Finite Group Databasebeta

← #29648616

Subgroups — #29648616

Order 15616.

1232 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840310C211yesmaximal, characteristic
2#21454449C211yesmaximal, characteristic, complemented
3#432×D61C211yesmaximal, characteristic, complemented
4#21444537C211yesmaximal, characteristic, complemented
5#21444535C211yesmaximal, characteristic, complemented
6#21444533C211yesmaximal, characteristic, complemented
7#21444521C214yesmaximal, complemented
8#21444493C214yesmaximal, complemented
9C61×#1555C211yesmaximal, characteristic, complemented
10C22×#11758224—21no—
11#16840310—21no—
12#16840310—21no—
13#16840310—21no—
14#16840310—21no—
15#16840310C412yes—
16#16840310C412yes—
17#16840310C2212yes—
18#16840310C2212yes—
19C2×#14519777C2211yescharacteristic
20C2×#14519764—21no—
21C2×#14519748—21no—
22C2×#14519748C412yes—
23C2×D4×Dic61—21no—
24C2×D4×Dic61C412yes—
25C2×C42×D61C2211yescharacteristic
26#16839819C2211yescharacteristic
27#16839819C2211yescharacteristic
28#16839818C2212yescomplemented
29#16839817—21no—
30#16839817—21no—
31#16839816—21no—
32#16839814C2212yescomplemented
33#16839812—21no—
34#16839784C2212yescomplemented
35C4⋊C8×D61C2212yescomplemented
36#16839724—21no—
37C22⋊C8×D61—21no—
38#16839714—21no—
39#97×D61—21no—
40#16839509C2214yescomplemented
41#16839507C2214yescomplemented
42#16839506C2214yescomplemented
43C2×#127×C61C2211yescharacteristic
44C61×#432C2211yescharacteristic
45C61×#432C2211yescharacteristic
46C61×#331C2214yescomplemented
47C2×#11758224—22no—
48C2×#11758224—22no—
49C2×#11758224—22no—
50C2×#11758224—22no—
51C2×#11758224—48no—
52C22×Dic122—21no—
53C22×C22⋊D61—22no—
54C23×Dic61—22no—
55C22×C4×D61C2×C411yescharacteristic
56C22×C4×D61D411yescharacteristic
57C22×C4×D61D411yescharacteristic
58C2×#11758189—22no—
59C2×C8×D61—22no—
60#14519777—21no—
61#14519777—21no—
62#14519777C2×C412yes—
63#14519777C2312yes—
64#14519764—22no—
65#14519764—22no—
66#14519764—22no—
67#14519764—22no—
68#14519748—22no—
69#14519748—22no—
70#14519748—22no—
71#14519748—22no—
72#14519748—22no—
73#14519748—22no—
74#14519748C2×C414yes—
75#14519748C2×C414yes—
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—22no—
79D4×Dic61—22no—
80D4×Dic61—22no—
81D4×Dic61—22no—
82D4×Dic61C2×C414yes—
83D4×Dic61C2×C414yes—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2×C412yes—
86C2×C22⋊Dic61—22no—
87C2×C22⋊Dic61—22no—
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61D411yescharacteristic
90C2×C4×Dic61D411yescharacteristic
91C42×D61C2×C411yescharacteristic
92C42×D61C2×C411yescharacteristic
93C42×D61D411yescharacteristic
94C42×D61D411yescharacteristic
95C42×D61D411yescharacteristic
96C42×D61D411yescharacteristic
97C42×D61C2311yescharacteristic
98C42×D61C2311yescharacteristic
99C4⋊#92418C2312yes—
100C4⋊#92418C2312yes—
101#14519706—22no—
102#14519706—22no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519689—22no—
106#14519689—22no—
107#14519688—21no—
108#14519688—21no—
109#14519687—22no—
110C8×Dic61—22no—
111C22×D4×C61—21no—
112C22×D4×C61—21no—
113C22×D4×C61C2×C412yes—
114#127×C61—21no—
115#127×C61—21no—
116#127×C61C2×C412yes—
117#127×C61C2312yes—
118C2×C42×C61C2311yescharacteristic
119C4⋊C8×C61C2312yes—
120C4⋊C8×C61C2312yes—
121C22⋊C8×C61—21no—
122C22⋊C8×C61—21no—
123#97×C61—21no—
124#97×C61—21no—
125C24×C61—22no—
126C23×D61C22⋊C411yescharacteristic
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61—22no—
132C2×D4×C61—44no—
133C2×D4×C61—44no—
134C2×D4×C61—44no—
135C2×D4×C61C22×C414yes—
136C22×C4×C61C22×C411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61—24no—
140C2×C22⋊D61—24no—
141C2×C22⋊D61—48no—
142C22×Dic61—22no—
143C22×Dic61—24no—
144C22×Dic61—24no—
145C22×Dic61—48no—
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149#11758224—48no—
150#11758224—48no—
151#11758224—48no—
152#11758224—48no—
153C2×C4×D61—24no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C22×C411yescharacteristic
163C2×C4×D61C22×C411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×C4×D61C2×D411yescharacteristic
167C2×C4×D61C2×D411yescharacteristic
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—44no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C42×C61C22×C411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2411yescharacteristic
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181C4⋊Dic61—22no—
182C4⋊Dic61—22no—
183C4⋊Dic61—22no—
184C4⋊Dic61C22×C414yes—
185C4×Dic61C22⋊C411yescharacteristic
186C4×Dic61C22⋊C411yescharacteristic
187C4×Dic61C22⋊C411yescharacteristic
188C4×Dic61C22⋊C411yescharacteristic
189C4×Dic61C22⋊C411yescharacteristic
190C4×Dic61C22⋊C411yescharacteristic
191C4×Dic61C22×C411yescharacteristic
192C4×Dic61C22×C411yescharacteristic
193C4×Dic61C2×D411yescharacteristic
194C4×Dic61C2×D411yescharacteristic
195C4×Dic61C2×D411yescharacteristic
196C4×Dic61C2×D411yescharacteristic
197C2×#92418—22no—
198C2×#92418—22no—
199#11758189—44no—
200C8×D61—44no—
201C23×C61—24no—
202C23×C61—44no—
203C23×C61—44no—
204C23×C61—44no—
205C23×C61C2×C22⋊C411yescharacteristic
206C22×D61—22no—
207C22×D61—22no—
208C22×D61—41no—
209C22×D61C2×C22⋊C411yescharacteristic
210C22×D61C2×C22⋊C411yescharacteristic
211D4×C61—44no—
212D4×C61—44no—
213D4×C61—44no—
214D4×C61—44no—
215D4×C61—44no—
216D4×C61—44no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61C2×C22⋊C411yescharacteristic
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61—48no—
227C22⋊D61—48no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—24no—
231C2×Dic61—41no—
232C2×Dic61—48no—
233C2×Dic61—44no—
234C2×Dic61—48no—
235C2×Dic61C2×C22⋊C411yescharacteristic
236C2×Dic61C2×C22⋊C411yescharacteristic
237C2×Dic61C2×C22⋊C411yescharacteristic
238C2×Dic61C2×C22⋊C411yescharacteristic
239C2×Dic61C2×C22⋊C411yescharacteristic
240C2×Dic61C2×C22⋊C411yescharacteristic
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—41no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—41no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61AGL(1,ℤ/8)14yes—
253Dic122—44no—
254Dic122—44no—
255C8×C61—42no—
256C8×C61—42no—
257#92418—42no—
258#92418—42no—
259#17744—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—44no—
263C22×C61—44no—
264C22×C61—44no—
265C22×C61—88no—
266C22×C61#41312yes—
267C22×C61C22×C22⋊C411yescharacteristic
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—44no—
274C2×D61#41812yes—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61C2×AGL(1,ℤ/8)12yes—
280C4×C61#57512yes—
281Dic61—21no—
282Dic61—21no—
283Dic61—21no—
284Dic61—21no—
285Dic61—21no—
286Dic61—44no—
287Dic61#41812yes—
288C2×#532—611no—
289#2920—611no—
290C2×#432—611no—
291#1556—611no—
292#1556—611no—
293#1555—611no—
294#1555D6111yescharacteristic, complemented
295#1553—614no—
296#1547—614no—
297C2×C61—22no—
298C2×C61—84no—
299C2×C61C2×#41311yescharacteristic
300C2×C61C2×#41811yescharacteristic
301C2×C61#289611yescharacteristic
302D61—22no—
303D61—41no—
304C22×#41—1221no—
305#532—612no—
306#532—612no—
307#532—612no—
308#532—612no—
309#532—1221no—
310#532—1221no—
311#532—1221no—
312#532—1221no—
313C2×C4⋊Q8—611no—
314C2×#127C2×D6111yescharacteristic
315C2×#124—1221no—
316C2×#121—612no—
317C2×#121—1221no—
318C2×C4×D4—612no—
319C2×C4×D4—1221no—
320C22×C42—611no—
321#432—611no—
322#432—611no—
323#432—612no—
324#432—1221no—
325#432—1221no—
326#432—1221no—
327#432C2×D6111yescharacteristic
328#432C2×D6111yescharacteristic
329#431—612no—
330#431—1221no—
331#423—612no—
332C2×C4⋊C8—612no—
333#407—1221no—
334C2×C22⋊C8—1221no—
335C4×M16—1221no—
336C2×#97—1221no—
337#332—614no—
338#332—614no—
339#331—614no—
340#331C2×D6114yescomplemented
341C61#1774411yescharacteristic, complemented, Sylow
342C2×#41—1222no—
343C2×#41—1222no—
344C2×#41—1222no—
345C2×#41—1222no—
346C2×#41—2448no—
347C22×Q8—1221no—
348C22×D4—21no—
349C22×D4—21no—
350C22×D4—1222no—
351C22×D4C4×D6112yes—
352C23×C4—611no—
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—1222no—
356C2×M16—1222no—
357C22×C8—1222no—
358C4⋊Q8—612no—
359C4⋊Q8—612no—
360C4⋊Q8—1221no—
361C4⋊Q8—1221no—
362#127—21no—
363#127—21no—
364#127C4×D6112yes—
365#127C22×D6112yes—
366#124—1222no—
367#124—1222no—
368#124—1222no—
369#124—1222no—
370#121—614no—
371#121—614no—
372#121—1222no—
373#121—1222no—
374#121—1222no—
375#121—1222no—
376#121—1222no—
377#121—1222no—
378C4×D4—614no—
379C4×D4—614no—
380C4×D4—1222no—
381C4×D4—1222no—
382C4×D4—1222no—
383C4×D4—1222no—
384C4×D4—1222no—
385C4×D4—1222no—
386C2×C4⋊C4—612no—
387C2×C4⋊C4—1221no—
388C2×C22⋊C4—1222no—
389C2×C22⋊C4—1222no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42C22×D6111yescharacteristic
402C4⋊C8—612no—
403C4⋊C8—612no—
404C4⋊C8—1222no—
405C4⋊C8—1222no—
406C4⋊C8C22×D6112yes—
407C4⋊C8C22×D6112yes—
408C22⋊C8—21no—
409C22⋊C8—21no—
410C22⋊C8—1222no—
411C22⋊C8—1222no—
412C22⋊C8—1221no—
413C22⋊C8—1221no—
414#97—21no—
415#97—21no—
416#97—1222no—
417#97—1221no—
418#97—1221no—
419C4×C8—1222no—
420C24—22no—
421C24—611no—
422#41—2448no—
423#41—2448no—
424#41—2448no—
425#41—2448no—
426C2×Q8—1222no—
427C2×Q8—1222no—
428C2×Q8—2444no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—1224no—
438C2×D4—1224no—
439C2×D4—2448no—
440C2×D4C2×C4×D6114yes—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—1224no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2448no—
463C22×C4C2×C4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466M16—2444no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—2444no—
472C4⋊C4—614no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C22⋊C4—1224no—
477C22⋊C4—1224no—
478C22⋊C4—1224no—
479C22⋊C4—1224no—
480C42—611no—
481C42—611no—
482C42—611no—
483C42—611no—
484C42—611no—
485C42—611no—
486C42—611no—
487C42—611no—
488C42—611no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42C2×C4×D6111yescharacteristic
493C42D4×D6111yescharacteristic
494C42D4×D6111yescharacteristic
495C42C23×D6111yescharacteristic
496C23—24no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—611no—
501C23—611no—
502C23—1222no—
503C23—1222no—
504C23—2441no—
505C23C22⋊C4×D6111yescharacteristic
506Q8—2444no—
507Q8—2444no—
508D4—44no—
509D4—44no—
510D4—44no—
511D4—44no—
512D4—44no—
513D4—44no—
514D4—2448no—
515D4—2448no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—611no—
520C2×C4—614no—
521C2×C4—611no—
522C2×C4—611no—
523C2×C4—611no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1224no—
534C2×C4—2441no—
535C2×C4—2441no—
536C2×C4—2441no—
537C2×C4—2441no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2448no—
541C2×C4—2441no—
542C2×C4—2448no—
543C2×C4—2444no—
544C2×C4C22⋊C4×D6111yescharacteristic
545C2×C4C22⋊C4×D6111yescharacteristic
546C2×C4C22⋊C4×D6111yescharacteristic
547C2×C4C22×C4×D6111yescharacteristic
548C2×C4C2×D4×D6111yescharacteristic
549C2×C4C2×D4×D6111yescharacteristic
550C8—42no—
551C8—42no—
552C8—2442no—
553C8—2442no—
554C22—21no—
555C22—21no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—88no—
560C22—612no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—1221no—
565C22—1221no—
566C22—2444no—
567C22#1683974312yes—
568C22C2×C22⋊C4×D6111yescharacteristic
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—612no—
574C4—1221no—
575C4—1221no—
576C4—1221no—
577C4—1221no—
578C4—1221no—
579C4—2444no—
580C4AGL(1,ℤ/8)×D6112yes—
581C4#1684055812yes—
582C2—22no—
583C2—84no—
584C2—1222no—
585C2—2441no—
586C2C2×#1683974311yescharacteristic
587C2#418×D6111yescharacteristic
588C2#2145433311yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1555C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840310C2
cosocleC42×C61C24
hypercenter#1555D61
hypocenterC61#17744
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
2. #21454449C2semidirect
3. #432×D61C2semidirect
4. #21444537C2semidirect
5. #21444535C2semidirect
6. #21444533C2semidirect
7. #21444521C2semidirect
8. #21444493C2semidirect
9. C61×#1555C2semidirect
28. #16839818C22semidirect
32. #16839814C22semidirect
34. #16839784C22semidirect
35. C4⋊C8×D61C22semidirect
40. #16839509C22semidirect
41. #16839507C22semidirect
42. #16839506C22semidirect
46. C61×#331C22semidirect
294. #1555D61semidirect
340. #331C2×D61semidirect
341. C61#17744semidirect

Maximal subgroups

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

subgroupindex
C2×#168403102
#214544492
#432×D612
#214445372
#214445352
#214445332
#214445212
#214444932
C61×#15552
#1774461