Finite Group Databasebeta

← #29835262

Subgroups — #29835262

Order 15616.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456549C211yesmaximal, characteristic, complemented
2#21454729C211yesmaximal, characteristic, complemented
3#21448036C211yesmaximal, characteristic, complemented
4#21448034C211yesmaximal, characteristic, complemented
5#21447993C212yesmaximal, complemented
6#21447890C212yesmaximal, complemented
7#21447765C211yesmaximal, characteristic
8#21447714C212yesmaximal, complemented
9#21447711C212yesmaximal, complemented
10#21447628C211yesmaximal, characteristic, complemented
11C61×#2191C211yesmaximal, characteristic, complemented
12#16840529—21no—
13#16840298—21no—
14#16840197C2212yes—
15#16840162C2211yescharacteristic, complemented
16#16840147C2211yescharacteristic, complemented
17#16840118—21no—
18#16840015C2211yescharacteristic
19#16840010—21no—
20#16839884—21no—
21#16839884C2212yes—
22#16839879—21no—
23#16839879C2212yes—
24#16839873C2211yescharacteristic, complemented
25#16839869C2211yescharacteristic, complemented
26#16839865—21no—
27#16839811C2211yescharacteristic
28C2×#14519704—21no—
29M16×Dic61C2211yescharacteristic
30#16839585C2212yescomplemented
31#16839584C2212yescomplemented
32#16839582C2212yescomplemented
33#16839578C2212yescomplemented
34#16839576C2212yescomplemented
35#16839572C2212yes—
36#16839544C2211yescharacteristic
37#16839539C2212yescomplemented
38C61×#575C2211yescharacteristic
39C61×#510C2212yes—
40C61×#470C2211yescharacteristic
41C61×#360C2212yes—
42C61×#349C2211yescharacteristic
43C2×#11758201—21no—
44C2×D488C2311yescharacteristic
45#14519784—22no—
46#14519784—22no—
47#14519784—42no—
48C2×#11758194—21no—
49C2×#11758194—42no—
50C2×#11758194D411yescharacteristic
51#14519772—41no—
52#14519768—21no—
53#14519766—21no—
54#14519761—41no—
55#14519751—21no—
56Q8×Dic61—21no—
57#14519739—21no—
58#14519737—41no—
59#14519735—41no—
60C2×C4×Dic61D411yescharacteristic
61#14519720—22no—
62#14519718—22no—
63#14519717C2312yes—
64#14519713—21no—
65#14519713C2312yes—
66#14519710—21no—
67#14519709C2311yescharacteristic
68#14519704—22no—
69#14519704—22no—
70#14519701—21no—
71#14519699—21no—
72#14519697—21no—
73#14519687D411yescharacteristic
74#14519687C2311yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C2×#41×C61—21no—
78#137×C61—22no—
79AGL(1,ℤ/8)×C61—22no—
80#135×C61—21no—
81#135×C61C2312yes—
82C2×Q16×C61C2311yescharacteristic
83C2×M16×C61C2311yescharacteristic
84C22⋊Q8×C61—21no—
85Q32×C61—22no—
86SD32×C61—22no—
87M32×C61C2312yes—
88#106×C61C2311yescharacteristic
89#103×C61—21no—
90#41×C61—22no—
91#41×C61—22no—
92#41×C61—41no—
93#41×C61—41no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×D4×C61—21no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61C2×D411yescharacteristic
100C2×C22⋊D61—41no—
101C22×Dic61—21no—
102C2×D244—21no—
103C2×C4×D61—41no—
104Q16×C61—21no—
105Q16×C61—22no—
106Q16×C61#4112yes—
107SD16×C61—22no—
108SD16×C61—22no—
109D8×C61—21no—
110D8×C61#4112yes—
111M16×C61—22no—
112C2×C8×C61C2×D411yescharacteristic
113C2×C8×C61C2411yescharacteristic
114C4⋊C4×C61—21no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117#11758201—42no—
118#11758198—41no—
119#11758198—41no—
120#11758198—41no—
121#11758198—41no—
122C4⋊Dic61—21no—
123C4⋊Dic61—41no—
124C4⋊Dic61—41no—
125#11758196—41no—
126C4×Dic61—21no—
127C4×Dic61—41no—
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130#11758194—22no—
131#11758194—22no—
132#11758194—42no—
133#11758194—41no—
134#11758194—42no—
135C2×#92418—42no—
136C2×#92418—41no—
137C2×#92418—42no—
138C2×#92418C2×D411yescharacteristic
139C2×#92418C2×D411yescharacteristic
140D488—22no—
141C16×C61—22no—
142#11758185—22no—
143C23×C61—41no—
144C23×C61AGL(1,ℤ/8)11yescharacteristic
145C22×D61—41no—
146Q8×C61—22no—
147Q8×C61—41no—
148Q8×C61—41no—
149Q8×C61—41no—
150D4×C61—22no—
151D4×C61—41no—
152D4×C61—42no—
153D4×C61—42no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—42no—
157C2×C4×C61—41no—
158C2×C4×C61—42no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C22⋊D61—82no—
164C2×Dic61—21no—
165C2×Dic61—21no—
166C2×Dic61—41no—
167C2×Dic61—81no—
168D244—41no—
169D244—41no—
170C4×D61—82no—
171C8×C61—21no—
172C8×C61C2×#4112yes—
173#92418—21no—
174#92418—42no—
175#92418AGL(1,ℤ/8)12yes—
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—42no—
187C4×C61—81no—
188C4×C61—81no—
189C4×C61#52511yescharacteristic
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191Dic61—41no—
192Dic61—41no—
193Dic61—81no—
194#3183—611no—
195#2956—611no—
196#2239—611no—
197#2238—611no—
198#2231—612no—
199#2212—612no—
200#2191D6111yescharacteristic, complemented
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—1221no—
212#529—1221no—
213#510C2×D6112yes—
214#509—612no—
215#499—611no—
216#495—611no—
217#488—1221no—
218#470C2×D6111yescharacteristic
219#469—611no—
220#468—1221no—
221#444—612no—
222#444—1221no—
223#443—612no—
224#443—1221no—
225#442—611no—
226#441—611no—
227#440—1221no—
228#430—611no—
229C2×#104—1221no—
230C4×M16—611no—
231#365—612no—
232#365—612no—
233#364—612no—
234#362—612no—
235#361—612no—
236#360—612no—
237#360C2×D6112yes—
238#349—611no—
239#349C2×D6111yescharacteristic
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×Q16C22×D6111yescharacteristic
248C2×SD16—1221no—
249C2×D8—611no—
250#131—1222no—
251#131—1222no—
252#131—2442no—
253C2×M16—611no—
254C2×M16—1221no—
255C2×M16—2442no—
256C2×M16C22×D6111yescharacteristic
257#126—2441no—
258#125—1221no—
259#124—1221no—
260#123—2441no—
261C22⋊Q8—21no—
262#121—1221no—
263C4×Q8—1221no—
264C4×D4—1221no—
265C4×D4—2441no—
266#117—2441no—
267C2×C42—611no—
268Q32—22no—
269SD32—22no—
270SD32—1222no—
271D16—1222no—
272M32—612no—
273M32C22×D6112yes—
274#108—612no—
275#108—1221no—
276#107—1221no—
277#106—611no—
278#106C22×D6111yescharacteristic
279#104—1222no—
280#104—1222no—
281#103—21no—
282#103—1221no—
283#102—1221no—
284#102—1221no—
285#97—611no—
286#97—611no—
287C4×C8—611no—
288C4×C8—611no—
289#41—22no—
290#41—22no—
291#41—41no—
292#41—41no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—1221no—
298C2×D4—2441no—
299C22×C4—41no—
300C22×C4—1221no—
301C22×C4—2441no—
302C22×C4D4×D6111yescharacteristic
303Q16—21no—
304Q16—22no—
305Q16#1175822412yes—
306SD16—22no—
307SD16—22no—
308SD16—2442no—
309D8—21no—
310D8—1222no—
311D8#1175822412yes—
312M16—22no—
313M16—1222no—
314M16—1222no—
315M16—2442no—
316M16—2442no—
317M16—2441no—
318C2×C8—611no—
319C2×C8—611no—
320C2×C8—2442no—
321C2×C8—2442no—
322C2×C8—2441no—
323C2×C8D4×D6111yescharacteristic
324C2×C8C23×D6111yescharacteristic
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—1221no—
328C4⋊C4—2441no—
329C4⋊C4—2441no—
330C4⋊C4—2441no—
331C22⋊C4—41no—
332C22⋊C4—2441no—
333C22⋊C4—2441no—
334C22⋊C4—2441no—
335C22⋊C4—2441no—
336C42—611no—
337C42—611no—
338C42—1221no—
339C42—2441no—
340C16—22no—
341C16—1222no—
342C23—41no—
343C23—2441no—
344C23#1451980111yescharacteristic
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—41no—
349D4—22no—
350D4—41no—
351D4—42no—
352D4—42no—
353D4—4882no—
354D4—2441no—
355D4—2441no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—1221no—
364C2×C4—1221no—
365C2×C4—4882no—
366C2×C4—4881no—
367C2×C4—2441no—
368C2×C4#1451980111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—21no—
371C8—612no—
372C8—1221no—
373C8—2442no—
374C8C2×#1175822412yes—
375C22—21no—
376C22—41no—
377C22—42no—
378C22—161no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22C2×#1451980111yescharacteristic
383C4—21no—
384C4—42no—
385C4—81no—
386C4—81no—
387C4—4881no—
388C4—2441no—
389C4—2441no—
390C4#1684026811yescharacteristic
391C4AGL(1,ℤ/8)×D6111yescharacteristic
392C2—21no—
393C2—41no—
394C2—82no—
395C2—9761no—
396C2#2145590511yescharacteristic
subgroupG/H
centerC2#21455905
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2191C2
socleC2×C61#3109
prime radical#21447765C2
hypercenter#2191D61
hypocenterC61#33690
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2191

Complements

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

NH ≅ G/Ntype
1. #21456549C2semidirect
2. #21454729C2semidirect
3. #21448036C2semidirect
4. #21448034C2semidirect
5. #21447993C2semidirect
6. #21447890C2semidirect
8. #21447714C2semidirect
9. #21447711C2semidirect
10. #21447628C2semidirect
11. C61×#2191C2semidirect
15. #16840162C22semidirect
16. #16840147C22semidirect
24. #16839873C22semidirect
25. #16839869C22semidirect
30. #16839585C22semidirect
31. #16839584C22semidirect
32. #16839582C22semidirect
33. #16839578C22semidirect
34. #16839576C22semidirect
37. #16839539C22semidirect
200. #2191D61semidirect
241. C61#33690semidirect

Maximal subgroups

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

subgroupindex
#214565492
#214547292
#214480362
#214480342
#214479932
#214478902
#214477652
#214477142
#214477112
#214476282
C61×#21912
#3369061