Finite Group Databasebeta

← #29653983

Subgroups — #29653983

Order 15616.

1312 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459152C211yesmaximal, characteristic, complemented
2#21456737C212yesmaximal, complemented
3#21444753C214yesmaximal, complemented
4#21444736C214yesmaximal, complemented
5#21444378C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic, complemented
7C61×#2619C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840343—21no—
10#16840343C2212yes—
11#16840338—22no—
12#16840331—22no—
13#16840324—21no—
14#16840324C2212yes—
15#16840312—21no—
16#16840312C2212yes—
17#16840274—24no—
18#16840138—22no—
19#16840108—22no—
20#16839954—24no—
21#16839896C2218yescomplemented
22#16839815C2212yescomplemented
23#16839495C2218yescomplemented
24#16839488C2212yescomplemented
25#16839486C2212yes—
26C2×#127×C61C2211yescharacteristic, complemented
27C2×#121×C61—21no—
28C2×#117×C61C2211yescharacteristic
29C61×#394—21no—
30C61×#394C2212yescomplemented
31C61×#392—21no—
32C61×#392C2212yes—
33C61×#390—21no—
34C61×#382C2211yescharacteristic, complemented
35C2×#11758222—22no—
36C2×#11758222—22no—
37C2×#11758222—22no—
38C2×#11758222—22no—
39C2×#11758222—44no—
40C2×#11758222—44no—
41C22×Dic122—21no—
42C22×D244—21no—
43C22×C22⋊D61—22no—
44C22×C4×D61—22no—
45C2×#11758199—28no—
46#14519777—21no—
47#14519777D412yescomplemented
48#14519774—21no—
49#14519774D412yescomplemented
50#14519765—22no—
51#14519765—22no—
52#14519761—24no—
53#14519761—24no—
54#14519757—24no—
55#14519757—24no—
56#14519752—24no—
57#14519752—24no—
58#14519751—22no—
59#14519751—22no—
60#14519751—22no—
61#14519751C2314yes—
62#14519747—24no—
63#14519747—24no—
64C4×D244—22no—
65C4×D244—22no—
66C4×D244—22no—
67C4×D244D414yescomplemented
68C4⋊#92418—22no—
69#14519698—28no—
70C22⋊#92418C2314yes—
71C4×#92418—22no—
72C22×D4×C61—22no—
73C22×D4×C61—21no—
74C22×D4×C61—21no—
75C22×D4×C61C2312yes—
76C23×C4×C61D411yescharacteristic
77#127×C61—22no—
78#127×C61—22no—
79#121×C61—44no—
80#117×C61—22no—
81#117×C61—22no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61D411yescharacteristic
89C2×C22⋊C4×C61C2311yescharacteristic
90C2×C42×C61D411yescharacteristic
91C2×C42×C61C2311yescharacteristic
92#95×C61—21no—
93#95×C61C2312yes—
94C24×C61—22no—
95C24×C61C2×D411yescharacteristic
96C23×D61—22no—
97C2×D4×C61—24no—
98C2×D4×C61—44no—
99C2×D4×C61—42no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—42no—
103C2×D4×C61—42no—
104C2×D4×C61—82no—
105C2×D4×C61—82no—
106C2×D4×C61—82no—
107C2×D4×C61—82no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61D814yescomplemented
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—24no—
121C2×C22⋊D61—44no—
122C2×C22⋊D61—44no—
123C22×Dic61—22no—
124#11758222—44no—
125#11758222—48no—
126#11758222—48no—
127#11758222—44no—
128#11758222—44no—
129#11758222—44no—
130C2×D244—22no—
131C2×D244—22no—
132C2×D244—22no—
133C2×D244—22no—
134C2×D244—22no—
135C2×D244—44no—
136C2×D244C2×D414yes—
137C2×C4×D61—24no—
138C2×C4×D61—24no—
139C2×C4×D61—48no—
140C2×Dic122—22no—
141C2×Dic122—22no—
142C2×Dic122—44no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—42no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—44no—
149C22⋊C4×C61—82no—
150C22⋊C4×C61—82no—
151C42×C61—21no—
152C42×C61—21no—
153C42×C61C2×D412yes—
154C42×C61C2×D412yes—
155#11758199—88no—
156#11758198—24no—
157#11758198—24no—
158#11758198—24no—
159#11758198—24no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61C2×D414yes—
164C2×#92418—24no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61—41no—
168C23×C61—41no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61C2×D812yes—
177C23×C61C22×D411yescharacteristic
178C22×D61—24no—
179C22×D61—44no—
180C22×D61—82no—
181C22×D61—82no—
182D4×C61—48no—
183D4×C61—84no—
184D4×C61—88no—
185D4×C61—82no—
186D4×C61—82no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—42no—
200C2×C4×C61#12012yes—
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61C2×D812yes—
204C2×C4×C61C2×D812yes—
205C2×C4×C61AGL(1,ℤ/8)12yes—
206C2×C4×C61C22×D412yes—
207C22⋊D61—48no—
208C22⋊D61—44no—
209C22⋊D61—44no—
210C2×Dic61—24no—
211C2×Dic61—44no—
212D244—44no—
213D244—44no—
214D244—44no—
215D244—82no—
216D244—82no—
217D244—82no—
218C4×D61—48no—
219C4×D61—48no—
220Dic122—44no—
221Dic122—44no—
222#92418—44no—
223#18300—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—81no—
231C22×C61—82no—
232C22×C61—82no—
233C22×C61—82no—
234C22×C61—82no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—82no—
238C22×C61C2×#12011yescharacteristic
239C22×C61C22×D811yescharacteristic
240C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
241C2×D61—44no—
242C2×D61—44no—
243C2×D61—44no—
244C2×D61—88no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—82no—
251C4×C61#44712yes—
252C4×C61#44712yes—
253Dic61—44no—
254#3449—611no—
255#3205—612no—
256#2619D6111yescharacteristic, complemented
257#1598—614no—
258#1595—614no—
259#1517—612no—
260#1458—611no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—42no—
264C2×C61—162no—
265C2×C61C2×#44711yescharacteristic
266C2×C61#302111yescharacteristic
267C2×C61#302711yescharacteristic
268D61—84no—
269C22×#41—1221no—
270#537—612no—
271#537—1221no—
272#536—1222no—
273#535—1222no—
274#534—612no—
275#534—1221no—
276#532—612no—
277#532—1221no—
278C2×#127C2×D6111yescharacteristic, complemented
279#525—1224no—
280C2×#121—21no—
281C2×#117C2×D6111yescharacteristic
282#493—1222no—
283#486—1222no—
284#459—1224no—
285#447—618no—
286#431—612no—
287#394—21no—
288#394C2×D6112yescomplemented
289#392—21no—
290#392C2×D6112yes—
291#390—21no—
292#382C2×D6111yescharacteristic, complemented
293#327—618no—
294#324—612no—
295#323—612no—
296C61#1830011yescharacteristic, complemented, Sylow
297C2×#41—1222no—
298C2×#41—1222no—
299C2×#41—1222no—
300C2×#41—1222no—
301C2×#41—2444no—
302C2×#41—2444no—
303C22×Q8—1221no—
304C22×D4—22no—
305C22×D4—21no—
306C22×D4—21no—
307C22×D4—1221no—
308C22×D4—1222no—
309C22×D4C22×D6112yes—
310C23×C4—1222no—
311C23×C4C22⋊D6111yescharacteristic
312C2×D8—1228no—
313C4⋊Q8—612no—
314C4⋊Q8—1221no—
315#127—22no—
316#127—22no—
317#127—612no—
318#127—1221no—
319#124—1222no—
320#124—1222no—
321#123—1224no—
322#123—1224no—
323C22⋊Q8—1224no—
324C22⋊Q8—1224no—
325#121—44no—
326#121—614no—
327#121—1224no—
328#121—1224no—
329#121—1222no—
330#121—1222no—
331#121—1222no—
332#120—1224no—
333#120—1224no—
334C4×D4—614no—
335C4×D4—1222no—
336C4×D4—1222no—
337C4×D4—1222no—
338#117—22no—
339#117—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4C22⋊D6111yescharacteristic
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4C22⋊D6111yescharacteristic
347C2×C22⋊C4C22×D6111yescharacteristic
348C2×C42C22⋊D6111yescharacteristic
349C2×C42C22×D6111yescharacteristic
350C4⋊C8—1222no—
351#102—1228no—
352C22⋊C8—614no—
353C4×C8—1222no—
354#95—21no—
355#95C22×D6112yes—
356C24—22no—
357C24—1222no—
358C24C2×C22⋊D6111yescharacteristic
359#41—2444no—
360#41—2448no—
361#41—2448no—
362#41—2444no—
363#41—2444no—
364#41—2444no—
365C2×Q8—1222no—
366C2×Q8—1222no—
367C2×Q8—2444no—
368C2×D4—24no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—82no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—82no—
379C2×D4—614no—
380C2×D4—1222no—
381C2×D4—1222no—
382C2×D4—1222no—
383C2×D4—1224no—
384C2×D4—1224no—
385C2×D4—1222no—
386C2×D4—1222no—
387C2×D4—2444no—
388C2×D4—2444no—
389C2×D4—2444no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—1224no—
395C22×C4—1222no—
396C22×C4—1224no—
397C22×C4—2448no—
398C22×C4#1175819914yescomplemented
399C22×C4C2×C22⋊D6111yescharacteristic
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C23×D6111yescharacteristic
405D8—4888no—
406C2×C8—1224no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—614no—
411C4⋊C4—1222no—
412C4⋊C4—1222no—
413C4⋊C4—1222no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—1224no—
420C22⋊C4—1224no—
421C22⋊C4—1224no—
422C22⋊C4—1224no—
423C42—21no—
424C42—21no—
425C42D4×D6112yes—
426C42D4×D6112yes—
427C23—21no—
428C23—21no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—1224no—
439C23—4882no—
440C23—4882no—
441C23—2444no—
442C23C2×#1175819912yes—
443C23C22×C22⋊D6111yescharacteristic
444Q8—2444no—
445Q8—2444no—
446D4—48no—
447D4—84no—
448D4—88no—
449D4—82no—
450D4—82no—
451D4—4882no—
452D4—4882no—
453D4—4882no—
454D4—2444no—
455D4—2444no—
456D4—2448no—
457D4—2444no—
458D4—2444no—
459D4—2444no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—1224no—
474C2×C4—2448no—
475C2×C4—2444no—
476C2×C4—2448no—
477C2×C4#1451974512yes—
478C2×C4#1451974511yescharacteristic
479C2×C4#1451974511yescharacteristic
480C2×C4C2×#1175819912yes—
481C2×C4C2×#1175819912yes—
482C2×C4#1451980212yes—
483C2×C4C2×D4×D6112yes—
484C8—2444no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—42no—
489C22—41no—
490C22—41no—
491C22—81no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—4888no—
500C22—2444no—
501C22—2444no—
502C22—2444no—
503C22C2×#1451974511yescharacteristic
504C22C22×#1175819911yescharacteristic
505C22C2×#1451980211yescharacteristic
506C4—22no—
507C4—22no—
508C4—22no—
509C4—41no—
510C4—41no—
511C4—82no—
512C4—2444no—
513C4#1683989512yes—
514C4#1683989512yes—
515C2—21no—
516C2—21no—
517C2—42no—
518C2—162no—
519C2—4884no—
520C2C2×#1683989511yescharacteristic
521C2#2145519011yescharacteristic
522C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2619C2
socleC22×C61C2×#120
hypercenter#2619D61
hypocenterC61#18300
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #21459152C2semidirect
2. #21456737C2semidirect
3. #21444753C2semidirect
4. #21444736C2semidirect
5. #21444378C2semidirect
6. #21444133C2semidirect
7. C61×#2619C2semidirect
21. #16839896C22semidirect
22. #16839815C22semidirect
23. #16839495C22semidirect
24. #16839488C22semidirect
26. C2×#127×C61C22semidirect
30. C61×#394C22semidirect
34. C61×#382C22semidirect
47. #14519777D4semidirect
49. #14519774D4semidirect
67. C4×D244D4semidirect
112. C22×C4×C61D8semidirect
256. #2619D61semidirect
278. C2×#127C2×D61semidirect
288. #394C2×D61semidirect
292. #382C2×D61semidirect
296. C61#18300semidirect
398. C22×C4#11758199semidirect

Maximal subgroups

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

subgroupindex
#214591522
#214567372
#214447532
#214447362
#214443782
#214441332
C61×#26192
#1830061