Finite Group Databasebeta

← #29647835

Subgroups — #29647835

Order 15616.

1254 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459581C211yesmaximal, characteristic
2#21454859C211yesmaximal, characteristic, complemented
3#21448794C211yesmaximal, characteristic, complemented
4#21444378C211yesmaximal, characteristic, complemented
5#21444346C214yesmaximal, complemented
6#21444333C214yesmaximal, complemented
7#21444315C211yesmaximal, characteristic, complemented
8#21444314C211yesmaximal, characteristic, complemented
9C61×#1517C211yesmaximal, characteristic, complemented
10C2×#14519823—21no—
11#16840417C412yescomplemented
12#16840415C412yescomplemented
13#16840408—22no—
14#16840405—22no—
15D4×D244C412yescomplemented
16#16840397C412yescomplemented
17#16840340—21no—
18#16840320—22no—
19#16840312C2211yescharacteristic
20#121×D61C2214yes—
21C2×C4×D244C2211yescharacteristic
22#16839841—22no—
23#16839839—22no—
24#16839815C2211yescharacteristic, complemented
25#16839724C2214yescomplemented
26#16839663—21no—
27#16839654—22no—
28#16839624C2212yescomplemented
29C4⋊C4×Dic61—21no—
30C4×C4⋊Dic61C2211yescharacteristic, complemented
31#16839609C2212yescomplemented
32#16839497C2212yescomplemented
33#16839497C2212yescomplemented
34#16839496C2212yescomplemented
35#16839496C2212yescomplemented
36#16839495C2214yescomplemented
37C61×#532C2211yescharacteristic
38C61×#431C2211yescharacteristic, complemented
39C61×#382C2211yescharacteristic, complemented
40C61×#327C2214yescomplemented
41#14519823—44no—
42#14519823—44no—
43C2×#11758226—22no—
44C2×#11758222—21no—
45C2×#11758222—22no—
46C22×D244—22no—
47C22×D244D411yescharacteristic
48C2×D4×D61—24no—
49C2×D4×D61—22no—
50C2×D4×D61C2×C414yes—
51C22×C4×D61C2312yes—
52C2×#11758189—24no—
53#14519777C2×C411yescharacteristic
54#14519774C2×C411yescharacteristic
55#14519765—21no—
56#14519762—24no—
57#14519759—24no—
58#14519756—22no—
59#14519753—22no—
60#14519752C2×C414yes—
61#14519751—21no—
62#14519751—22no—
63#14519751C2312yes—
64#14519750—24no—
65#14519749—24no—
66#14519748—22no—
67#14519748C2×C414yes—
68#14519747—24no—
69#14519745—24no—
70C4×D244—21no—
71C4×D244C2×C412yes—
72C4×D244C2×C412yes—
73C4×D244C2×C412yes—
74C4×D244D412yes—
75C4×D244D412yes—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61D411yescharacteristic
78C4⋊C4×D61C2×C414yes—
79C2×#11758198—22no—
80C2×#11758198—22no—
81C2×#11758198D412yes—
82C22⋊C4×D61—24no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—22no—
85C4⋊#92418—21no—
86C22⋊#92418C2312yes—
87#14519689—24no—
88C4×#92418—21no—
89#14519684—22no—
90#14519684C2311yescharacteristic
91#14519684C2311yescharacteristic
92C2×#41×C61—21no—
93C4⋊Q8×C61C2×C411yescharacteristic
94#127×C61C2×C411yescharacteristic
95#124×C61—21no—
96#121×C61—21no—
97#121×C61C2312yes—
98C4×D4×C61—21no—
99C4×D4×C61C2×C412yes—
100C2×C4⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C22⋊C8×C61C2312yes—
104C4×C8×C61—21no—
105#95×C61—21no—
106#95×C61C2312yes—
107C23×D61—24no—
108C23×D61C2×D412yes—
109#41×C61—42no—
110#41×C61—42no—
111C2×Q8×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61—21no—
114C2×D4×C61—21no—
115C2×D4×C61C22×C412yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2411yescharacteristic
121C2×C22⋊D61—22no—
122C2×C22⋊D61—24no—
123C2×C22⋊D61—24no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61C2×D412yes—
127#11758226—48no—
128D4×D61—48no—
129D4×D61—48no—
130D4×D61—48no—
131D4×D61—84no—
132#11758222—48no—
133#11758222—42no—
134#11758222—42no—
135C2×D244—22no—
136C2×D244—22no—
137C2×D244—24no—
138C2×D244—21no—
139C2×D244—21no—
140C2×D244—48no—
141C2×D244C22⋊C412yes—
142C2×D244C22⋊C412yes—
143C2×D244C22×C412yes—
144C2×C4×D61—24no—
145C2×C4×D61—22no—
146C2×C4×D61—82no—
147C2×C4×D61C22×C414yes—
148C2×C4×D61C2×D414yes—
149C2×Dic122—21no—
150C2×C8×C61—22no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—42no—
153C4⋊C4×C61C22×C412yes—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61C22×C411yescharacteristic
157C42×C61C22×C411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C22⋊Dic61—24no—
161#11758198—22no—
162#11758198—22no—
163#11758198—24no—
164#11758198—44no—
165#11758198—82no—
166#11758198C22⋊C414yes—
167#11758198C22⋊C414yes—
168C4⋊Dic61—21no—
169C4⋊Dic61—42no—
170C4⋊Dic61C22⋊C412yes—
171C4⋊Dic61C22⋊C412yes—
172C4⋊Dic61C22×C412yes—
173#11758196—24no—
174C4×Dic61—42no—
175C4×Dic61—82no—
176C2×#92418—22no—
177#11758189—84no—
178C23×C61—22no—
179C23×C61C22×D411yescharacteristic
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61—48no—
186C22×D61—84no—
187C22×D61—84no—
188C22×D61C2×C22⋊C414yes—
189Q8×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—81no—
195C2×C4×C61—22no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—44no—
199C2×C4×C61—42no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61C2×C22⋊C411yescharacteristic
203C2×C4×C61C2×C22⋊C411yescharacteristic
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61AGL(1,ℤ/8)12yes—
207C2×C4×C61AGL(1,ℤ/8)12yes—
208C2×C4×C61C23×C411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C22⋊D61—48no—
211C22⋊D61—42no—
212C22⋊D61—42no—
213C22⋊D61—84no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—42no—
217C2×Dic61—41no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—41no—
222C2×Dic61C2×C22⋊C414yes—
223D244—24no—
224D244—24no—
225D244—42no—
226D244—48no—
227D244—42no—
228D244—81no—
229C4×D61—44no—
230C4×D61—44no—
231C4×D61—48no—
232C4×D61—44no—
233Dic122—42no—
234C8×C61—42no—
235#92418—42no—
236#17626—611nomaximal, Sylow
237C22×C61—21no—
238C22×C61—21no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61C22×C22⋊C411yescharacteristic
243C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
244C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
245C2×D61—24no—
246C2×D61—24no—
247C2×D61—24no—
248C2×D61—42no—
249C2×D61—42no—
250C2×D61—42no—
251C2×D61—82no—
252C2×D61—82no—
253C2×D61—168no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—21no—
257C4×C61—21no—
258C4×C61—42no—
259C4×C61—82no—
260C4×C61#41812yes—
261C4×C61#41812yes—
262Dic61—24no—
263Dic61—42no—
264Dic61—42no—
265Dic61—81no—
266#3496—611no—
267#2975—611no—
268#2357—611no—
269#1517—611no—
270#1517D6111yescharacteristic, complemented
271#1510—614no—
272#1507—614no—
273#1502—611no—
274#1502—611no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—82no—
278C2×C61#288311yescharacteristic
279C2×C61C2×#41811yescharacteristic
280C2×C61#290111yescharacteristic
281D61—44no—
282D61—82no—
283C2×H(5,𝔽2)—1221no—
284#547—612no—
285#547—612no—
286#546—1222no—
287#546—1222no—
288D42—612no—
289D42—612no—
290#537—1221no—
291#534—1222no—
292#532—611no—
293#532C2×D6111yescharacteristic
294C2×#121—614no—
295C2×C4×D4—611no—
296#436—1222no—
297#436—1222no—
298#431—611no—
299#431C2×D6111yescharacteristic, complemented
300#407—614no—
301#390—1221no—
302#388—1222no—
303#382C2×D6111yescharacteristic, complemented
304#381—612no—
305C4×C4⋊C4—611no—
306C4×C4⋊C4—1221no—
307C4×C22⋊C4—612no—
308#327—612no—
309#327—612no—
310#327—612no—
311#327—612no—
312#327—614no—
313#327C2×D6114yescomplemented
314C61#1762611yescharacteristic, complemented, Sylow
315H(5,𝔽2)—2444no—
316H(5,𝔽2)—2444no—
317C2×#41—21no—
318C2×#41—1222no—
319C2×#41—1222no—
320C2×#41—1221no—
321C22×D4—614no—
322C22×D4—611no—
323C22×D4—1222no—
324C22×D4—1222no—
325C22×D4—1224no—
326C23×C4—612no—
327C2×M16—1224no—
328C4⋊Q8—611no—
329C4⋊Q8C4×D6111yescharacteristic
330#127—611no—
331#127C4×D6111yescharacteristic
332#124—21no—
333#124—1221no—
334#123—1224no—
335#123—1224no—
336C22⋊Q8—1222no—
337C22⋊Q8—1222no—
338#121—21no—
339#121—614no—
340#121—612no—
341#121—614no—
342#121—1224no—
343#121—1222no—
344#121—1222no—
345#121—1224no—
346#121—1221no—
347#121C22×D6112yes—
348#120—1224no—
349#120—1224no—
350C4×D4—21no—
351C4×D4—612no—
352C4×D4—612no—
353C4×D4—612no—
354C4×D4—612no—
355C4×D4—612no—
356C4×D4—1221no—
357C4×D4C4×D6112yes—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—611no—
360C2×C4⋊C4—614no—
361C2×C4⋊C4—1221no—
362C2×C22⋊C4—612no—
363C2×C22⋊C4—1224no—
364C2×C22⋊C4—1222no—
365C2×C22⋊C4—1222no—
366C2×C42—1222no—
367C2×C42—1221no—
368C2×C42C22×D6111yescharacteristic
369C4⋊C8—21no—
370C4⋊C8—1221no—
371C22⋊C8—612no—
372C22⋊C8—1224no—
373C22⋊C8C22×D6112yes—
374C4×C8—21no—
375C4×C8—1221no—
376#95—21no—
377#95—611no—
378#95—611no—
379#95—1222no—
380#95C22×D6112yes—
381C24—612no—
382C24—1224no—
383#41—42no—
384#41—42no—
385#41—2448no—
386#41—2448no—
387#41—2442no—
388#41—2442no—
389C2×Q8—21no—
390C2×Q8—1221no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—612no—
395C2×D4—612no—
396C2×D4—612no—
397C2×D4—1224no—
398C2×D4—1224no—
399C2×D4—1224no—
400C2×D4—1222no—
401C2×D4—1221no—
402C2×D4—1221no—
403C2×D4—1222no—
404C2×D4—1222no—
405C2×D4—4884no—
406C2×D4—2448no—
407C2×D4—2448no—
408C2×D4—2448no—
409C2×D4—2448no—
410C2×D4C2×C4×D6112yes—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—612no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—1224no—
417C22×C4—1221no—
418C22×C4—1222no—
419C22×C4—1221no—
420C22×C4—4882no—
421C22×C4D4×D6111yescharacteristic
422C22×C4D4×D6111yescharacteristic
423C22×C4C23×D6111yescharacteristic
424M16—4884no—
425C2×C8—22no—
426C2×C8—1222no—
427C4⋊C4—21no—
428C4⋊C4—42no—
429C4⋊C4—612no—
430C4⋊C4—612no—
431C4⋊C4—612no—
432C4⋊C4—1224no—
433C4⋊C4—1221no—
434C4⋊C4—2442no—
435C4⋊C4C2×C4×D6112yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—614no—
439C22⋊C4—614no—
440C22⋊C4—1224no—
441C22⋊C4—1224no—
442C22⋊C4—1222no—
443C22⋊C4—1222no—
444C22⋊C4—4882no—
445C22⋊C4—2444no—
446C42—4882no—
447C42—2442no—
448C42C2×C4×D6111yescharacteristic
449C42C2×C4×D6111yescharacteristic
450C42D4×D6111yescharacteristic
451C42D4×D6111yescharacteristic
452C23—22no—
453C23—614no—
454C23—1222no—
455C23—1222no—
456C23—1222no—
457C23—4884no—
458C23—4884no—
459C23—2442no—
460C23—2442no—
461C23—2448no—
462C23C2×D4×D6111yescharacteristic
463Q8—42no—
464Q8—2442no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—81no—
470D4—1224no—
471D4—1224no—
472D4—4881no—
473D4—4884no—
474D4—2442no—
475D4—2448no—
476D4—2448no—
477D4—2442no—
478D4—2442no—
479D4—2442no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—614no—
488C2×C4—1222no—
489C2×C4—1222no—
490C2×C4—2444no—
491C2×C4—2444no—
492C2×C4—2448no—
493C2×C4—2442no—
494C2×C4—2441no—
495C2×C4—2441no—
496C2×C4—2441no—
497C2×C4—2441no—
498C2×C4—2441no—
499C2×C4—2444no—
500C2×C4C22⋊C4×D6111yescharacteristic
501C2×C4C22⋊C4×D6111yescharacteristic
502C2×C4C22⋊C4×D6111yescharacteristic
503C2×C4C22⋊C4×D6111yescharacteristic
504C2×C4#1451980112yes—
505C2×C4#1451980412yes—
506C2×C4C22×C4×D6111yescharacteristic
507C2×C4C2×D4×D6111yescharacteristic
508C8—42no—
509C8—2442no—
510C22—21no—
511C22—21no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—1224no—
516C22—1224no—
517C22—1224no—
518C22—9768no—
519C22—4882no—
520C22—4882no—
521C22—2442no—
522C22—2442no—
523C22—2442no—
524C22C2×C22⋊C4×D6111yescharacteristic
525C22C2×#1451980111yescharacteristic
526C22C2×#1451980411yescharacteristic
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—42no—
532C4—82no—
533C4—1224no—
534C4—4881no—
535C4—2442no—
536C4—2442no—
537C4#1683976512yes—
538C4#1683976512yes—
539C2—21no—
540C2—21no—
541C2—82no—
542C2—4882no—
543C2—2444no—
544C2#2145427811yescharacteristic
545C2C2×#1683976511yescharacteristic
546C2#2145436511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1517C2
socleC22×C61C22×C22⋊C4
prime radical#21459581C2
cosocleC22×C4×C61C24
hypercenter#1517D61
hypocenterC61#17626
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1517

Complements

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

NH ≅ G/Ntype
2. #21454859C2semidirect
3. #21448794C2semidirect
4. #21444378C2semidirect
5. #21444346C2semidirect
6. #21444333C2semidirect
7. #21444315C2semidirect
8. #21444314C2semidirect
9. C61×#1517C2semidirect
11. #16840417C4semidirect
12. #16840415C4semidirect
15. D4×D244C4semidirect
16. #16840397C4semidirect
24. #16839815C22semidirect
25. #16839724C22semidirect
28. #16839624C22semidirect
30. C4×C4⋊Dic61C22semidirect
31. #16839609C22semidirect
32. #16839497C22semidirect
33. #16839497C22semidirect
34. #16839496C22semidirect
35. #16839496C22semidirect
36. #16839495C22semidirect
38. C61×#431C22semidirect
39. C61×#382C22semidirect
40. C61×#327C22semidirect
270. #1517D61semidirect
299. #431C2×D61semidirect
303. #382C2×D61semidirect
313. #327C2×D61semidirect
314. C61#17626semidirect

Maximal subgroups

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

subgroupindex
#214595812
#214548592
#214487942
#214443782
#214443462
#214443332
#214443152
#214443142
C61×#15172
#1762661