Finite Group Databasebeta

← #29646521

Subgroups — #29646521

Order 15616.

1032 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#21459114C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3#431×D61C211yesmaximal, characteristic, complemented
4#21444224C214yesmaximal, complemented
5#21444223C211yesmaximal, characteristic, complemented
6#21444221C211yesmaximal, characteristic, complemented
7#21444218C211yesmaximal, characteristic, complemented
8#21444198C214yesmaximal, complemented
9C61×#1480C211yesmaximal, 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#16839815C2211yescharacteristic
29#16839814—21no—
30#16839814—21no—
31#16839814—21no—
32#16839814C2212yes—
33#16839814C2212yes—
34#16839813C2212yes—
35#16839812—21no—
36C2×C4⋊#92418C2211yescharacteristic
37C4⋊C8×D61—21no—
38C2×#14519689—21no—
39C22⋊C8×D61—21no—
40C2×C8×Dic61—21no—
41C4×C8×D61C2212yes—
42#16839494C2214yescomplemented
43#16839492C2214yescomplemented
44D4⋊#92418C2214yescomplemented
45C61×#532C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C2×C4⋊C8×C61C2211yescharacteristic
48C61×D4⋊C8C2214yescomplemented
49C2×#11758224—21no—
50C2×#11758224C2×C412yes—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×#92418—21no—
55C2×C8×D61—22no—
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⋊#92418—21no—
115C4⋊#92418C2312yes—
116#14519706—22no—
117C22⋊#92418—21no—
118#14519689—22no—
119#14519689—22no—
120#14519689—22no—
121C4×#92418C2312yes—
122C8×Dic61—22no—
123C8×Dic61—22no—
124C8×Dic61—22no—
125C2×#41×C61—21no—
126C22×C8×C61—21no—
127C4⋊Q8×C61C2×C411yescharacteristic
128#127×C61C2×C411yescharacteristic
129#124×C61—21no—
130#121×C61—21no—
131#121×C61C2×C412yes—
132C4×D4×C61—21no—
133C4×D4×C61C2312yes—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C22⋊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—28no—
160#11758224—44no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—24no—
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—
179C2×C8×C61—22no—
180C2×C8×C61—22no—
181C2×C8×C61—42no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61C22×C412yes—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—22no—
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2411yescharacteristic
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195#11758198—22no—
196#11758198—22no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—22no—
201C4⋊Dic61—22no—
202C4⋊Dic61C22×C412yes—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—22no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C22×C411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C2×#92418—22no—
223C2×#92418—22no—
224C2×#92418—42no—
225C8×D61—44no—
226C23×C61—22no—
227C23×C61C2×C22⋊C411yescharacteristic
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—41no—
231C22×D61C2×C22⋊C411yescharacteristic
232C22×D61C2×C22⋊C411yescharacteristic
233Q8×C61—42no—
234D4×C61—24no—
235D4×C61—42no—
236D4×C61—44no—
237D4×C61—42no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—42no—
242C2×C4×C61—42no—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61#13512yes—
247C2×C4×C61#13512yes—
248C2×C4×C61C23×C411yescharacteristic
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C22⋊D61—44no—
252C22⋊D61—44no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—44no—
259C2×Dic61—44no—
260C2×Dic61C2×C22⋊C411yescharacteristic
261C2×Dic61C2×C22⋊C411yescharacteristic
262C2×Dic61C2×C22⋊C411yescharacteristic
263C2×Dic61C2×C22⋊C411yescharacteristic
264C2×Dic61C2×C22⋊C411yescharacteristic
265C2×Dic61C2×C22⋊C411yescharacteristic
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—21no—
269C4×D61—21no—
270C4×D61—21no—
271C4×D61—21no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—41no—
278C4×D61—41no—
279C4×D61#13114yes—
280Dic122—24no—
281Dic122—42no—
282C8×C61—42no—
283C8×C61—42no—
284#92418—42no—
285#92418—42no—
286#17458—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61—42no—
292C22×C61#41612yes—
293C22×C61C22×C22⋊C411yescharacteristic
294C2×D61—21no—
295C2×D61—21no—
296C2×D61—21no—
297C2×D61—21no—
298C2×D61—21no—
299C2×D61—44no—
300C2×D61#40812yes—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—42no—
305C4×C61#41612yes—
306C4×C61C2×#13112yes—
307C4×C61C2×#13511yescharacteristic
308C4×C61C2×#13511yescharacteristic
309Dic61—21no—
310Dic61—21no—
311Dic61—21no—
312Dic61—21no—
313Dic61—42no—
314Dic61—42no—
315Dic61#40812yes—
316Dic61#42112yes—
317#3444—611no—
318C2×#431—611no—
319C2×#431—611no—
320#1482—614no—
321#1481—611no—
322#1481—611no—
323#1480—611no—
324#1480D6111yescharacteristic, complemented
325#1476—614no—
326C2×C61—22no—
327C2×C61—82no—
328C2×C61C2×#40811yescharacteristic
329C2×C61C2×#41611yescharacteristic
330C2×C61C2×#42111yescharacteristic
331D61—22no—
332D61—41no—
333#533—611no—
334#533—611no—
335#533—611no—
336#532—611no—
337#532—611no—
338#532C2×D6111yescharacteristic
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#431—611no—
353#431—612no—
354#431—612no—
355#431—612no—
356#431—1221no—
357#431—1221no—
358#431—1221no—
359#431—1221no—
360#431C2×D6111yescharacteristic
361C2×C4⋊C8—611no—
362C2×C4⋊C8—1221no—
363C2×C4⋊C8C2×D6111yescharacteristic
364C2×C22⋊C8—1221no—
365C2×C22⋊C8—1221no—
366C2×C4×C8—612no—
367C2×C4×C8—1221no—
368Q8⋊C8—614no—
369Q8⋊C8—614no—
370D4⋊C8—614no—
371D4⋊C8C2×D6114yescomplemented
372C61#1745811yescharacteristic, complemented, Sylow
373C2×#41—21no—
374C2×#41—612no—
375C2×#41—1221no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4—611no—
379C22×C8—21no—
380C22×C8—1222no—
381C22×C8—1221no—
382C4⋊Q8—611no—
383C4⋊Q8—611no—
384C4⋊Q8C4×D6111yescharacteristic
385#127C4×D6111yescharacteristic
386#126—1222no—
387#126—1222no—
388#125—612no—
389#125—1221no—
390#125—1221no—
391#124—21no—
392#124—612no—
393#124—1221no—
394#123—1222no—
395#123—1222no—
396#123—1222no—
397C22⋊Q8—612no—
398C22⋊Q8—612no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C22⋊Q8—1221no—
402C22⋊Q8—1221no—
403#121—21no—
404#121—612no—
405#121—1221no—
406#121—1221no—
407#121—1221no—
408#121C4×D6112yes—
409#120—1222no—
410C4×Q8—612no—
411C4×Q8—612no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416C4×D4—21no—
417C4×D4—612no—
418C4×D4—1221no—
419C4×D4—1221no—
420C4×D4—1221no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—1222no—
427C4×D4C22×D6112yes—
428#117—612no—
429#117—1222no—
430#117—1222no—
431#117—1222no—
432#117—1221no—
433#117—1221no—
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×C42C22×D6111yescharacteristic
447C4⋊C8—21no—
448C4⋊C8—21no—
449C4⋊C8—612no—
450C4⋊C8—1222no—
451C4⋊C8—1221no—
452C4⋊C8—1221no—
453C4⋊C8C22×D6112yes—
454C22⋊C8—21no—
455C22⋊C8—1222no—
456C22⋊C8—1222no—
457C22⋊C8—1222no—
458C22⋊C8—1221no—
459C4×C8—612no—
460C4×C8—1222no—
461C4×C8—1222no—
462C4×C8—1222no—
463C4×C8C22×D6112yes—
464C24—611no—
465#41—42no—
466#41—42no—
467#41—1228no—
468#41—2444no—
469C2×Q8—21no—
470C2×Q8—612no—
471C2×Q8—1221no—
472C2×D4—22no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—1222no—
476C2×D4—1222no—
477C2×D4C2×C4×D6112yes—
478C22×C4—22no—
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—611no—
494C22×C4—1224no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—2442no—
500C22×C4C2×C4×D6111yescharacteristic
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—42no—
506C2×C8—1222no—
507C2×C8—1222no—
508C2×C8—2444no—
509C2×C8—2442no—
510C4⋊C4—21no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1222no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4C2×C4×D6112yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—1222no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—1222no—
544C42—1222no—
545C42—1222no—
546C42C2×C4×D6111yescharacteristic
547C42D4×D6111yescharacteristic
548C42D4×D6111yescharacteristic
549C42C23×D6111yescharacteristic
550C23—22no—
551C23—611no—
552C23—611no—
553C23—1222no—
554C23—1222no—
555C23—2441no—
556C23C22⋊C4×D6111yescharacteristic
557Q8—42no—
558Q8—1224no—
559Q8—2442no—
560D4—24no—
561D4—42no—
562D4—44no—
563D4—42no—
564D4—2444no—
565D4—2444no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—611no—
572C2×C4—611no—
573C2×C4—614no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—611no—
578C2×C4—1222no—
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—1222no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4C22⋊C4×D6111yescharacteristic
599C2×C4C22⋊C4×D6111yescharacteristic
600C2×C4C22⋊C4×D6111yescharacteristic
601C2×C4#1451979612yes—
602C2×C4#1451979812yes—
603C2×C4C22×C4×D6111yescharacteristic
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4C2×D4×D6111yescharacteristic
606C8—42no—
607C8—42no—
608C8—2442no—
609C8—2442no—
610C22—21no—
611C22—21no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—612no—
616C22—1221no—
617C22—1221no—
618C22—1221no—
619C22—1221no—
620C22—1221no—
621C22—2444no—
622C22#1683975512yes—
623C22C2×C22⋊C4×D6111yescharacteristic
624C4—21no—
625C4—21no—
626C4—21no—
627C4—42no—
628C4—612no—
629C4—612no—
630C4—1221no—
631C4—1221no—
632C4—1221no—
633C4—1221no—
634C4—2442no—
635C4—2442no—
636C4#1683975512yes—
637C4#131×D6112yes—
638C4C2×#1451979611yescharacteristic
639C4C2×#1451979811yescharacteristic
640C2—22no—
641C2—82no—
642C2—1222no—
643C2—2441no—
644C2#408×D6111yescharacteristic
645C2C2×#1683975511yescharacteristic
646C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1480C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC42×C61C24
hypercenter#1480D61
hypocenterC61#17458
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. C2×#16839814C2semidirect
3. #431×D61C2semidirect
4. #21444224C2semidirect
5. #21444223C2semidirect
6. #21444221C2semidirect
7. #21444218C2semidirect
8. #21444198C2semidirect
9. C61×#1480C2semidirect
42. #16839494C22semidirect
43. #16839492C22semidirect
44. D4⋊#92418C22semidirect
48. C61×D4⋊C8C22semidirect
324. #1480D61semidirect
371. D4⋊C8C2×D61semidirect
372. C61#17458semidirect

Maximal subgroups

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

subgroupindex
#214591142
C2×#168398142
#431×D612
#214442242
#214442232
#214442212
#214442182
#214441982
C61×#14802
#1745861