1222no—
395C22⋊Q8—21no—
396C22⋊Q8—612no—
397C22⋊Q8—1221no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8C22×D6112yes—
401#121—612no—
402#121—612no—
403#121—1221no—
404#121—1221no—
405#121—1221no—
406#121—1221no—
407#120—1222no—
408C4×Q8—21no—
409C4×Q8—612no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×Q8C4×D6112yes—
414C4×D4—612no—
415C4×D4—612no—
416C4×D4—1222no—
417C4×D4—1222no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1221no—
426#117—21no—
427#117—612no—
428#117—1222no—
429#117—1222no—
430#117—1222no—
431#117—1221no—
432C2×C4⋊C4—1222no—
433C2×C22⋊C4—1222no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—1222no—
446C2×C42—1222no—
447C2×C42C22×D6111yescharacteristic
448C4⋊C8—21no—
449C4⋊C8—1222no—
450C4⋊C8—1221no—
451C22⋊C8—612no—
452C22⋊C8—1222no—
453C22⋊C8C22×D6112yes—
454#97—21no—
455#97—1221no—
456C4×C8—1222no—
457#95—21no—
458#95—21no—
459#95—612no—
460#95—1222no—
461#95—1221no—
462#95—1221no—
463#95C22×D6112yes—
464C24—611no—
465#41—2444no—
466#41—2444no—
467#41—2444no—
468C2×Q8—21no—
469C2×Q8—1221no—
470C2×Q8C2×C4×D6112yes—
471C2×D4—612no—
472C2×D4—1222no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—1221no—
476C2×D4—1221no—
477C22×C4—22no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4—2442no—
500C22×C4D4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4C23×D6111yescharacteristic
503M16—2444no—
504C2×C8—22no—
505C2×C8—1222no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1221no—
517C4⋊C4—2444no—
518C4⋊C4C2×C4×D6112yes—
519C22⋊C4—22no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—2444no—
529C42—22no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—1222no—
543C42—1222no—
544C42—2444no—
545C42C2×C4×D6111yescharacteristic
546C42C2×C4×D6111yescharacteristic
547C42D4×D6111yescharacteristic
548C42D4×D6111yescharacteristic
549C23—611no—
550C23—611no—
551C23—1222no—
552C23—1222no—
553C23—2441no—
554C23—2441no—
555C23C2×D4×D6111yescharacteristic
556Q8—42no—
557Q8—42no—
558Q8—2442no—
559D4—2442no—
560D4—2442no—
561D4—2442no—
562D4—2444no—
563D4—2444no—
564D4—2444no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2444no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22⋊C4×D6111yescharacteristic
604C2×C4C22⋊C4×D6111yescharacteristic
605C2×C4C22×C4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C8—42no—
608C8—2442no—
609C22—21no—
610C22—21no—
611C22—1221no—
612C22—1221no—
613C22—1221no—
614C22—1221no—
615C22—1221no—
616C22—1221no—
617C22—2444no—
618C22—2442no—
619C22—2442no—
620C22—2442no—
621C22C2×C22⋊C4×D6111yescharacteristic
622C22#1684056112yes—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—42no—
630C4—42no—
631C4—82no—
632C4—612no—
633C4—612no—
634C4—1221no—
635C4—1221no—
636C4—1221no—
637C4—1221no—
638C4—4882no—
639C4—2442no—
640C2—22no—
641C2—4882no—
642C2—2441no—
643C2—2441no—
644C2#410×D6111yescharacteristic
645C2#2145433711yescharacteristic
646C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1515C2
socleC22×C61C22×C22⋊C4
prime radical#21459116C2
cosocleC22×C4×C61C24
hypercenter#1515D61
hypocenterC61#17558
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1515

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. #21448216C2semidirect
4. #21444370C2semidirect
5. #21444367C2semidirect
6. #21444365C2semidirect
7. #21444361C2semidirect
8. #21444304C2semidirect
9. C61×#1515C2semidirect
10. #16840317C4semidirect
11. #16840317C4semidirect
12. #16840313C4semidirect
13. #16840313C4semidirect
16. #16840299C4semidirect
25. C4×#11758226C4semidirect
321. #1515D61semidirect
372. C61#17558semidirect

Maximal subgroups

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

subgroupindex
#214591162
#214544492
#214482162
#214443702
#214443672
#214443652
#214443612
#214443042
C61×#15152
#1755861