Finite Group Databasebeta

← #29646735

Subgroups — #29646735

Order 15616.

1002 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454451C211yesmaximal, characteristic, complemented
3C2×#16839813C211yesmaximal, characteristic, complemented
4#21444275C211yesmaximal, characteristic, complemented
5#21444273C211yesmaximal, characteristic, complemented
6#21444270C211yesmaximal, characteristic, complemented
7#21444250C214yesmaximal, complemented
8#21444224C214yesmaximal, complemented
9C61×#1492C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840317C411yescharacteristic
12#16840317C411yescharacteristic
13#16840310C411yescharacteristic
14#16840310C411yescharacteristic
15#16840294—21no—
16#16840294C2212yes—
17#16840293—21no—
18#16840293C412yes—
19#16840291—21no—
20#16840291—21no—
21#16840272—21no—
22#16840272C412yes—
23#16840269—21no—
24#41×Dic61—21no—
25C4×#11758224—21no—
26C4×#11758224C2212yes—
27C2×C42×D61C2211yescharacteristic
28#16839818—21no—
29#16839818C2212yes—
30#16839814—21no—
31#16839814C2212yes—
32#16839813—21no—
33#16839813—21no—
34#16839813C2212yes—
35#16839813C2212yes—
36#16839786C2211yescharacteristic
37C2×#14519706—21no—
38#16839725—21no—
39C2×#14519689—21no—
40M16×Dic61—21no—
41C2×C4×#92418C2211yescharacteristic
42#16839494C2214yescomplemented
43#16839492C2214yescomplemented
44D4⋊#92418C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#423C2211yescharacteristic
47C2×C4×C8×C61C2211yescharacteristic
48C61×D4⋊C8C2214yescomplemented
49C2×#11758224—21no—
50C2×#11758224C2×C412yes—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C2×#11758194—21no—
55C22×#92418—21no—
56#14519777C2×C411yescharacteristic
57#14519777C2×C411yescharacteristic
58#14519770—22no—
59#14519770—22no—
60#14519769—21no—
61#14519768—21no—
62#14519768C2×C412yes—
63#14519764—21no—
64#14519764C2×C412yes—
65#14519759—22no—
66#14519759—22no—
67#14519758—22no—
68#14519757—21no—
69#14519757C2×C412yes—
70#14519756—21no—
71#14519756C2×C412yes—
72#14519753—21no—
73#14519753—21no—
74#14519748—21no—
75#14519748—21no—
76#14519748—21no—
77#14519748C2×C412yes—
78#14519745—22no—
79Q8×Dic61—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2312yes—
83#14519742—21no—
84#14519742C2×C412yes—
85D4×Dic61—22no—
86D4×Dic61—22no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89D4×Dic61—21no—
90D4×Dic61C2×C412yes—
91C4×C22⋊D61—22no—
92C4×C22⋊D61—22no—
93#14519737—22no—
94#14519737—22no—
95#14519736—21no—
96#14519736—22no—
97#14519735—21no—
98#14519735C2×C412yes—
99#14519733—22no—
100#14519733—22no—
101C2×C4×Dic61—22no—
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⋊#92418C2312yes—
115#14519706—22no—
116#14519706—22no—
117#14519689—22no—
118#14519689—22no—
119#14519689—22no—
120#14519689—22no—
121#14519687—22no—
122C4×#92418—21no—
123C4×#92418C2312yes—
124C8×Dic61—22no—
125C2×#41×C61—21no—
126C2×M16×C61—21no—
127C22×C8×C61—21no—
128C4⋊Q8×C61C2×C411yescharacteristic
129#127×C61C2×C411yescharacteristic
130#124×C61—21no—
131#121×C61—21no—
132#121×C61C2×C412yes—
133C4×D4×C61—21no—
134C4×D4×C61C2312yes—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C4×C8×C61—21no—
139C4×C8×C61C2312yes—
140C23×D61C22⋊C411yescharacteristic
141#41×C61—42no—
142#41×C61—42no—
143C2×Q8×C61—21no—
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—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×Dic122C22×C412yes—
179M16×C61—42no—
180C2×C8×C61—22no—
181C2×C8×C61—22no—
182C2×C8×C61—42no—
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—
196#11758198—22no—
197#11758198—22no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61C22×C412yes—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208C4×Dic61—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—
226C2×#92418—42no—
227C23×C61—22no—
228C23×C61C2×C22⋊C411yescharacteristic
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61C2×C22⋊C411yescharacteristic
233C22×D61C2×C22⋊C411yescharacteristic
234Q8×C61—42no—
235D4×C61—24no—
236D4×C61—44no—
237D4×C61—42no—
238D4×C61—42no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—42no—
243C2×C4×C61—42no—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61#13512yes—
248C2×C4×C61#13512yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C22⋊D61—44no—
253C22⋊D61—44no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—44no—
260C2×Dic61—44no—
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C2×Dic61C2×C22⋊C411yescharacteristic
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—21no—
270C4×D61—21no—
271C4×D61—21no—
272C4×D61—21no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61—41no—
280C4×D61—41no—
281Dic122—24no—
282Dic122—42no—
283C8×C61—42no—
284C8×C61—42no—
285#92418—42no—
286#92418—42no—
287#17489—611nomaximal, Sylow
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61—42no—
293C22×C61#41612yes—
294C22×C61C22×C22⋊C411yescharacteristic
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—21no—
300C2×D61—21no—
301C2×D61—44no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61#41612yes—
306C4×C61#42112yes—
307C4×C61#56812yes—
308C4×C61C2×#13511yescharacteristic
309C4×C61C2×#13511yescharacteristic
310Dic61—21no—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—42no—
317Dic61—42no—
318#3444—611no—
319#2920—611no—
320C2×#431—611no—
321#1493—611no—
322#1493—611no—
323#1492—611no—
324#1492D6111yescharacteristic, complemented
325#1488—614no—
326#1482—614no—
327C2×C61—22no—
328C2×C61—82no—
329C2×C61#287911yescharacteristic
330C2×C61C2×#41611yescharacteristic
331C2×C61C2×#42111yescharacteristic
332D61—41no—
333D61—41no—
334#533—611no—
335#533—611no—
336#533—611no—
337#532—611no—
338#532—611no—
339#532C2×D6111yescharacteristic
340#529—612no—
341#529—612no—
342#529—1221no—
343#529—1221no—
344#529—1221no—
345#529—1221no—
346#525—612no—
347#525—1221no—
348#525—1221no—
349C4×#41—612no—
350C4×#41—1221no—
351C4×#41—1221no—
352C22×C42—611no—
353#432—612no—
354#432—1221no—
355#431—612no—
356#431—612no—
357#431—612no—
358#431—1221no—
359#431—1221no—
360#431—1221no—
361#423—611no—
362#423C2×D6111yescharacteristic
363C2×C4⋊C8—1221no—
364#407—1221no—
365C2×C22⋊C8—1221no—
366C4×M16—1221no—
367C2×C4×C8—611no—
368C2×C4×C8C2×D6111yescharacteristic
369Q8⋊C8—614no—
370Q8⋊C8—614no—
371D4⋊C8—614no—
372D4⋊C8C2×D6114yescomplemented
373C61#1748911yescharacteristic, complemented, Sylow
374C2×#41—21no—
375C2×#41—612no—
376C2×#41—1221no—
377C23×C4—611no—
378C23×C4—611no—
379C23×C4—611no—
380C2×M16—21no—
381C2×M16—1221no—
382C22×C8—21no—
383C22×C8—1221no—
384C4⋊Q8—611no—
385C4⋊Q8—611no—
386C4⋊Q8C4×D6111yescharacteristic
387#127C4×D6111yescharacteristic
388#126—1222no—
389#126—1222no—
390#125—612no—
391#125—1221no—
392#125—1221no—
393#124—21no—
394#124—612no—
395#124—1221no—
396#123—1222no—
397#123—1222no—
398#123—1222no—
399C22⋊Q8—612no—
400C22⋊Q8—612no—
401C22⋊Q8—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C22⋊Q8—1221no—
405#121—21no—
406#121—612no—
407#121—1221no—
408#121—1221no—
409#121—1221no—
410#121C4×D6112yes—
411#120—1222no—
412C4×Q8—612no—
413C4×Q8—612no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416C4×Q8—1221no—
417C4×Q8—1221no—
418C4×D4—21no—
419C4×D4—612no—
420C4×D4—1221no—
421C4×D4—1221no—
422C4×D4—1221no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4C22×D6112yes—
430#117—612no—
431#117—1222no—
432#117—1222no—
433#117—1222no—
434#117—1221no—
435#117—1221no—
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—611no—
446C2×C42—611no—
447C2×C42—1222no—
448C2×C42C22×D6111yescharacteristic
449C4⋊C8—21no—
450C4⋊C8—612no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453C4⋊C8—1221no—
454C4⋊C8C22×D6112yes—
455C22⋊C8—1222no—
456C22⋊C8—1222no—
457C22⋊C8—1222no—
458C22⋊C8—1222no—
459#97—1222no—
460C4×C8—21no—
461C4×C8—612no—
462C4×C8—1222no—
463C4×C8—1221no—
464C4×C8C22×D6112yes—
465C24—611no—
466#41—42no—
467#41—42no—
468#41—2444no—
469#41—2444no—
470C2×Q8—21no—
471C2×Q8—612no—
472C2×Q8—1221no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—1222no—
477C2×D4—1222no—
478C2×D4C2×C4×D6112yes—
479C22×C4—22no—
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—611no—
494C22×C4—611no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4C2×C4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4D4×D6111yescharacteristic
504M16—42no—
505M16—2442no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—42no—
509C2×C8—1222no—
510C2×C8—1222no—
511C2×C8—2442no—
512C4⋊C4—21no—
513C4⋊C4—612no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C22⋊C4—1222no—
532C22⋊C4—1222no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—1222no—
546C42—1222no—
547C42—1222no—
548C42C2×C4×D6111yescharacteristic
549C42D4×D6111yescharacteristic
550C42D4×D6111yescharacteristic
551C42C23×D6111yescharacteristic
552C23—22no—
553C23—611no—
554C23—611no—
555C23—1222no—
556C23—2441no—
557C23—2441no—
558C23C22⋊C4×D6111yescharacteristic
559Q8—42no—
560Q8—1224no—
561Q8—2442no—
562D4—24no—
563D4—44no—
564D4—42no—
565D4—42no—
566D4—2444no—
567D4—2444no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
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—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2444no—
599C2×C4—2444no—
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4#1451979612yes—
604C2×C4#1451979812yes—
605C2×C4C22×C4×D6111yescharacteristic
606C2×C4C2×D4×D6111yescharacteristic
607C2×C4C2×D4×D6111yescharacteristic
608C8—42no—
609C8—42no—
610C8—2442no—
611C8—2442no—
612C22—21no—
613C22—21no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—1221no—
618C22—1221no—
619C22—1221no—
620C22—1221no—
621C22—1221no—
622C22—1221no—
623C22—2444no—
624C22#1683975512yes—
625C22C2×C22⋊C4×D6111yescharacteristic
626C4—21no—
627C4—21no—
628C4—42no—
629C4—1221no—
630C4—1221no—
631C4—1221no—
632C4—1221no—
633C4—1221no—
634C4—1221no—
635C4—2442no—
636C4—2442no—
637C4#1683975512yes—
638C4#1683977712yes—
639C4#1684052712yes—
640C4C2×#1451979611yescharacteristic
641C4C2×#1451979811yescharacteristic
642C2—22no—
643C2—82no—
644C2—2441no—
645C2—2441no—
646C2#2145426111yescharacteristic
647C2C2×#1683975511yescharacteristic
648C2C2×#1683977711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1492C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1492D61
hypocenterC61#17489
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #21454451C2semidirect
3. C2×#16839813C2semidirect
4. #21444275C2semidirect
5. #21444273C2semidirect
6. #21444270C2semidirect
7. #21444250C2semidirect
8. #21444224C2semidirect
9. C61×#1492C2semidirect
42. #16839494C22semidirect
43. #16839492C22semidirect
44. D4⋊#92418C22semidirect
48. C61×D4⋊C8C22semidirect
324. #1492D61semidirect
372. D4⋊C8C2×D61semidirect
373. C61#17489semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544512
C2×#168398132
#214442752
#214442732
#214442702
#214442502
#214442242
C61×#14922
#1748961