Finite Group Databasebeta

← #29570361

Subgroups — #29570361

Order 15616.

914 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#407×D61C211yesmaximal, characteristic
2#407×D61C211yesmaximal, characteristic
3#21448201C211yesmaximal, characteristic
4#21443726C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443724C211yesmaximal, characteristic, complemented
7C61×#1305C211yesmaximal, characteristic, complemented
8C23×C4×D61C2211yescharacteristic
9C2×M16×D61—21no—
10C2×M16×D61C412yescomplemented
11C2×#14519735—21no—
12#16839726C2211yescharacteristic
13#16839726C2211yescharacteristic
14#16839725—21no—
15#16839725C412yescomplemented
16#16839724—21no—
17#16839724—21no—
18#16839724—21no—
19#16839724C412yescomplemented
20C22⋊C8×D61—21no—
21C22⋊C8×D61—21no—
22C22⋊C8×D61—21no—
23C22⋊C8×D61C412yescomplemented
24#16839624—21no—
25#16839622C2211yescharacteristic
26#16839619—21no—
27#16839617C412yes—
28C22⋊C4×Dic61C412yes—
29#16839609—21no—
30C61×#407C2211yescharacteristic
31C61×#407C2211yescharacteristic
32C61×#380C2211yescharacteristic
33C24×D61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C22×C4×D61—44no—
36C22×C4×D61C2×C411yescharacteristic
37C22×C4×D61C2×C411yescharacteristic
38C22×C4×D61C2×C411yescharacteristic
39C22×C4×D61C2×C411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61D411yescharacteristic
42C22×C4×D61D411yescharacteristic
43C22×C4×D61D411yescharacteristic
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61Q811yescharacteristic
47C22×C4×D61Q811yescharacteristic
48C2×#11758194—21no—
49C2×#11758194C2×C412yes—
50C2×#11758189—22no—
51C2×#11758189—22no—
52M16×D61—44no—
53M16×D61—44no—
54C2×C8×D61—22no—
55C2×C8×D61—22no—
56#14519735—44no—
57C2×C4⋊Dic61—21no—
58C2×C22⋊Dic61C2×C412yes—
59C2×#11758198—22no—
60C2×#11758198—22no—
61C2×C4×Dic61—22no—
62C22⋊#92418—21no—
63C22⋊#92418—21no—
64C22⋊#92418—21no—
65C22⋊#92418C2×C412yes—
66#14519689—22no—
67#14519689—22no—
68#14519689—22no—
69#14519689—22no—
70#14519684—21no—
71#14519684—22no—
72C23×C4×C61C2311yescharacteristic
73C2×M16×C61—21no—
74C2×M16×C61C2×C412yes—
75C2×C4⋊C4×C61—21no—
76C2×C22⋊C4×C61C2×C412yes—
77C22⋊C8×C61—21no—
78C22⋊C8×C61—21no—
79C22⋊C8×C61—21no—
80C22⋊C8×C61C2×C412yes—
81#95×C61—21no—
82C24×C61C22×C411yescharacteristic
83C23×D61—42no—
84C23×D61—42no—
85C23×D61—41no—
86C23×D61—41no—
87C23×D61C4211yescharacteristic
88C23×D61C4211yescharacteristic
89C23×D61C22⋊C411yescharacteristic
90C23×D61C22⋊C411yescharacteristic
91C23×D61C4⋊C411yescharacteristic
92C23×D61C4⋊C411yescharacteristic
93C22×C4×C61—22no—
94C22×C4×C61—42no—
95C22×C4×C61C22×C411yescharacteristic
96C22×C4×C61C22×C411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×Q811yescharacteristic
101C22×Dic61—22no—
102C22×Dic61—42no—
103C22×Dic61C4211yescharacteristic
104C22×Dic61C4211yescharacteristic
105C22×Dic61C22⋊C411yescharacteristic
106C22×Dic61C22⋊C411yescharacteristic
107C22×Dic61C4⋊C411yescharacteristic
108C22×Dic61C4⋊C411yescharacteristic
109C2×C4×D61—24no—
110C2×C4×D61—24no—
111C2×C4×D61—24no—
112C2×C4×D61—42no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61—42no—
117C2×C4×D61—42no—
118C2×C4×D61—42no—
119C2×C4×D61—42no—
120C2×C4×D61—42no—
121C2×C4×D61C22⋊C411yescharacteristic
122C2×C4×D61C22⋊C411yescharacteristic
123C2×C4×D61C22⋊C411yescharacteristic
124C2×C4×D61C22⋊C411yescharacteristic
125C2×C4×D61C22⋊C411yescharacteristic
126C2×C4×D61C22⋊C411yescharacteristic
127C2×C4×D61C22⋊C411yescharacteristic
128C2×C4×D61C22⋊C411yescharacteristic
129C2×C4×D61C4⋊C411yescharacteristic
130C2×C4×D61C4⋊C411yescharacteristic
131C2×C4×D61C4⋊C411yescharacteristic
132C2×C4×D61C4⋊C411yescharacteristic
133C2×C4×D61C4⋊C411yescharacteristic
134C2×C4×D61C4⋊C411yescharacteristic
135C2×C4×D61C4⋊C411yescharacteristic
136C2×C4×D61C4⋊C411yescharacteristic
137M16×C61—42no—
138M16×C61—42no—
139C2×C8×C61—22no—
140C2×C8×C61—22no—
141C4⋊C4×C61—42no—
142C22⋊C4×C61—42no—
143C22⋊Dic61—42no—
144#11758198—44no—
145#11758198—44no—
146C4⋊Dic61—42no—
147C4×Dic61—44no—
148#11758194—42no—
149#11758194—42no—
150C2×#92418—22no—
151C2×#92418—22no—
152#11758189—44no—
153#11758189—44no—
154C8×D61—44no—
155C8×D61—44no—
156C23×C61—21no—
157C23×C61—21no—
158C23×C61—41no—
159C23×C61—41no—
160C23×C61C2×C4211yescharacteristic
161C23×C61C2×C22⋊C411yescharacteristic
162C23×C61C2×C4⋊C411yescharacteristic
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—21no—
166C22×D61—21no—
167C22×D61—22no—
168C22×D61—22no—
169C22×D61—22no—
170C22×D61—22no—
171C22×D61—22no—
172C22×D61—22no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—48no—
184C22×D61—48no—
185C22×D61#9511yescharacteristic
186C22×D61#9511yescharacteristic
187C22×D61#9511yescharacteristic
188C22×D61#9511yescharacteristic
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C2×C4×C61C2×C4⋊C411yescharacteristic
201C2×C4×C61C2×C4⋊C411yescharacteristic
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—22no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61#9511yescharacteristic
212C2×Dic61#9511yescharacteristic
213C2×Dic61#9511yescharacteristic
214C2×Dic61#9511yescharacteristic
215C4×D61—22no—
216C4×D61—22no—
217C4×D61—22no—
218C4×D61—22no—
219C4×D61—22no—
220C4×D61—22no—
221C4×D61—22no—
222C4×D61—22no—
223C4×D61—22no—
224C4×D61—41no—
225C4×D61—41no—
226C4×D61—41no—
227C4×D61—41no—
228C4×D61—41no—
229C4×D61—41no—
230C4×D61#9914yes—
231C8×C61—42no—
232C8×C61—42no—
233#92418—42no—
234#92418—42no—
235#8108—611nomaximal, Sylow
236C22×C61—21no—
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61C2×#9511yescharacteristic
247C22×C61#41311yescharacteristic
248C22×C61#41311yescharacteristic
249C2×D61—21no—
250C2×D61—21no—
251C2×D61—21no—
252C2×D61—21no—
253C2×D61—21no—
254C2×D61—21no—
255C2×D61—21no—
256C2×D61—21no—
257C2×D61—21no—
258C2×D61—21no—
259C2×D61—44no—
260C2×D61—42no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—44no—
265C2×D61—44no—
266C2×D61—44no—
267C2×D61#34212yes—
268C2×D61#34412yes—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—82no—
273C4×C61C2×#9912yes—
274Dic61—21no—
275Dic61—21no—
276Dic61—21no—
277Dic61—82no—
278Dic61#34212yes—
279C2×#407—611no—
280C2×#407—611no—
281#2280—611no—
282#1305—611no—
283#1305—611no—
284#1305—611no—
285#1305D6111yescharacteristic, complemented
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—41no—
289C2×C61—41no—
290C2×C61C2×#34211yescharacteristic
291C2×C61#174211yescharacteristic
292C2×C61C2×#34411yescharacteristic
293D61—22no—
294D61—22no—
295D61—41no—
296D61—41no—
297C24×C4—611no—
298C22×M16—612no—
299C22×M16—1221no—
300C2×#117—1221no—
301#407—611no—
302#407—612no—
303#407—611no—
304#407—612no—
305#407—1221no—
306#407—1221no—
307#407—1221no—
308#407—1221no—
309#407C2×D6111yescharacteristic
310#407C2×D6111yescharacteristic
311C2×C22⋊C8—612no—
312C2×C22⋊C8—1221no—
313C2×C22⋊C8—1221no—
314C2×C22⋊C8—1221no—
315#381—1221no—
316#380—611no—
317#380—1221no—
318#380C2×D6111yescharacteristic
319#379—612no—
320C4×C22⋊C4—612no—
321C4×C22⋊C4—1221no—
322C61#810811yescharacteristic, complemented, Sylow
323C25—611no—
324C23×C4—611no—
325C23×C4—611no—
326C23×C4—611no—
327C23×C4—611no—
328C23×C4—611no—
329C23×C4—611no—
330C23×C4—611no—
331C23×C4—611no—
332C23×C4—611no—
333C23×C4—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—611no—
337C23×C4—2444no—
338C23×C4C22×D6111yescharacteristic
339C2×M16—21no—
340C2×M16—612no—
341C2×M16—1222no—
342C2×M16—1222no—
343C2×M16—1221no—
344C2×M16—2444no—
345C2×M16—2444no—
346C2×M16C4×D6112yes—
347C22×C8—1222no—
348C22×C8—1222no—
349#117—2444no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—1221no—
352C2×C22⋊C4—612no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4C4×D6112yes—
356C2×C42—1222no—
357C22⋊C8—21no—
358C22⋊C8—21no—
359C22⋊C8—21no—
360C22⋊C8—612no—
361C22⋊C8—1222no—
362C22⋊C8—1222no—
363C22⋊C8—1222no—
364C22⋊C8—1222no—
365C22⋊C8—1221no—
366C22⋊C8—1221no—
367C22⋊C8—1221no—
368C22⋊C8C4×D6112yes—
369#95—21no—
370#95—1222no—
371#95—1221no—
372C24—611no—
373C24—611no—
374C24—611no—
375C24—611no—
376C24—611no—
377C24—611no—
378C24—2441no—
379C24—2441no—
380C24—2442no—
381C24—2442no—
382C24C2×C4×D6111yescharacteristic
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—611no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—611no—
391C22×C4—611no—
392C22×C4—611no—
393C22×C4—611no—
394C22×C4—611no—
395C22×C4—611no—
396C22×C4—611no—
397C22×C4—611no—
398C22×C4—611no—
399C22×C4—611no—
400C22×C4—611no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—611no—
404C22×C4—611no—
405C22×C4—611no—
406C22×C4—611no—
407C22×C4—1224no—
408C22×C4—1224no—
409C22×C4—1224no—
410C22×C4—1222no—
411C22×C4—2442no—
412C22×C4—2442no—
413C22×C4—2442no—
414C22×C4—2442no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2442no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4C2×C4×D6111yescharacteristic
422C22×C4C2×C4×D6111yescharacteristic
423C22×C4D4×D6111yescharacteristic
424C22×C4D4×D6111yescharacteristic
425C22×C4D4×D6111yescharacteristic
426C22×C4Q8×D6111yescharacteristic
427M16—42no—
428M16—42no—
429M16—2444no—
430M16—2444no—
431M16—2442no—
432M16—2442no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1222no—
436C2×C8—1222no—
437C2×C8—2444no—
438C2×C8—2444no—
439C4⋊C4—42no—
440C4⋊C4—2442no—
441C22⋊C4—42no—
442C22⋊C4—2444no—
443C22⋊C4—2444no—
444C22⋊C4—2442no—
445C42—2444no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—611no—
451C23—611no—
452C23—611no—
453C23—611no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—1222no—
458C23—1222no—
459C23—1222no—
460C23—1221no—
461C23—1221no—
462C23—1221no—
463C23—1221no—
464C23—2441no—
465C23—2441no—
466C23—2441no—
467C23—2441no—
468C23—2441no—
469C23—2441no—
470C23—2441no—
471C23—2441no—
472C23—2441no—
473C23—2441no—
474C23—2448no—
475C23—2448no—
476C23C42×D6111yescharacteristic
477C23C22⋊C4×D6111yescharacteristic
478C23C4⋊C4×D6111yescharacteristic
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—611no—
489C2×C4—614no—
490C2×C4—611no—
491C2×C4—611no—
492C2×C4—611no—
493C2×C4—1222no—
494C2×C4—1222no—
495C2×C4—1222no—
496C2×C4—1222no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—1222no—
500C2×C4—1222no—
501C2×C4—1222no—
502C2×C4—1222no—
503C2×C4—1222no—
504C2×C4—1222no—
505C2×C4—2441no—
506C2×C4—2441no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4—2441no—
512C2×C4—2441no—
513C2×C4—2441no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4C22⋊C4×D6111yescharacteristic
518C2×C4C22⋊C4×D6111yescharacteristic
519C2×C4C4⋊C4×D6111yescharacteristic
520C2×C4C4⋊C4×D6111yescharacteristic
521C8—42no—
522C8—42no—
523C8—2442no—
524C8—2442no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—612no—
536C22—612no—
537C22—1221no—
538C22—1221no—
539C22—1221no—
540C22—1221no—
541C22—1221no—
542C22—1221no—
543C22—1221no—
544C22—1221no—
545C22—1221no—
546C22—1221no—
547C22—2442no—
548C22—2442no—
549C22—2442no—
550C22—2442no—
551C22—2444no—
552C22—2444no—
553C22—2444no—
554C22—2444no—
555C22#95×D6111yescharacteristic
556C22#1683974311yescharacteristic
557C22#1683974511yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—82no—
562C4—612no—
563C4—1221no—
564C4—1221no—
565C4—1221no—
566C4—4882no—
567C4#99×D6112yes—
568C2—21no—
569C2—21no—
570C2—41no—
571C2—41no—
572C2—1222no—
573C2—1222no—
574C2—2441no—
575C2—2441no—
576C2#342×D6111yescharacteristic
577C2#2144555611yescharacteristic
578C2#344×D6111yescharacteristic
subgroupG/H
centerC22#95×D61
commutatorC23×C61C2×C42
FrattiniC23×C4C22×D61
FittingC61×#1305C2
socleC22×C61C2×#95
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1305D61
hypocenterC61#8108
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
4. #21443726C2semidirect
5. #21443725C2semidirect
6. #21443724C2semidirect
7. C61×#1305C2semidirect
10. C2×M16×D61C4semidirect
15. #16839725C4semidirect
19. #16839724C4semidirect
23. C22⋊C8×D61C4semidirect
285. #1305D61semidirect
322. C61#8108semidirect

Maximal subgroups

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

subgroupindex
#407×D612
#214482012
#214437262
#214437252
#214437242
C61×#13052
#810861