Finite Group Databasebeta

← #29835261

Subgroups — #29835261

Order 15616.

526 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456548C211yesmaximal, characteristic, complemented
2#21448038C211yesmaximal, characteristic, complemented
3#21448033C211yesmaximal, characteristic, complemented
4#21447995C212yesmaximal, complemented
5#21447888C212yesmaximal, complemented
6#21447770C211yesmaximal, characteristic, complemented
7#21447765C211yesmaximal, characteristic
8#21447717C212yesmaximal, complemented
9#21447710C212yesmaximal, complemented
10#21447631C211yesmaximal, characteristic, complemented
11C61×#2956C211yesmaximal, characteristic, complemented
12#16840296—21no—
13#16840202C2212yes—
14#16840197C2212yes—
15#16840164C2211yescharacteristic, complemented
16#16840146C2211yescharacteristic, complemented
17#16840117—21no—
18#16840020C2211yescharacteristic
19#16840015C2211yescharacteristic
20#16840008—21no—
21#16839875C2211yescharacteristic, complemented
22#16839870C2211yescharacteristic, complemented
23#16839864—21no—
24#16839586C2212yescomplemented
25#16839586C2212yescomplemented
26#16839583C2212yescomplemented
27#16839577C2212yescomplemented
28#16839574C2212yescomplemented
29#16839572C2212yes—
30#16839572C2212yes—
31#16839544C2211yescharacteristic
32#16839544C2211yescharacteristic
33#16839540C2212yescomplemented
34C61×#575C2211yescharacteristic
35C61×#568—21no—
36C61×#444—21no—
37C61×#444C2212yes—
38C61×#443—21no—
39C61×#443C2212yes—
40C61×#430C2211yescharacteristic
41C2×#104×C61—21no—
42C4×M16×C61C2211yescharacteristic
43C2×Dic244C2311yescharacteristic
44C2×#11758190—21no—
45C2×D488C2311yescharacteristic
46#14519771—41no—
47#14519769—21no—
48#14519765—21no—
49#14519760—41no—
50#14519756—21no—
51#14519751—21no—
52C4×Dic122—21no—
53C4×D244—21no—
54C4×C22⋊D61—41no—
55#14519734—41no—
56#14519723—22no—
57#14519721—22no—
58#14519720—22no—
59#14519718—22no—
60#14519717C2312yes—
61#14519717C2312yes—
62#14519711—21no—
63#14519709C2311yescharacteristic
64#14519709C2311yescharacteristic
65#14519702—21no—
66#14519702—21no—
67#14519699—21no—
68#14519699—21no—
69C2×#41×C61—21no—
70#137×C61—22no—
71AGL(1,ℤ/8)×C61—22no—
72#135×C61—21no—
73#135×C61C2312yes—
74#131×C61—22no—
75#131×C61—22no—
76#131×C61—42no—
77C2×M16×C61—21no—
78C2×M16×C61—42no—
79C2×M16×C61D411yescharacteristic
80C2×M16×C61C2311yescharacteristic
81C2×C42×C61D411yescharacteristic
82#108×C61—21no—
83#108×C61C2312yes—
84#104×C61—22no—
85#104×C61—22no—
86#97×C61D411yescharacteristic
87#97×C61C2311yescharacteristic
88C4×C8×C61D411yescharacteristic
89C4×C8×C61C2311yescharacteristic
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—41no—
93#41×C61—41no—
94C2×Q8×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—41no—
97C22×C4×C61—21no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C2×C22⋊D61—41no—
101C2×D244—21no—
102C2×C4×D61—41no—
103C2×Dic122—21no—
104Q16×C61—21no—
105Q16×C61#4112yes—
106SD16×C61—22no—
107SD16×C61—22no—
108D8×C61—21no—
109D8×C61#4112yes—
110M16×C61—22no—
111M16×C61—22no—
112M16×C61—22no—
113M16×C61—42no—
114M16×C61—41no—
115M16×C61—42no—
116C2×C8×C61—42no—
117C2×C8×C61—41no—
118C2×C8×C61—42no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2411yescharacteristic
123C42×C61—21no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C22⋊Dic61—41no—
127#11758198—41no—
128#11758198—41no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133#11758196—41no—
134#11758196—41no—
135#11758196—41no—
136#11758196—41no—
137C4×Dic61—41no—
138Dic244—22no—
139D488—22no—
140#11758190—42no—
141#11758185—22no—
142#11758185—22no—
143C23×C61—41no—
144C23×C61AGL(1,ℤ/8)11yescharacteristic
145C22×D61—41no—
146Q8×C61—22no—
147Q8×C61—41no—
148D4×C61—22no—
149D4×C61—41no—
150D4×C61—42no—
151D4×C61—42no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—81no—
160C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
161C2×C4×C61C22×D411yescharacteristic
162C22⋊D61—82no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166D244—41no—
167D244—41no—
168C4×D61—82no—
169Dic122—41no—
170Dic122—41no—
171C8×C61—21no—
172C8×C61—21no—
173C8×C61—42no—
174C8×C61AGL(1,ℤ/8)12yes—
175C8×C61C2×#4112yes—
176#33690—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—41no—
179C22×C61—42no—
180C22×C61—161no—
181C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—41no—
187C4×C61—41no—
188C4×C61—42no—
189C4×C61#52511yescharacteristic
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191Dic61—81no—
192Dic61—81no—
193Dic61—81no—
194#3183—611no—
195#2956D6111yescharacteristic, complemented
196#2239—611no—
197#2238—611no—
198#2231—612no—
199#2212—612no—
200#2191—611no—
201#2190—611no—
202AGL(1,ℤ/16)—612no—
203#2180—612no—
204#2165—611no—
205C2×C61—21no—
206C2×C61—41no—
207C2×C61—82no—
208C2×C61#310911yescharacteristic
209D61—161no—
210#575C2×D6111yescharacteristic
211#568—21no—
212#529—1221no—
213#510—612no—
214#509—612no—
215#499—611no—
216#495—611no—
217#488—1221no—
218#470—611no—
219#469—611no—
220#468—1221no—
221#444—21no—
222#444C2×D6112yes—
223#443—21no—
224#443C2×D6112yes—
225#442—611no—
226#441—611no—
227#440—1221no—
228#430C2×D6111yescharacteristic
229C2×#104—21no—
230C4×M16C2×D6111yescharacteristic
231#365—612no—
232#365—612no—
233#364—612no—
234#362—612no—
235#361—612no—
236#360—612no—
237#360—612no—
238#349—611no—
239#349—611no—
240#347—612no—
241C61#3369011yescharacteristic, complemented, Sylow
242C2×#41—21no—
243#137—22no—
244AGL(1,ℤ/8)—22no—
245#135—21no—
246#135C22×D6112yes—
247C2×Q16—611no—
248C2×SD16—1221no—
249C2×D8—611no—
250#131—22no—
251#131—22no—
252#131—42no—
253C2×M16—21no—
254C2×M16—42no—
255C2×M16C22⋊D6111yescharacteristic
256C2×M16C22×D6111yescharacteristic
257#126—2441no—
258#125—1221no—
259#124—1221no—
260#123—2441no—
261C22⋊Q8—1221no—
262#121—1221no—
263C4×Q8—1221no—
264C4×D4—1221no—
265C4×D4—2441no—
266#117—2441no—
267C2×C42C22⋊D6111yescharacteristic
268Q32—1222no—
269SD32—1222no—
270SD32—1222no—
271D16—1222no—
272M32—612no—
273M32—612no—
274#108—21no—
275#108C22×D6112yes—
276#107—1221no—
277#106—611no—
278#106—611no—
279#104—22no—
280#104—22no—
281#103—1221no—
282#103—1221no—
283#102—1221no—
284#102—1221no—
285#97C22⋊D6111yescharacteristic
286#97C22×D6111yescharacteristic
287C4×C8C22⋊D6111yescharacteristic
288C4×C8C22×D6111yescharacteristic
289#41—22no—
290#41—22no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—1221no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1221no—
298C2×D4—2441no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—2441no—
302C22×C4C2×C22⋊D6111yescharacteristic
303Q16—21no—
304Q16—1222no—
305Q16#1175822212yes—
306SD16—22no—
307SD16—22no—
308SD16—2442no—
309D8—21no—
310D8—1222no—
311D8#1175822212yes—
312M16—22no—
313M16—22no—
314M16—22no—
315M16—42no—
316M16—41no—
317M16—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8C2×C22⋊D6111yescharacteristic
322C2×C8C2×C22⋊D6111yescharacteristic
323C2×C8C2×C22⋊D6111yescharacteristic
324C2×C8C23×D6111yescharacteristic
325C4⋊C4—1221no—
326C4⋊C4—1221no—
327C4⋊C4—2441no—
328C4⋊C4—2441no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C22⋊C4—2441no—
336C42—21no—
337C42—2441no—
338C42C2×C22⋊D6111yescharacteristic
339C42C2×C22⋊D6111yescharacteristic
340C16—1222no—
341C16—1222no—
342C23—41no—
343C23—2441no—
344C23#1451980211yescharacteristic
345Q8—22no—
346Q8—41no—
347Q8—2441no—
348Q8—2441no—
349D4—22no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—4882no—
354D4—2441no—
355D4—2441no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—4882no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4#1451980211yescharacteristic
369C2×C4C22×C22⋊D6111yescharacteristic
370C8—21no—
371C8—21no—
372C8—42no—
373C8#1451980312yes—
374C8C2×#1175822212yes—
375C22—21no—
376C22—41no—
377C22—42no—
378C22—161no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22C2×#1451980211yescharacteristic
383C4—21no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—4881no—
388C4—4881no—
389C4—4881no—
390C4#1684027311yescharacteristic
391C4C2×#1451980311yescharacteristic
392C2—21no—
393C2—41no—
394C2—82no—
395C2—9761no—
396C2#2145591011yescharacteristic
subgroupG/H
centerC2#21455910
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2956C2
socleC2×C61#3109
prime radical#21447765C2
hypercenter#2956D61
hypocenterC61#33690
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #2956

Complements

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

NH ≅ G/Ntype
1. #21456548C2semidirect
2. #21448038C2semidirect
3. #21448033C2semidirect
4. #21447995C2semidirect
5. #21447888C2semidirect
6. #21447770C2semidirect
8. #21447717C2semidirect
9. #21447710C2semidirect
10. #21447631C2semidirect
11. C61×#2956C2semidirect
15. #16840164C22semidirect
16. #16840146C22semidirect
21. #16839875C22semidirect
22. #16839870C22semidirect
24. #16839586C22semidirect
25. #16839586C22semidirect
26. #16839583C22semidirect
27. #16839577C22semidirect
28. #16839574C22semidirect
33. #16839540C22semidirect
195. #2956D61semidirect
241. C61#33690semidirect

Maximal subgroups

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

subgroupindex
#214565482
#214480382
#214480332
#214479952
#214478882
#214477702
#214477652
#214477172
#214477102
#214476312
C61×#29562
#3369061