Finite Group Databasebeta

← #29655659

Subgroups — #29655659

Order 15616.

1082 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456245C211yesmaximal, characteristic, complemented
2#21455879C212yesmaximal, complemented
3#21444953C212yesmaximal, complemented
4#21444940C211yesmaximal, characteristic, complemented
5#21444935C211yesmaximal, characteristic, complemented
6#21444891C212yesmaximal, complemented
7#21444878C211yesmaximal, characteristic, complemented
8#21444873C211yesmaximal, characteristic, complemented
9#21444481C211yesmaximal, characteristic, complemented
10#21444201C212yesmaximal, complemented
11C61×#3481C211yesmaximal, characteristic, complemented
12C2×#14519797—22no—
13#16840312C2211yescharacteristic, complemented
14#16840144—22no—
15#16840141C2211yescharacteristic, complemented
16#16840138C2211yescharacteristic
17#16840136C2211yescharacteristic, complemented
18#16840134C2211yescharacteristic
19#16840086—22no—
20#16840057C2212yescomplemented
21#16840052C2212yes—
22#16839962C2212yescomplemented
23#16839957C2212yes—
24#16839907—22no—
25C4×#11758201—22no—
26C4×#11758199—22no—
27#16839815C2212yescomplemented
28C2×C4×#92418C2211yescharacteristic, complemented
29#16839511C2212yescomplemented
30#16839506C2211yescharacteristic
31#16839506C2211yescharacteristic
32Q8⋊#92418C2212yescomplemented
33Q8⋊#92418C2212yescomplemented
34D4⋊#92418C2212yes—
35D4⋊#92418C2212yes—
36C61×#559C2211yescharacteristic
37C61×#554C2212yes—
38C61×#553—22no—
39C61×#551—22no—
40C61×#546—22no—
41D42×C61C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C61×#529—22no—
44C61×#525—22no—
45C4×#41×C61C2212yes—
46C2×C4×D4×C61—22no—
47C2×#11758222—21no—
48#14519797—48no—
49C2×#11758202—22no—
50C2×#11758200—22no—
51C2×#11758201—22no—
52C2×#11758201—22no—
53C2×#11758199—22no—
54C2×#11758199—22no—
55C22×#92418—21no—
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519765—21no—
59#14519751—22no—
60C4×D244—22no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65#14519701—22no—
66#14519698—22no—
67#14519697—22no—
68#14519697—22no—
69C22⋊#92418—22no—
70C4×#92418—21no—
71C4×#92418—22no—
72C4×#92418C2311yescharacteristic
73C4×#92418C2311yescharacteristic
74C2×#41×C61D412yes—
75C2×#41×C61D412yes—
76C22×D4×C61—22no—
77C23×C4×C61—22no—
78C4⋊Q8×C61C2311yescharacteristic
79#127×C61C2311yescharacteristic
80#126×C61—24no—
81#125×C61—22no—
82#124×C61—22no—
83#124×C61D412yes—
84#123×C61—24no—
85#123×C61—22no—
86#123×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—24no—
89#121×C61—24no—
90#121×C61—22no—
91#121×C61—22no—
92#120×C61—22no—
93#120×C61—22no—
94C4×Q8×C61D412yes—
95C4×Q8×C61C2312yes—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61D412yes—
104C4×D4×C61C2312yes—
105#117×C61—22no—
106#117×C61—22no—
107#117×C61D412yes—
108C2×C4⋊C4×C61—22no—
109C2×C22⋊C4×C61—24no—
110C2×C42×C61—22no—
111C2×C42×C61C2311yescharacteristic
112C24×C61—22no—
113#41×C61—24no—
114#41×C61—42no—
115#41×C61—44no—
116C2×Q8×C61C2×D412yes—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—24no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—22no—
122C2×D4×C61—44no—
123C2×D4×C61—44no—
124C2×D4×C61C2×D412yes—
125C2×D4×C61C2×D412yes—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—84no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C2×C22⋊D61—41no—
136#11758222—44no—
137C2×D244—21no—
138C2×D244—22no—
139C2×C4×D61—41no—
140C2×Dic122—21no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—22no—
156C42×C61—22no—
157C42×C61—22no—
158C42×C61—22no—
159C42×C61C2×D412yes—
160C42×C61C2×D411yescharacteristic
161C42×C61C2411yescharacteristic
162#11758202—44no—
163#11758201—44no—
164#11758201—44no—
165#11758200—44no—
166#11758199—44no—
167#11758199—44no—
168#11758198—42no—
169C4⋊Dic61—22no—
170C2×#92418—24no—
171C2×#92418—21no—
172C2×#92418—21no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—48no—
178C23×C61—82no—
179C23×C61#12011yescharacteristic
180C22×D61—41no—
181Q8×C61—22no—
182Q8×C61—22no—
183Q8×C61—42no—
184D4×C61—22no—
185D4×C61—42no—
186D4×C61—44no—
187D4×C61—48no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—82no—
191D4×C61#13514yescomplemented
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—44no—
200C2×C4×C61—42no—
201C2×C4×C61—42no—
202C2×C4×C61—44no—
203C2×C4×C61—44no—
204C2×C4×C61—82no—
205C2×C4×C61—84no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D412yes—
210C22⋊D61—82no—
211C2×Dic61—41no—
212D244—42no—
213D244—42no—
214D244—42no—
215C4×D61—82no—
216Dic122—42no—
217#92418—22no—
218#92418—22no—
219#92418—42no—
220#92418—42no—
221#18480—611nomaximal, Sylow
222C22×C61—22no—
223C22×C61—24no—
224C22×C61—21no—
225C22×C61—42no—
226C22×C61—42no—
227C22×C61—44no—
228C22×C61—42no—
229C22×C61—88no—
230C22×C61C2×#12011yescharacteristic
231C2×D61—82no—
232C2×D61—81no—
233C4×C61—22no—
234C4×C61—21no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61#45312yes—
240C4×C61C2×#13512yes—
241C4×C61#57512yes—
242Dic61—81no—
243#3481D6111yescharacteristic, complemented
244#3147—611no—
245#3106—612no—
246#1632—612no—
247#1630—611no—
248#1629—611no—
249#1622—612no—
250#1620—611no—
251#1619—611no—
252#1543—611no—
253#1476—612no—
254C2×C61—41no—
255C2×C61—44no—
256C2×C61—82no—
257C2×C61#300212yes—
258C2×C61C2×#45311yescharacteristic
259D61—161no—
260C2×#135—1222no—
261#559C2×D6111yescharacteristic
262#554C2×D6112yes—
263#553—22no—
264#551—22no—
265#546—22no—
266D42C2×D6111yescharacteristic
267#532—611no—
268#532C2×D6111yescharacteristic
269#529—22no—
270#525—22no—
271C4×#41C2×D6112yes—
272C2×C4×D4—22no—
273#495—1222no—
274#494—611no—
275#493—611no—
276#492—611no—
277#492—611no—
278#482—1222no—
279#477—612no—
280#476—612no—
281#460—612no—
282#459—612no—
283#449—1222no—
284C4×SD16—1222no—
285C4×D8—1222no—
286#431—612no—
287C2×C4×C8—611no—
288#333—612no—
289#331—611no—
290#331—611no—
291Q8⋊C8—612no—
292Q8⋊C8—612no—
293D4⋊C8—612no—
294D4⋊C8—612no—
295C61#1848011yescharacteristic, complemented, Sylow
296C2×#41—1221no—
297C2×#41C22⋊D6112yes—
298C2×#41C22⋊D6112yes—
299C22×D4—22no—
300C23×C4—22no—
301#135—2448no—
302C2×Q16—1222no—
303C2×SD16—1222no—
304C2×SD16—1222no—
305C2×SD16—1222no—
306C2×D8—1222no—
307C2×D8—1222no—
308C22×C8—1221no—
309C4⋊Q8—611no—
310C4⋊Q8C22×D6111yescharacteristic
311#127—611no—
312#127C22×D6111yescharacteristic
313#126—24no—
314#125—22no—
315#124—22no—
316#124—1221no—
317#124C22⋊D6112yes—
318#123—22no—
319#123—24no—
320#123—22no—
321C22⋊Q8—24no—
322C22⋊Q8—22no—
323#121—24no—
324#121—22no—
325#121—22no—
326#121—1222no—
327#120—22no—
328#120—22no—
329C4×Q8C22⋊D6112yes—
330C4×Q8C22×D6112yes—
331C4×D4—24no—
332C4×D4—24no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—22no—
337C4×D4—22no—
338C4×D4—1222no—
339C4×D4C22⋊D6112yes—
340C4×D4C22×D6112yes—
341#117—22no—
342#117—22no—
343#117C22⋊D6112yes—
344C2×C4⋊C4—22no—
345C2×C22⋊C4—24no—
346C2×C42—22no—
347C2×C42C22×D6111yescharacteristic
348#107—1222no—
349#106—1222no—
350C4⋊C8—612no—
351C4⋊C8—612no—
352#103—1222no—
353#102—1222no—
354#102—1222no—
355#102—1222no—
356C22⋊C8—1222no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—1221no—
360C4×C8—1222no—
361C24—22no—
362#41—24no—
363#41—42no—
364#41—44no—
365#41—2444no—
366C2×Q8—1221no—
367C2×Q8C2×C22⋊D6112yes—
368C2×Q8C2×C22⋊D6112yes—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—1222no—
376C2×D4—1221no—
377C2×D4—2441no—
378C2×D4C2×C22⋊D6112yes—
379C2×D4C2×C22⋊D6112yes—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—84no—
387C22×C4—2441no—
388C22×C4C2×C22⋊D6112yes—
389C22×C4C2×C22⋊D6111yescharacteristic
390Q16—2444no—
391SD16—2444no—
392SD16—2444no—
393SD16—2444no—
394D8—2444no—
395D8—2444no—
396C2×C8—1224no—
397C2×C8—1221no—
398C2×C8—1221no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—1222no—
407C4⋊C4C2×C22⋊D6112yes—
408C4⋊C4C2×C22⋊D6112yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—24no—
417C22⋊C4—2442no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42C2×C22⋊D6111yescharacteristic
422C42C2×C22⋊D6112yes—
423C42C23×D6111yescharacteristic
424C23—22no—
425C23—22no—
426C23—48no—
427C23—82no—
428C23—2441no—
429C23#1451974611yescharacteristic
430Q8—22no—
431Q8—22no—
432Q8—42no—
433Q8—2442no—
434D4—22no—
435D4—42no—
436D4—44no—
437D4—48no—
438D4—42no—
439D4—42no—
440D4—82no—
441D4—4882no—
442D4—2442no—
443D4—2442no—
444D4—2442no—
445D4#1451979714yescomplemented
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—82no—
459C2×C4—84no—
460C2×C4—4882no—
461C2×C4—2441no—
462C2×C4#1451974612yes—
463C2×C4#1451974611yescharacteristic
464C2×C4C22×C22⋊D6111yescharacteristic
465C2×C4C22×C22⋊D6112yes—
466C8—1222no—
467C8—1222no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—22no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—44no—
477C22—88no—
478C22—4881no—
479C22—4882no—
480C22C2×#1451974611yescharacteristic
481C4—21no—
482C4—22no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—4881no—
488C4#1683993012yes—
489C4C2×#1451979712yes—
490C4#1684056012yes—
491C2—41no—
492C2—44no—
493C2—82no—
494C2—9761no—
495C2#2145503712yes—
496C2C2×#1683993011yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3481C2
socleC22×C61C2×#120
hypercenter#3481D61
hypocenterC61#18480
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3481

Complements

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

NH ≅ G/Ntype
1. #21456245C2semidirect
2. #21455879C2semidirect
3. #21444953C2semidirect
4. #21444940C2semidirect
5. #21444935C2semidirect
6. #21444891C2semidirect
7. #21444878C2semidirect
8. #21444873C2semidirect
9. #21444481C2semidirect
10. #21444201C2semidirect
11. C61×#3481C2semidirect
13. #16840312C22semidirect
15. #16840141C22semidirect
17. #16840136C22semidirect
20. #16840057C22semidirect
22. #16839962C22semidirect
27. #16839815C22semidirect
28. C2×C4×#92418C22semidirect
29. #16839511C22semidirect
32. Q8⋊#92418C22semidirect
33. Q8⋊#92418C22semidirect
191. D4×C61#135semidirect
243. #3481D61semidirect
295. C61#18480semidirect
445. D4#14519797semidirect

Maximal subgroups

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

subgroupindex
#214562452
#214558792
#214449532
#214449402
#214449352
#214448912
#214448782
#214448732
#214444812
#214442012
C61×#34812
#1848061