Finite Group Databasebeta

← #29647986

Subgroups — #29647986

Order 15616.

1060 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#432×D61C211yesmaximal, characteristic, complemented
3C2×#16839714C211yesmaximal, characteristic, complemented
4#21444435C211yesmaximal, characteristic, complemented
5#21444433C211yesmaximal, characteristic, complemented
6#21444430C211yesmaximal, characteristic, complemented
7#21444423C214yesmaximal, complemented
8#21444404C214yesmaximal, complemented
9C61×#1530C211yesmaximal, characteristic, complemented
10C22×#11758189—21no—
11#16840319C2211yescharacteristic
12#16840317C411yescharacteristic
13#16840317C411yescharacteristic
14#16840310C411yescharacteristic
15#16840310C411yescharacteristic
16#16840294—21no—
17#16840294—21no—
18#16840293—21no—
19#16840293—21no—
20#16840291C2212yes—
21#16840291C2212yes—
22#16840272—21no—
23#16840272—21no—
24#16840269C412yes—
25#41×Dic61C412yes—
26C4×#11758224—21no—
27C4×#11758224—21no—
28C2×C42×D61C2211yescharacteristic
29#16839819C2211yescharacteristic
30#16839818—21no—
31#16839817C2212yes—
32#16839816—21no—
33C4⋊C8×D61—21no—
34C22⋊C8×D61—21no—
35#16839714—21no—
36#16839714—21no—
37#16839714C2212yes—
38#16839714C2212yes—
39C2×#14519688C2211yescharacteristic
40#97×D61C2212yes—
41C2×C8×Dic61—21no—
42#16839503C2214yescomplemented
43#16839503C2214yescomplemented
44#16839502C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C2×#97×C61C2211yescharacteristic
48C61×#329C2214yescomplemented
49C2×#11758224—21no—
50C2×#11758224—21no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758189—28no—
55C2×#11758189—22no—
56C2×#11758189—22no—
57C22×#92418—21no—
58C2×C8×D61—22no—
59#14519777C2×C411yescharacteristic
60#14519777C2×C411yescharacteristic
61#14519770—22no—
62#14519770—22no—
63#14519769C2312yes—
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#14519736C2×C412yes—
100#14519735—21no—
101#14519735—21no—
102#14519733—22no—
103#14519733—22no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D411yescharacteristic
108C42×D61C2×C411yescharacteristic
109C42×D61C2×C411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61C2311yescharacteristic
115C42×D61C2311yescharacteristic
116C4⋊#92418—21no—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120#14519688—21no—
121#14519688C2312yes—
122#14519688C2312yes—
123#14519687—22no—
124C8×Dic61—22no—
125C8×Dic61—22no—
126C2×#41×C61C2×C412yes—
127C22×C8×C61—21no—
128C4⋊Q8×C61C2×C411yescharacteristic
129#127×C61C2×C411yescharacteristic
130#124×C61C2312yes—
131#121×C61—21no—
132#121×C61—21no—
133C4×D4×C61—21no—
134C4×D4×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C22⋊C8×C61—21no—
138#97×C61—21no—
139#97×C61C2312yes—
140#97×C61C2312yes—
141C23×D61C22⋊C411yescharacteristic
142#41×C61—44no—
143C2×Q8×C61C22×C412yes—
144C2×D4×C61—22no—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61C22×C412yes—
148C22×C4×C61—22no—
149C22×C4×C61C22×C411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61C22⋊C411yescharacteristic
157C22×Dic61C22⋊C411yescharacteristic
158C22×Dic61C22⋊C411yescharacteristic
159#11758224—44no—
160#11758224—44no—
161C2×C4×D61—24no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61C22⋊C411yescharacteristic
165C2×C4×D61C22⋊C411yescharacteristic
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×D61C2×D411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×C8×C61—24no—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2411yescharacteristic
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194#11758198—22no—
195#11758198—22no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22×C412yes—
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C2×#92418—24no—
222C2×#92418—22no—
223C2×#92418—22no—
224#11758189—28no—
225#11758189—28no—
226C8×D61—44no—
227C23×C61—22no—
228C23×C61C2×C22⋊C411yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—41no—
232C22×D61C2×C22⋊C411yescharacteristic
233C22×D61C2×C22⋊C411yescharacteristic
234Q8×C61—42no—
235D4×C61—42no—
236D4×C61—42no—
237D4×C61—44no—
238D4×C61—42no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—44no—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C22×D411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C22⋊D61—44no—
251C22⋊D61—44no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—44no—
258C2×Dic61—44no—
259C2×Dic61C2×C22⋊C411yescharacteristic
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274C4×D61—41no—
275C4×D61—41no—
276C4×D61#10414yes—
277Dic122—42no—
278Dic122—42no—
279C8×C61—24no—
280C8×C61—42no—
281#92418—24no—
282#92418—42no—
283#17651—611nomaximal, Sylow
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61#41312yes—
290C22×C61C22×C22⋊C411yescharacteristic
291C2×D61—21no—
292C2×D61—21no—
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—44no—
297C2×D61C2×#10412yes—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—42no—
304C4×C61C2×#10412yes—
305Dic61—21no—
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—42no—
310Dic61—42no—
311Dic61C2×#10412yes—
312Dic61#42112yes—
313#3444—611no—
314C2×#432—611no—
315C2×C4×M16—611no—
316#1531—611no—
317#1531—611no—
318#1530—611no—
319#1530D6111yescharacteristic, complemented
320#1529—614no—
321#1524—614no—
322C2×C61—22no—
323C2×C61—82no—
324C2×C61C2×#41311yescharacteristic
325C2×C61C22×#10411yescharacteristic
326C2×C61C2×#42111yescharacteristic
327D61—22no—
328D61—41no—
329C22×M16—1221no—
330#533—611no—
331#533—611no—
332#533—611no—
333#532—611no—
334#532—611no—
335#532C2×D6111yescharacteristic
336#529—612no—
337#529—612no—
338#529—1221no—
339#529—1221no—
340#529—1221no—
341#529—1221no—
342#525—612no—
343#525—1221no—
344#525—1221no—
345C4×#41—612no—
346C4×#41—1221no—
347C4×#41—1221no—
348C22×C42—611no—
349#432—611no—
350#432—612no—
351#432—1221no—
352#432—1221no—
353#432C2×D6111yescharacteristic
354C2×C4⋊C8—1221no—
355C2×C22⋊C8—1221no—
356C4×M16—612no—
357C4×M16—612no—
358C4×M16—1221no—
359C4×M16—1221no—
360C2×#97—611no—
361C2×#97—612no—
362C2×#97C2×D6111yescharacteristic
363C2×C4×C8—1221no—
364#330—614no—
365#330—614no—
366#329—614no—
367#329C2×D6114yescomplemented
368C61#1765111yescharacteristic, complemented, Sylow
369C2×#41—1221no—
370C2×#41—1221no—
371C2×#41C4×D6112yes—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—611no—
375C2×M16—1222no—
376C2×M16—1222no—
377C2×M16—1228no—
378C22×C8—21no—
379C22×C8—1222no—
380C22×C8—1221no—
381C4⋊Q8—611no—
382C4⋊Q8—611no—
383C4⋊Q8C4×D6111yescharacteristic
384#127C4×D6111yescharacteristic
385#126—1222no—
386#126—1222no—
387#125—612no—
388#125—1221no—
389#125—1221no—
390#124—1221no—
391#124—1221no—
392#124C22×D6112yes—
393#123—1222no—
394#123—1222no—
395#123—1222no—
396C22⋊Q8—612no—
397C22⋊Q8—612no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C22⋊Q8—1221no—
402#121—21no—
403#121—21no—
404#121—612no—
405#121—612no—
406#121—1221no—
407#121—1221no—
408#120—1222no—
409C4×Q8—612no—
410C4×Q8—612no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8—1221no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—612no—
418C4×D4—612no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427#117—612no—
428#117—1222no—
429#117—1222no—
430#117—1222no—
431#117—1221no—
432#117—1221no—
433C2×C42—611no—
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—1222no—
445C2×C42C22×D6111yescharacteristic
446C4⋊C8—21no—
447C4⋊C8—1222no—
448C4⋊C8—1221no—
449C22⋊C8—21no—
450C22⋊C8—1222no—
451C22⋊C8—1221no—
452#97—21no—
453#97—612no—
454#97—612no—
455#97—1222no—
456#97—1221no—
457#97C22×D6112yes—
458#97C22×D6112yes—
459C4×C8—1222no—
460C4×C8—1222no—
461C24—611no—
462#41—44no—
463#41—2444no—
464#41—2444no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8C2×C4×D6112yes—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—1222no—
472C2×D4—1222no—
473C2×D4C2×C4×D6112yes—
474C22×C4—22no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
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—1224no—
491C22×C4—1222no—
492C22×C4—1222no—
493C22×C4—2442no—
494C22×C4—2442no—
495C22×C4C2×C4×D6111yescharacteristic
496C22×C4D4×D6111yescharacteristic
497C22×C4D4×D6111yescharacteristic
498M16—1228no—
499M16—1228no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—1224no—
504C2×C8—1222no—
505C2×C8—1222no—
506C2×C8—2444no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—612no—
510C4⋊C4—612no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—1222no—
541C42—1222no—
542C42—1222no—
543C42C2×C4×D6111yescharacteristic
544C42D4×D6111yescharacteristic
545C42D4×D6111yescharacteristic
546C42C23×D6111yescharacteristic
547C23—22no—
548C23—611no—
549C23—611no—
550C23—1222no—
551C23—1222no—
552C23—2441no—
553C23C22⋊C4×D6111yescharacteristic
554Q8—42no—
555Q8—2442no—
556Q8—2442no—
557D4—42no—
558D4—42no—
559D4—44no—
560D4—42no—
561D4—2444no—
562D4—2444no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—611no—
569C2×C4—611no—
570C2×C4—614no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—611no—
574C2×C4—611no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2444no—
592C2×C4—2444no—
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—24no—
600C8—42no—
601C8—1224no—
602C8—2442no—
603C22—21no—
604C22—21no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—612no—
609C22—1221no—
610C22—1221no—
611C22—1221no—
612C22—1221no—
613C22—1221no—
614C22—2444no—
615C22#1683974312yes—
616C22C2×C22⋊C4×D6111yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—42no—
623C4—612no—
624C4—612no—
625C4—1221no—
626C4—1221no—
627C4—1221no—
628C4—1221no—
629C4—2442no—
630C4—2442no—
631C4#104×D6112yes—
632C2—22no—
633C2—82no—
634C2—1222no—
635C2—2441no—
636C2C2×#1683974311yescharacteristic
637C2C2×#104×D6111yescharacteristic
638C2#421×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1530C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1530D61
hypocenterC61#17651
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1530

Complements

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

NH ≅ G/Ntype
2. #432×D61C2semidirect
3. C2×#16839714C2semidirect
4. #21444435C2semidirect
5. #21444433C2semidirect
6. #21444430C2semidirect
7. #21444423C2semidirect
8. #21444404C2semidirect
9. C61×#1530C2semidirect
42. #16839503C22semidirect
43. #16839503C22semidirect
44. #16839502C22semidirect
48. C61×#329C22semidirect
319. #1530D61semidirect
367. #329C2×D61semidirect
368. C61#17651semidirect

Maximal subgroups

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

subgroupindex
#214591142
#432×D612
C2×#168397142
#214444352
#214444332
#214444302
#214444232
#214444042
C61×#15302
#1765161