Finite Group Databasebeta

← #29698655

Subgroups — #29698655

Order 15616.

938 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839827C211yesmaximal, characteristic
2#21448751C211yesmaximal, characteristic
3#21448745C211yesmaximal, characteristic
4#21446589C211yesmaximal, characteristic, complemented
5#21446021C211yesmaximal, characteristic, complemented
6#21446015C211yesmaximal, characteristic, complemented
7#21446013C211yesmaximal, characteristic, complemented
8#21446010C211yesmaximal, characteristic, complemented
9#21445949C211yesmaximal, characteristic, complemented
10#21445845C211yesmaximal, characteristic, complemented
11#21445841C211yesmaximal, characteristic, complemented
12C4×#14519708C211yesmaximal, characteristic, complemented
13#21445534C211yesmaximal, characteristic, complemented
14#340×D61C211yesmaximal, characteristic, complemented
15C61×#1949C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839827C418yescomplemented
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×C4×#92418C2211yescharacteristic
31#16839653—21no—
32#16839651—21no—
33#16839647—21no—
34#16839645—21no—
35#16839640—21no—
36#16839640—21no—
37#16839628C2211yescharacteristic
38#16839619C2211yescharacteristic
39#16839619C2211yescharacteristic
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43C4×#11758198—21no—
44#16839609—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53C2×#107×C61C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C61×#382C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C22×D244D411yescharacteristic
61C22×C4×D61—21no—
62C22×C4×D61—21no—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C8×D61—41no—
68C2×C8×D61—41no—
69C2×C8×D61—41no—
70C2×C8×D61—41no—
71C2×C8×D61C2×C414yes—
72C2×C8×D61C2×C414yescomplemented
73C4×D244C2×C414yes—
74C4×D244Q814yes—
75#14519735—22no—
76#14519735—22no—
77#14519735—41no—
78#14519735—41no—
79C2×C4⋊Dic61D411yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—41no—
89C4⋊C4×D61—41no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198D411yescharacteristic
97C2×#11758198D411yescharacteristic
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101#14519711—41no—
102#14519709—41no—
103#14519708—22no—
104#14519708—22no—
105#14519689C2×C414yes—
106#14519689C2×C414yescomplemented
107C4×#92418C2×C414yescomplemented
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#107×C61—41no—
124#106×C61—41no—
125C4⋊C8×C61C2×C414yescomplemented
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129C23×D61C2×D411yescharacteristic
130C23×D61C2×D411yescharacteristic
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2411yescharacteristic
138C22×Dic61—21no—
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61C2×D411yescharacteristic
143C22×Dic61C2×D411yescharacteristic
144C2×D244—22no—
145C2×D244—22no—
146C2×D244C4⋊C414yes—
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C22×C412yes—
163C2×C4×D61C2×Q812yes—
164C2×C4×D61C2×Q812yes—
165C2×C8×C61—21no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61C22×C412yes—
170C2×C8×C61C22×C412yes—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61#4112yes—
179C42×C61—81no—
180C42×C61C22×C412yes—
181C42×C61C2×Q812yes—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—42no—
186#11758198—81no—
187#11758198—81no—
188#11758198C4⋊C414yes—
189#11758198C4⋊C414yes—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61C4⋊C414yes—
198C4⋊Dic61#4112yes—
199#11758196—42no—
200#11758196—42no—
201C4×Dic61—42no—
202C4×Dic61—42no—
203C4×Dic61—81no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C22×C412yes—
209C2×#92418C22×C412yes—
210C8×D61—44no—
211C8×D61—44no—
212C23×C61C22×D411yescharacteristic
213C22×D61—21no—
214C22×D61—21no—
215C22×D61—21no—
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—21no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61C2×C4⋊C412yes—
224C22×D61C2×C4⋊C412yes—
225C22×D61#13512yes—
226C22×D61#13512yes—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—22no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61C2×C4⋊C412yes—
244C2×C4×C61C2×C4⋊C412yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×Q811yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61H(5,𝔽2)11yescharacteristic
250C2×C4×C61H(5,𝔽2)11yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—22no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61C2×C4⋊C412yes—
268C2×Dic61C2×C4⋊C412yes—
269D244—28no—
270C4×D61—22no—
271C4×D61—22no—
272C4×D61—22no—
273C4×D61—22no—
274C4×D61—22no—
275C4×D61—22no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—84no—
279C4×D61—84no—
280C8×C61—22no—
281C8×C61—22no—
282#92418—22no—
283#92418—22no—
284#22706—611nomaximal, Sylow
285C22×C61C22×C4⋊C411yescharacteristic
286C22×C61#54611yescharacteristic
287C22×C61#54611yescharacteristic
288C22×C61C2×#13511yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61#57711yescharacteristic
291C22×C61#57711yescharacteristic
292C2×D61—22no—
293C2×D61—22no—
294C2×D61—22no—
295C2×D61—22no—
296C2×D61—22no—
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—41no—
302C2×D61—41no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61#42714yes—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—81no—
310C4×C61—81no—
311C4×C61—81no—
312C4×C61C4×#4111yescharacteristic
313C4×C61#51811yescharacteristic
314C4×C61#55411yescharacteristic
315C4×C61#55411yescharacteristic
316Dic61—22no—
317Dic61—22no—
318Dic61—42no—
319Dic61—81no—
320Dic61—81no—
321Dic61—81no—
322C2×C8×D4—611no—
323#2352—611no—
324#2352—611no—
325#1949—611no—
326#1949D6111yescharacteristic, complemented
327#1854—611no—
328#1854—611no—
329#1853—611no—
330#1853—611no—
331#1835—611no—
332#1808—611no—
333#1807—611no—
334C4×#106—611no—
335#1735—611no—
336C2×#340—611no—
337C2×C61#234911yescharacteristic
338C2×C61C2×#42711yescharacteristic
339C2×C61#291211yescharacteristic
340C2×C61#329611yescharacteristic
341C2×C61#329611yescharacteristic
342C2×C61#330111yescharacteristic
343C2×C61#330111yescharacteristic
344D61—24no—
345D61—42no—
346C23×C8—611no—
347C23×C8—611no—
348C2×C4×D4—611no—
349C2×#117—611no—
350C2×#117—611no—
351C22×C4⋊C4—611no—
352C22×C4⋊C4—611no—
353C8×D4—618no—
354C2×#107—611no—
355C2×#107C2×D6111yescharacteristic
356C2×#106—611no—
357C2×#106—611no—
358C2×#106—611no—
359C2×#106C2×D6111yescharacteristic
360C2×C4⋊C8C2×D6111yescharacteristic
361C2×C22⋊C8—611no—
362C2×C22⋊C8—611no—
363C2×C4×C8—611no—
364#388—1221no—
365#388—1221no—
366#387—1221no—
367#387—1221no—
368#386—1221no—
369#386—1221no—
370#382—611no—
371#382C2×D6111yescharacteristic
372#380—611no—
373#380—611no—
374#380—611no—
375#380—611no—
376C4×C4⋊C4—611no—
377C4×C4⋊C4C2×D6111yescharacteristic
378C4×C22⋊C4—1221no—
379C4×C22⋊C4—1221no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340C2×D6111yescharacteristic
389#340C2×D6111yescharacteristic
390C61#2270611yescharacteristic, complemented, Sylow
391C22×D4—611no—
392C23×C4—611no—
393C23×C4—611no—
394C23×C4—1221no—
395C23×C4—1221no—
396C22×C8—611no—
397C22×C8—611no—
398C22×C8—614no—
399C22×C8—614no—
400C22×C8—2441no—
401C22×C8—2441no—
402C22×C8—2441no—
403C22×C8—2441no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4×D4—614no—
407C4×D4—614no—
408#117—1222no—
409#117—1222no—
410#117—2441no—
411#117—2441no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4—1222no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—2441no—
422C2×C4⋊C4—2441no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—21no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42—1221no—
439C2×C42C22×D6111yescharacteristic
440#107—41no—
441#107—2441no—
442#106—41no—
443#106—1222no—
444#106—1222no—
445#106—2441no—
446C4⋊C8C4×D6114yescomplemented
447C22⋊C8—614no—
448C22⋊C8—614no—
449C4×C8—614no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—611no—
461C24—611no—
462C2×D4—614no—
463C2×D4—1222no—
464C2×D4—1222no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—612no—
472C22×C4—612no—
473C22×C4—612no—
474C22×C4—612no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—612no—
501C2×C8—612no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—2444no—
507C2×C8—2444no—
508C2×C8C2×C4×D6112yes—
509C2×C8C2×C4×D6112yes—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—614no—
518C4⋊C4—612no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4#1175822612yes—
529C22⋊C4—614no—
530C22⋊C4—614no—
531C22⋊C4—4881no—
532C22⋊C4—4881no—
533C22⋊C4—2442no—
534C22⋊C4—2442no—
535C22⋊C4—2442no—
536C22⋊C4—2442no—
537C42—81no—
538C42—4881no—
539C42—2442no—
540C42—2442no—
541C42C2×C4×D6112yes—
542C42Q8×D6112yes—
543C23—612no—
544C23—612no—
545C23—612no—
546C23—612no—
547C23—1221no—
548C23—1221no—
549C23—1221no—
550C23—1221no—
551C23—1221no—
552C23—1221no—
553C23—1221no—
554C23—1221no—
555C23—1221no—
556C23—1221no—
557C23C2×D4×D6111yescharacteristic
558D4—1228no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1222no—
582C2×C4—1222no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—4884no—
591C2×C4—4884no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4C4⋊C4×D6112yes—
604C2×C4C4⋊C4×D6112yes—
605C2×C4C22×C4×D6111yescharacteristic
606C2×C4C2×Q8×D6111yescharacteristic
607C2×C4#41×D6111yescharacteristic
608C2×C4C2×#1175822611yescharacteristic
609C2×C4#1451982211yescharacteristic
610C2×C4#1451982311yescharacteristic
611C8—22no—
612C8—22no—
613C8—1222no—
614C8—1222no—
615C22—614no—
616C22—1222no—
617C22—1222no—
618C22—1222no—
619C22—1222no—
620C22—1222no—
621C22—1222no—
622C22—1222no—
623C22—1222no—
624C22—1222no—
625C22—2441no—
626C22—2441no—
627C22—2441no—
628C22—2441no—
629C22C2×C4⋊C4×D6111yescharacteristic
630C22#1684040311yescharacteristic
631C22#1684040911yescharacteristic
632C22#135×D6111yescharacteristic
633C22#135×D6111yescharacteristic
634C22#1684057111yescharacteristic
635C22#1684057211yescharacteristic
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—1222no—
643C4—1222no—
644C4—4881no—
645C4—4881no—
646C4—4881no—
647C4—2442no—
648C4C4×#1175822611yescharacteristic
649C4#1684023411yescharacteristic
650C4#1684046811yescharacteristic
651C4#1684047311yescharacteristic
652C2—1224no—
653C2—2442no—
654C2#2144872811yescharacteristic
655C2#427×D6111yescharacteristic
656C2#2145441311yescharacteristic
657C2#2145765111yescharacteristic
658C2#2145766111yescharacteristic
659C2#2145771811yescharacteristic
660C2#2145772811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1949C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1949D61
hypocenterC61#22706
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #21446589C2semidirect
5. #21446021C2semidirect
6. #21446015C2semidirect
7. #21446013C2semidirect
8. #21446010C2semidirect
9. #21445949C2semidirect
10. #21445845C2semidirect
11. #21445841C2semidirect
12. C4×#14519708C2semidirect
13. #21445534C2semidirect
14. #340×D61C2semidirect
15. C61×#1949C2semidirect
23. #16839827C4semidirect
72. C2×C8×D61C2×C4semidirect
106. #14519689C2×C4semidirect
107. C4×#92418C2×C4semidirect
125. C4⋊C8×C61C2×C4semidirect
326. #1949D61semidirect
390. C61#22706semidirect
446. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398272
#214487512
#214487452
#214465892
#214460212
#214460152
#214460132
#214460102
#214459492
#214458452
#214458412
C4×#145197082
#214455342
#340×D612
C61×#19492
#2270661