Finite Group Databasebeta

← #29646720

Subgroups — #29646720

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840294C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3#431×D61C211yesmaximal, characteristic, complemented
4#21444276C214yesmaximal, complemented
5#21444249C211yesmaximal, characteristic, complemented
6#21444247C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444224C214yesmaximal, complemented
9C61×#1486C211yesmaximal, characteristic, complemented
10#16840294—21no—
11#16840294—21no—
12#16840294—21no—
13#16840294—21no—
14#16840294C412yes—
15#16840294C412yes—
16#16840294C2212yes—
17#16840294C2212yes—
18C2×#14519770—21no—
19C2×#14519768C411yescharacteristic
20C2×#14519764C411yescharacteristic
21C2×#14519759—21no—
22C2×#14519757C411yescharacteristic
23C2×#14519748C411yescharacteristic
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×C22⋊D61—21no—
26C2×#14519736—21no—
27C2×C42×D61C2211yescharacteristic
28#16839818—21no—
29#16839818C2212yescomplemented
30#16839815C2211yescharacteristic
31#16839814—21no—
32#16839814—21no—
33#16839814C2212yescomplemented
34#16839813C2212yescomplemented
35#16839812—21no—
36#16839786C2211yescharacteristic
37C4⋊C8×D61—21no—
38#16839725—21no—
39C22⋊C8×D61—21no—
40M16×Dic61—21no—
41C4×C8×D61C2212yescomplemented
42#16839494C2214yescomplemented
43#16839492C2214yescomplemented
44D4⋊#92418C2214yescomplemented
45C2×C4×D4×C61C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C61×#423C2211yescharacteristic
48C61×D4⋊C8C2214yescomplemented
49C22×Dic122C2×C411yescharacteristic
50C22×C22⋊D61—21no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758194—21no—
55C2×C8×D61—22no—
56#14519770—22no—
57#14519770—22no—
58#14519770—22no—
59#14519770—22no—
60#14519768—21no—
61#14519768—21no—
62#14519768C2×C412yes—
63#14519768C2×C412yes—
64#14519764—21no—
65#14519764—21no—
66#14519764C2×C412yes—
67#14519764C2×C412yes—
68#14519759—22no—
69#14519759—22no—
70#14519759—22no—
71#14519759—22no—
72#14519757—21no—
73#14519757—21no—
74#14519757C2×C412yes—
75#14519757C2×C412yes—
76#14519748—21no—
77#14519748—21no—
78#14519748C2×C412yes—
79#14519748C2×C412yes—
80C4×Dic122—21no—
81C4×Dic122—21no—
82C4×Dic122C2×C412yes—
83C4×Dic122C2312yes—
84C4×C22⋊D61—22no—
85C4×C22⋊D61—22no—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88#14519736—22no—
89#14519736—22no—
90#14519736—22no—
91#14519736—22no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×C22⋊Dic61—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×#11758198—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2×C411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C42×D61C2×C411yescharacteristic
106C42×D61C2×C411yescharacteristic
107C42×D61D411yescharacteristic
108C42×D61D411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61C2311yescharacteristic
112C42×D61C2311yescharacteristic
113C4⋊#92418—21no—
114C4⋊#92418—21no—
115C4⋊#92418C2312yes—
116#14519706—22no—
117C22⋊#92418—21no—
118#14519689—22no—
119#14519689—22no—
120#14519689—22no—
121#14519687—22no—
122C4×#92418C2312yes—
123C8×Dic61—22no—
124C8×Dic61—22no—
125C22×D4×C61C2×C411yescharacteristic
126C23×C4×C61—21no—
127C2×M16×C61—21no—
128C4×D4×C61—21no—
129C4×D4×C61—21no—
130C4×D4×C61C2×C412yes—
131C4×D4×C61C2312yes—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C22⋊C8×C61—21no—
139C4×C8×C61C2312yes—
140C24×C61—21no—
141C23×D61C22⋊C411yescharacteristic
142C2×D4×C61—21no—
143C2×D4×C61—42no—
144C2×D4×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—22no—
147C22×C4×C61—22no—
148C22×C4×C61—42no—
149C22×C4×C61—42no—
150C22×C4×C61C22×C411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—44no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61C22⋊C411yescharacteristic
159C22×Dic61C22⋊C411yescharacteristic
160C22×Dic61C22⋊C411yescharacteristic
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
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×D61C22×C411yescharacteristic
173C2×C4×D61C2×D411yescharacteristic
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61C2×D411yescharacteristic
177C2×Dic122—21no—
178C2×Dic122—42no—
179C2×Dic122C22×C412yes—
180M16×C61—42no—
181C2×C8×C61—22no—
182C2×C8×C61—22no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C42×C61C22×C411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2411yescharacteristic
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196C22⋊Dic61—22no—
197#11758198—22no—
198#11758198—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—22no—
204C4⋊Dic61C22×C412yes—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
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×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223#11758194—42no—
224C2×#92418—22no—
225C2×#92418—22no—
226C8×D61—44no—
227C23×C61—22no—
228C23×C61—42no—
229C23×C61—42no—
230C23×C61—42no—
231C23×C61C2×C22⋊C411yescharacteristic
232C22×D61—22no—
233C22×D61—22no—
234C22×D61—41no—
235C22×D61C2×C22⋊C411yescharacteristic
236C22×D61C2×C22⋊C411yescharacteristic
237D4×C61—24no—
238D4×C61—42no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—44no—
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61#13512yes—
250C2×C4×C61#13512yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×D411yescharacteristic
254C22⋊D61—44no—
255C22⋊D61—44no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—42no—
262C2×Dic61—42no—
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C2×Dic61C2×C22⋊C411yescharacteristic
268C2×Dic61C2×C22⋊C411yescharacteristic
269C4×D61—22no—
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—41no—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283Dic122—24no—
284Dic122—42no—
285C8×C61—42no—
286C8×C61—42no—
287#92418—42no—
288#92418—42no—
289#17487—611nomaximal, Sylow
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—84no—
296C22×C61#41612yes—
297C22×C61C22×C22⋊C411yescharacteristic
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—21no—
302C2×D61—21no—
303C2×D61—44no—
304C2×D61#42112yes—
305C4×C61—21no—
306C4×C61—21no—
307C4×C61—21no—
308C4×C61—42no—
309C4×C61#41612yes—
310C4×C61#56812yes—
311C4×C61C2×#13511yescharacteristic
312C4×C61C2×#13511yescharacteristic
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—21no—
317Dic61—21no—
318Dic61—21no—
319Dic61—42no—
320Dic61—42no—
321C2×#529—611no—
322#2920—611no—
323C2×#431—611no—
324#1494—614no—
325#1487—611no—
326#1487—611no—
327#1486—611no—
328#1486D6111yescharacteristic, complemented
329#1482—614no—
330C2×C61—22no—
331C2×C61—82no—
332C2×C61#287911yescharacteristic
333C2×C61C2×#41611yescharacteristic
334C2×C61C2×#42111yescharacteristic
335D61—22no—
336D61—41no—
337#529—612no—
338#529—612no—
339#529—612no—
340#529—612no—
341#529—1221no—
342#529—1221no—
343#529—1221no—
344#529—1221no—
345C2×#126—1221no—
346C2×#125—611no—
347C2×#124—611no—
348C2×#123—1221no—
349C2×C22⋊Q8—611no—
350C2×#121—611no—
351C2×C4×Q8—611no—
352C2×C4×D4—1221no—
353C2×C4×D4C2×D6111yescharacteristic
354C2×#117—1221no—
355C22×C42—611no—
356#432—612no—
357#432—1221no—
358#431—611no—
359#431—612no—
360#431—612no—
361#431—1221no—
362#431—1221no—
363#431—1221no—
364#431C2×D6111yescharacteristic
365#423—611no—
366#423C2×D6111yescharacteristic
367C2×C4⋊C8—1221no—
368#407—1221no—
369C2×C22⋊C8—1221no—
370C4×M16—1221no—
371C2×C4×C8—612no—
372Q8⋊C8—614no—
373Q8⋊C8—614no—
374D4⋊C8—614no—
375D4⋊C8C2×D6114yescomplemented
376C61#1748711yescharacteristic, complemented, Sylow
377C22×Q8—611no—
378C22×D4—1221no—
379C22×D4C4×D6111yescharacteristic
380C23×C4—21no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C2×M16—21no—
385C2×M16—1221no—
386C22×C8—1222no—
387#126—1222no—
388#126—1222no—
389#126—1222no—
390#126—1222no—
391#125—612no—
392#125—612no—
393#125—1221no—
394#125—1221no—
395#124—612no—
396#124—612no—
397#124—1221no—
398#124—1221no—
399#123—1222no—
400#123—1222no—
401#123—1222no—
402#123—1222no—
403C22⋊Q8—612no—
404C22⋊Q8—612no—
405C22⋊Q8—1221no—
406C22⋊Q8—1221no—
407#121—612no—
408#121—612no—
409#121—1221no—
410#121—1221no—
411C4×Q8—612no—
412C4×Q8—612no—
413C4×Q8—1221no—
414C4×Q8—1221no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—1222no—
418C4×D4—1222no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4C4×D6112yes—
422C4×D4C22×D6112yes—
423#117—1222no—
424#117—1222no—
425#117—1222no—
426#117—1222no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4C4×D6111yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451C4⋊C8—21no—
452C4⋊C8—21no—
453C4⋊C8—612no—
454C4⋊C8—1222no—
455C4⋊C8—1221no—
456C4⋊C8—1221no—
457C4⋊C8C22×D6112yes—
458C22⋊C8—21no—
459C22⋊C8—1222no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462C22⋊C8—1221no—
463#97—1222no—
464C4×C8—612no—
465C4×C8—1222no—
466C4×C8—1222no—
467C4×C8C22×D6112yes—
468C24—21no—
469C24—611no—
470C2×Q8—612no—
471C2×Q8—1221no—
472C2×Q8—2442no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4—2444no—
478C2×D4C2×C4×D6112yes—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—42no—
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—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1222no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508M16—42no—
509M16—2442no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—1222no—
513C2×C8—1222no—
514C2×C8—2444no—
515C4⋊C4—21no—
516C4⋊C4—612no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1221no—
525C4⋊C4—1222no—
526C4⋊C4C2×C4×D6112yes—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1222no—
535C22⋊C4—1222no—
536C22⋊C4—1222no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—611no—
548C42—611no—
549C42—1222no—
550C42—1222no—
551C42C2×C4×D6111yescharacteristic
552C42D4×D6111yescharacteristic
553C42D4×D6111yescharacteristic
554C42C23×D6111yescharacteristic
555C23—22no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—611no—
560C23—611no—
561C23—1222no—
562C23—1222no—
563C23—2441no—
564C23C22⋊C4×D6111yescharacteristic
565Q8—1224no—
566Q8—2442no—
567D4—24no—
568D4—42no—
569D4—2444no—
570D4—2444no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1221no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2442no—
604C2×C4—2442no—
605C2×C4C22⋊C4×D6111yescharacteristic
606C2×C4C22⋊C4×D6111yescharacteristic
607C2×C4C22⋊C4×D6111yescharacteristic
608C2×C4#1451979612yes—
609C2×C4#1451979812yes—
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×D4×D6111yescharacteristic
613C8—42no—
614C8—42no—
615C8—2442no—
616C8—2442no—
617C22—21no—
618C22—21no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—84no—
623C22—612no—
624C22—1221no—
625C22—1221no—
626C22—1221no—
627C22—1221no—
628C22—1221no—
629C22—2444no—
630C22#1683975512yes—
631C22C2×C22⋊C4×D6111yescharacteristic
632C4—21no—
633C4—21no—
634C4—21no—
635C4—42no—
636C4—1221no—
637C4—1221no—
638C4—1221no—
639C4—1221no—
640C4—1221no—
641C4—1221no—
642C4—2442no—
643C4—2442no—
644C4#1683975512yes—
645C4#1684052812yes—
646C4C2×#1451979611yescharacteristic
647C4C2×#1451979811yescharacteristic
648C2—22no—
649C2—82no—
650C2—1222no—
651C2—2441no—
652C2#2145426211yescharacteristic
653C2C2×#1683975511yescharacteristic
654C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1486C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519748C4
cosocleC42×C61C24
hypercenter#1486D61
hypocenterC61#17487
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. #431×D61C2semidirect
4. #21444276C2semidirect
5. #21444249C2semidirect
6. #21444247C2semidirect
7. #21444244C2semidirect
8. #21444224C2semidirect
9. C61×#1486C2semidirect
29. #16839818C22semidirect
33. #16839814C22semidirect
34. #16839813C22semidirect
41. C4×C8×D61C22semidirect
42. #16839494C22semidirect
43. #16839492C22semidirect
44. D4⋊#92418C22semidirect
48. C61×D4⋊C8C22semidirect
328. #1486D61semidirect
375. D4⋊C8C2×D61semidirect
376. C61#17487semidirect

Maximal subgroups

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

subgroupindex
C2×#168402942
#214544512
#431×D612
#214442762
#214442492
#214442472
#214442442
#214442242
C61×#14862
#1748761