Finite Group Databasebeta

← #29648421

Subgroups — #29648421

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3C2×#16839814C211yesmaximal, characteristic, complemented
4#21444526C212yesmaximal, complemented
5#21444521C212yesmaximal, complemented
6#21444510C211yesmaximal, characteristic, complemented
7#21444507C211yesmaximal, characteristic, complemented
8#21444496C211yesmaximal, characteristic, complemented
9#21444482C212yesmaximal, complemented
10#21444478C212yesmaximal, complemented
11C61×#1547C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840317C2211yescharacteristic
14#16840317C2211yescharacteristic
15#16840310C2211yescharacteristic
16#16840310C2211yescharacteristic
17#16840294—21no—
18#16840294C412yes—
19#16840293—21no—
20#16840293C412yes—
21#16840291—21no—
22#16840291—21no—
23#16840272—21no—
24#16840272C412yes—
25#16840269—21no—
26#41×Dic61—21no—
27C4×#11758224—21no—
28C4×#11758224C412yes—
29C2×C42×D61C2211yescharacteristic
30#16839818—21no—
31#16839818C2212yes—
32#16839814—21no—
33#16839814—21no—
34#16839814—21no—
35#16839814C2212yes—
36#16839814C2212yes—
37#16839814C2212yes—
38#16839786C2211yescharacteristic
39C2×C4⋊#92418C2211yescharacteristic
40#16839725—21no—
41C2×#14519689—21no—
42M16×Dic61—21no—
43C2×C8×Dic61—21no—
44#16839510C2212yescomplemented
45#16839510C2212yescomplemented
46#16839509C2212yescomplemented
47#16839508C2212yescomplemented
48#16839507C2212yescomplemented
49#16839506C2212yescomplemented
50C61×#532C2211yescharacteristic
51C61×#423C2211yescharacteristic
52C2×C4⋊C8×C61C2211yescharacteristic
53C61×#332C2212yescomplemented
54C61×#331C2212yescomplemented
55C2×#11758224—21no—
56C2×#11758224C2×C412yes—
57C22×C4×D61C2×C411yescharacteristic
58C22×C4×D61D411yescharacteristic
59C22×C4×D61D411yescharacteristic
60C2×#11758194—21no—
61C22×#92418—21no—
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519770—22no—
65#14519770—22no—
66#14519769—21no—
67#14519768—21no—
68#14519768C2×C412yes—
69#14519764—21no—
70#14519764C2×C412yes—
71#14519759—22no—
72#14519759—22no—
73#14519758—22no—
74#14519757—21no—
75#14519757C2×C412yes—
76#14519756—21no—
77#14519756C2×C412yes—
78#14519753—21no—
79#14519753—21no—
80#14519748—21no—
81#14519748—21no—
82#14519748—21no—
83#14519748C2×C412yes—
84#14519745—22no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87C4×Dic122—21no—
88C4×Dic122C2×C412yes—
89#14519742—21no—
90#14519742C2×C412yes—
91D4×Dic61—22no—
92D4×Dic61—22no—
93D4×Dic61—21no—
94D4×Dic61—21no—
95D4×Dic61—21no—
96D4×Dic61C2×C412yes—
97C4×C22⋊D61—22no—
98C4×C22⋊D61—22no—
99#14519737—22no—
100#14519737—22no—
101#14519736—21no—
102#14519736—22no—
103#14519735—21no—
104#14519735C2×C412yes—
105#14519733—22no—
106#14519733—22no—
107C2×C4×Dic61—22no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61D411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61C2311yescharacteristic
118C42×D61C2311yescharacteristic
119C4⋊#92418—21no—
120C4⋊#92418—21no—
121C4⋊#92418C2312yes—
122C4⋊#92418C2312yes—
123#14519689—22no—
124#14519689—22no—
125#14519689—22no—
126#14519689—22no—
127#14519687—22no—
128C8×Dic61—22no—
129C8×Dic61—22no—
130C8×Dic61—22no—
131C2×#41×C61—21no—
132C2×M16×C61—21no—
133C22×C8×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136#124×C61—21no—
137#121×C61—21no—
138#121×C61C2×C412yes—
139C4×D4×C61—21no—
140C4×D4×C61C2×C412yes—
141C2×C42×C61C2311yescharacteristic
142C4⋊C8×C61—21no—
143C4⋊C8×C61—21no—
144C4⋊C8×C61C2312yes—
145C4⋊C8×C61C2312yes—
146C23×D61C22⋊C411yescharacteristic
147#41×C61—42no—
148#41×C61—42no—
149C2×Q8×C61—21no—
150C2×D4×C61—22no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61C22×C412yes—
154C22×C4×C61—22no—
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2×D411yescharacteristic
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160C22×Dic61—22no—
161C22×Dic61—22no—
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165#11758224—42no—
166#11758224—42no—
167#11758224—44no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—42no—
171C2×C4×D61—41no—
172C2×C4×D61—41no—
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22×C411yescharacteristic
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×D411yescharacteristic
185C2×Dic122—21no—
186C2×Dic122C22×C412yes—
187M16×C61—42no—
188C2×C8×C61—22no—
189C2×C8×C61—22no—
190C2×C8×C61—42no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—22no—
195C42×C61C22×C411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2411yescharacteristic
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—22no—
203C22⋊Dic61—22no—
204#11758198—22no—
205#11758198—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—22no—
210C4⋊Dic61—22no—
211C4⋊Dic61C22×C412yes—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22⋊C411yescharacteristic
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22×C411yescharacteristic
226C4×Dic61C22×C411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228C4×Dic61C2×D411yescharacteristic
229C4×Dic61C2×D411yescharacteristic
230C4×Dic61C2×D411yescharacteristic
231#11758194—42no—
232C2×#92418—22no—
233C2×#92418—22no—
234C2×#92418—42no—
235C23×C61—22no—
236C23×C61C2×C22⋊C411yescharacteristic
237C22×D61—22no—
238C22×D61—41no—
239C22×D61—41no—
240C22×D61C2×C22⋊C411yescharacteristic
241C22×D61C2×C22⋊C411yescharacteristic
242Q8×C61—41no—
243Q8×C61—41no—
244D4×C61—42no—
245D4×C61—44no—
246D4×C61—41no—
247D4×C61—41no—
248D4×C61—41no—
249D4×C61—41no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—41no—
254C2×C4×C61—44no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61#13512yes—
259C2×C4×C61#13512yes—
260C2×C4×C61C23×C411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C22⋊D61—44no—
264C22⋊D61—44no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—44no—
271C2×Dic61—44no—
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C4×D61—21no—
279C4×D61—21no—
280C4×D61—21no—
281C4×D61—21no—
282C4×D61—21no—
283C4×D61—21no—
284C4×D61—21no—
285C4×D61—21no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—42no—
297C8×C61—42no—
298C8×C61—42no—
299#92418—42no—
300#92418—42no—
301#17716—611nomaximal, Sylow
302C22×C61—21no—
303C22×C61—21no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61#41612yes—
308C22×C61C22×C22⋊C411yescharacteristic
309C2×D61—21no—
310C2×D61—21no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61—44no—
316C4×C61—21no—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—42no—
320C4×C61#41612yes—
321C4×C61C2×#13511yescharacteristic
322C4×C61C2×#13511yescharacteristic
323C4×C61#57511yescharacteristic
324C4×C61#57511yescharacteristic
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—42no—
331Dic61—42no—
332Dic61#41312yes—
333#3444—611no—
334#2920—611no—
335C2×#431—611no—
336#1554—612no—
337#1553—612no—
338#1550—611no—
339#1550—611no—
340#1547—611no—
341#1547D6111yescharacteristic, complemented
342#1544—612no—
343#1543—612no—
344C2×C61—22no—
345C2×C61—82no—
346C2×C61C2×#41311yescharacteristic
347C2×C61C2×#41611yescharacteristic
348C2×C61#289611yescharacteristic
349D61—41no—
350D61—41no—
351#533—611no—
352#533—611no—
353#533—611no—
354#532—611no—
355#532—611no—
356#532C2×D6111yescharacteristic
357#529—612no—
358#529—612no—
359#529—1221no—
360#529—1221no—
361#529—1221no—
362#529—1221no—
363#525—612no—
364#525—1221no—
365#525—1221no—
366C4×#41—612no—
367C4×#41—1221no—
368C4×#41—1221no—
369C22×C42—611no—
370#432—612no—
371#432—1221no—
372#431—612no—
373#431—612no—
374#431—612no—
375#431—1221no—
376#431—1221no—
377#431—1221no—
378#423—611no—
379#423C2×D6111yescharacteristic
380C2×C4⋊C8—611no—
381C2×C4⋊C8C2×D6111yescharacteristic
382#407—1221no—
383C2×C22⋊C8—1221no—
384C4×M16—1221no—
385C2×C4×C8—1221no—
386#333—612no—
387#333—612no—
388#332—612no—
389#332—612no—
390#332—612no—
391#332C2×D6112yescomplemented
392#331—612no—
393#331C2×D6112yescomplemented
394C61#1771611yescharacteristic, complemented, Sylow
395C2×#41—21no—
396C2×#41—612no—
397C2×#41—1221no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C2×M16—21no—
402C2×M16—1221no—
403C22×C8—21no—
404C22×C8—1221no—
405C4⋊Q8—611no—
406C4⋊Q8—611no—
407C4⋊Q8C22×D6111yescharacteristic
408#127C22×D6111yescharacteristic
409#126—1222no—
410#126—1222no—
411#125—612no—
412#125—1221no—
413#125—1221no—
414#124—21no—
415#124—612no—
416#124—1221no—
417#123—1222no—
418#123—1222no—
419#123—1222no—
420C22⋊Q8—612no—
421C22⋊Q8—612no—
422C22⋊Q8—1221no—
423C22⋊Q8—1221no—
424C22⋊Q8—1221no—
425C22⋊Q8—1221no—
426#121—21no—
427#121—612no—
428#121—1221no—
429#121—1221no—
430#121—1221no—
431#121C4×D6112yes—
432#120—1222no—
433C4×Q8—612no—
434C4×Q8—612no—
435C4×Q8—1221no—
436C4×Q8—1221no—
437C4×Q8—1221no—
438C4×Q8—1221no—
439C4×D4—21no—
440C4×D4—612no—
441C4×D4—1221no—
442C4×D4—1221no—
443C4×D4—1221no—
444C4×D4—1222no—
445C4×D4—1222no—
446C4×D4—1222no—
447C4×D4—1222no—
448C4×D4—1222no—
449C4×D4—1222no—
450C4×D4C4×D6112yes—
451#117—612no—
452#117—1222no—
453#117—1222no—
454#117—1222no—
455#117—1221no—
456#117—1221no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—1222no—
469C2×C42C22×D6111yescharacteristic
470C4⋊C8—21no—
471C4⋊C8—21no—
472C4⋊C8—612no—
473C4⋊C8—612no—
474C4⋊C8—1221no—
475C4⋊C8—1221no—
476C4⋊C8C22×D6112yes—
477C4⋊C8C22×D6112yes—
478C22⋊C8—1222no—
479C22⋊C8—1222no—
480C22⋊C8—1222no—
481C22⋊C8—1222no—
482#97—1222no—
483C4×C8—1222no—
484C4×C8—1222no—
485C4×C8—1222no—
486C24—611no—
487#41—42no—
488#41—42no—
489#41—2442no—
490#41—2442no—
491#41—2444no—
492C2×Q8—21no—
493C2×Q8—612no—
494C2×Q8—1221no—
495C2×D4—22no—
496C2×D4—21no—
497C2×D4—21no—
498C2×D4—1222no—
499C2×D4—1222no—
500C2×D4C2×C4×D6112yes—
501C22×C4—22no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—2441no—
522C22×C4—2441no—
523C22×C4—2442no—
524C22×C4C2×C4×D6111yescharacteristic
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527M16—42no—
528M16—2442no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—42no—
532C2×C8—1222no—
533C2×C8—1222no—
534C2×C8—2442no—
535C4⋊C4—21no—
536C4⋊C4—612no—
537C4⋊C4—1222no—
538C4⋊C4—1222no—
539C4⋊C4—1222no—
540C4⋊C4—1222no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4C2×C4×D6112yes—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C22⋊C4—1222no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—1222no—
569C42—1222no—
570C42—1222no—
571C42C2×C4×D6111yescharacteristic
572C42D4×D6111yescharacteristic
573C42D4×D6111yescharacteristic
574C42C23×D6111yescharacteristic
575C23—22no—
576C23—611no—
577C23—611no—
578C23—1222no—
579C23—2441no—
580C23—2441no—
581C23C22⋊C4×D6111yescharacteristic
582Q8—41no—
583Q8—41no—
584Q8—2442no—
585Q8—2441no—
586Q8—2441no—
587D4—42no—
588D4—44no—
589D4—41no—
590D4—41no—
591D4—41no—
592D4—41no—
593D4—2444no—
594D4—2444no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—611no—
601C2×C4—611no—
602C2×C4—611no—
603C2×C4—611no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2444no—
627C2×C4—2441no—
628C2×C4—2444no—
629C2×C4C22⋊C4×D6111yescharacteristic
630C2×C4C22⋊C4×D6111yescharacteristic
631C2×C4C22⋊C4×D6111yescharacteristic
632C2×C4#1451979612yes—
633C2×C4#1451979812yes—
634C2×C4C22×C4×D6111yescharacteristic
635C2×C4C2×D4×D6111yescharacteristic
636C2×C4C2×D4×D6111yescharacteristic
637C8—42no—
638C8—42no—
639C8—2442no—
640C8—2442no—
641C22—21no—
642C22—21no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—1221no—
647C22—1221no—
648C22—1221no—
649C22—1221no—
650C22—1221no—
651C22—1221no—
652C22—2444no—
653C22#1683975512yes—
654C22C2×C22⋊C4×D6111yescharacteristic
655C4—21no—
656C4—21no—
657C4—21no—
658C4—42no—
659C4—612no—
660C4—1221no—
661C4—1221no—
662C4—1221no—
663C4—1221no—
664C4—1221no—
665C4—2442no—
666C4—2442no—
667C4#1683975512yes—
668C4C2×#1451979611yescharacteristic
669C4C2×#1451979811yescharacteristic
670C4#1684055811yescharacteristic
671C4#1684056111yescharacteristic
672C2—22no—
673C2—82no—
674C2—2441no—
675C2—2441no—
676C2#413×D6111yescharacteristic
677C2C2×#1683975511yescharacteristic
678C2#2145433511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1547C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1547D61
hypocenterC61#17716
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1547

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. C2×#16839814C2semidirect
4. #21444526C2semidirect
5. #21444521C2semidirect
6. #21444510C2semidirect
7. #21444507C2semidirect
8. #21444496C2semidirect
9. #21444482C2semidirect
10. #21444478C2semidirect
11. C61×#1547C2semidirect
44. #16839510C22semidirect
45. #16839510C22semidirect
46. #16839509C22semidirect
47. #16839508C22semidirect
48. #16839507C22semidirect
49. #16839506C22semidirect
53. C61×#332C22semidirect
54. C61×#331C22semidirect
341. #1547D61semidirect
391. #332C2×D61semidirect
393. #331C2×D61semidirect
394. C61#17716semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544512
C2×#168398142
#214445262
#214445212
#214445102
#214445072
#214444962
#214444822
#214444782
C61×#15472
#1771661