Finite Group Databasebeta

← #29656116

Subgroups — #29656116

Order 15616.

1630 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C2×#16840397C211yesmaximal, characteristic
2#21455914C212yesmaximal, complemented
3#21444995C218yesmaximal, complemented
4#21444533C211yesmaximal, characteristic, complemented
5#21444263C212yesmaximal, complemented
6C61×#3138C211yesmaximal, characteristic, complemented
7C22×D4×D61—22no—
8#16840397—24no—
9#16840397—22no—
10#16840397C2214yes—
11C2×#14519748—22no—
12C2×#14519745—22no—
13C2×C4×D244C2212yes—
14#16840095—22no—
15#16840039C2214yescomplemented
16#16839964C2214yescomplemented
17#16839912—22no—
18#16839860—24no—
19#16839819C2212yescomplemented
20#16839506C2214yescomplemented
21#16839491C2218yescomplemented
22C61×#533C2211yescharacteristic, complemented
23C2×#127×C61C2211yescharacteristic
24C61×#490—22no—
25C61×#486C2214yescomplemented
26C61×#418—22no—
27C4×M16×C61C2211yescharacteristic, complemented
28C24×D61—22no—
29C22×D244D412yes—
30C22×C22⋊D61—24no—
31C2×D4×D61—28no—
32C2×D4×D61—28no—
33C2×D4×D61—416no—
34C22×C4×D61—22no—
35C2×#11758201—24no—
36#14519748—28no—
37#14519748—28no—
38#14519745—28no—
39#14519745—28no—
40C4×D244—22no—
41C4×D244—22no—
42C4×D244D414yes—
43C4×D244C2314yes—
44C2×C4⋊Dic61D412yes—
45C2×C22⋊Dic61—22no—
46C2×#11758198—22no—
47#14519710—24no—
48C4⋊#92418C2314yescomplemented
49#14519701—24no—
50#14519697—24no—
51C22⋊#92418—22no—
52#14519688—22no—
53C22×D4×C61—22no—
54C22×D4×C61D412yes—
55C2×M16×C61—21no—
56C4⋊Q8×C61C2312yescomplemented
57#127×C61—22no—
58#127×C61D412yes—
59#127×C61C2312yes—
60#125×C61—21no—
61C22⋊Q8×C61—22no—
62C4×Q8×C61—22no—
63#117×C61—21no—
64C2×C42×C61C2311yescharacteristic
65#102×C61—28no—
66#97×C61—21no—
67C4×C8×C61C2312yescomplemented
68C24×C61—22no—
69C23×D61—22no—
70C23×D61—28no—
71C23×D61—416no—
72C23×D61—84no—
73C2×Q8×C61—22no—
74C2×D4×C61—24no—
75C2×D4×C61—22no—
76C2×D4×C61—24no—
77C2×D4×C61—48no—
78C2×D4×C61—44no—
79C2×D4×C61C2×D414yes—
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D411yescharacteristic
82C2×C22⋊D61—216no—
83C2×C22⋊D61—416no—
84C22×Dic61—22no—
85D4×D61—432no—
86D4×D61—416no—
87D4×D61—416no—
88C2×D244—22no—
89C2×D244—28no—
90C2×D244C2×D414yes—
91C2×C4×D61—24no—
92C2×C4×D61—24no—
93C2×C4×D61—48no—
94M16×C61—42no—
95C2×C8×C61—22no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—42no—
99C22⋊C4×C61—41no—
100C42×C61—41no—
101C42×C61C2×D412yes—
102C42×C61C2×D411yescharacteristic
103C42×C61C2411yescharacteristic
104C22⋊Dic61—28no—
105#11758201—48no—
106#11758198—24no—
107#11758198—24no—
108C4⋊Dic61—22no—
109C4⋊Dic61C2×D414yes—
110C2×#92418—24no—
111C23×C61—24no—
112C23×C61—48no—
113C23×C61—44no—
114C23×C61#12011yescharacteristic
115C22×D61—28no—
116C22×D61—24no—
117C22×D61—432no—
118C22×D61—44no—
119C22×D61—44no—
120C22×D61—88no—
121C22×D61—832no—
122Q8×C61—44no—
123D4×C61—44no—
124D4×C61—48no—
125D4×C61—48no—
126D4×C61—82no—
127C2×C4×C61—22no—
128C2×C4×C61—24no—
129C2×C4×C61—42no—
130C2×C4×C61#12012yes—
131C2×C4×C61#12011yescharacteristic
132C2×C4×C61C22×D411yescharacteristic
133C2×C4×C61C22×D412yes—
134C22⋊D61—432no—
135C2×Dic61—24no—
136C2×Dic61—44no—
137D244—28no—
138D244—24no—
139D244AGL(1,ℤ/8)18yes—
140C4×D61—48no—
141C4×D61—48no—
142C8×C61—42no—
143#92418—44no—
144#18529—611nomaximal, Sylow
145C22×C61—22no—
146C22×C61—48no—
147C22×C61—44no—
148C22×C61—88no—
149C22×C61#45512yes—
150C22×C61C2×#12011yescharacteristic
151C2×D61—28no—
152C2×D61—416no—
153C2×D61—44no—
154C2×D61—44no—
155C2×D61—832no—
156C4×C61—22no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—82no—
160C4×C61C2×AGL(1,ℤ/8)14yes—
161Dic61—44no—
162C2×D42—611no—
163#3138D6111yescharacteristic, complemented
164#3110—612no—
165#1639—618no—
166#1555—611no—
167#1490—612no—
168C2×C61—22no—
169C2×C61—84no—
170C2×C61#300312yes—
171C2×C61C2×#45511yescharacteristic
172D61—48no—
173C23×D4—1222no—
174D42—614no—
175D42—1224no—
176D42—1222no—
177#533C2×D6111yescharacteristic, complemented
178C2×#127C2×D6111yescharacteristic
179C2×#121—1222no—
180C2×#120—1222no—
181C2×C4×D4—612no—
182#490—22no—
183#486C2×D6114yescomplemented
184#484—1222no—
185#474—614no—
186#461—614no—
187#450—1222no—
188#440—1224no—
189#432—612no—
190#418—22no—
191C4×M16C2×D6111yescharacteristic, complemented
192#331—614no—
193D4⋊C8—618no—
194C61#1852911yescharacteristic, complemented, Sylow
195C25—1222no—
196C22×D4—22no—
197C22×D4—612no—
198C22×D4—1228no—
199C22×D4—1228no—
200C22×D4—1224no—
201C22×D4—24416no—
202C22×D4C22⋊D6112yes—
203C23×C4—1222no—
204C2×SD16—1224no—
205C2×M16—21no—
206C4⋊Q8C22×D6112yescomplemented
207#127—22no—
208#127C22⋊D6112yes—
209#127C22×D6112yes—
210#125—21no—
211C22⋊Q8—22no—
212#121—1228no—
213#121—1228no—
214#120—1228no—
215#120—1228no—
216C4×Q8—22no—
217C4×D4—614no—
218C4×D4—614no—
219C4×D4—1222no—
220C4×D4—1222no—
221#117—21no—
222C2×C4⋊C4—612no—
223C2×C22⋊C4—1222no—
224C2×C22⋊C4—1222no—
225C2×C42C22×D6111yescharacteristic
226#107—1224no—
227C4⋊C8—614no—
228#103—1224no—
229#102—28no—
230#102—1224no—
231C22⋊C8—1222no—
232#97—21no—
233#97—1222no—
234C4×C8C22×D6112yescomplemented
235C24—22no—
236C24—1228no—
237C24—1222no—
238C24—4884no—
239C24—24416no—
240C2×Q8—22no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—48no—
245C2×D4—44no—
246C2×D4—614no—
247C2×D4—1228no—
248C2×D4—12216no—
249C2×D4—1222no—
250C2×D4—24416no—
251C2×D4—24432no—
252C2×D4—24416no—
253C2×D4—24416no—
254C2×D4C2×C22⋊D6114yes—
255C22×C4—1222no—
256C22×C4—1224no—
257C22×C4—1224no—
258C22×C4—2448no—
259C22×C4D4×D6112yes—
260C22×C4C2×C22⋊D6111yescharacteristic
261SD16—2448no—
262M16—42no—
263C2×C8—22no—
264C2×C8—1224no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—42no—
268C4⋊C4—614no—
269C4⋊C4—1222no—
270C22⋊C4—41no—
271C22⋊C4—1228no—
272C22⋊C4—1224no—
273C22⋊C4—1224no—
274C42—41no—
275C42D4×D6112yes—
276C42C2×C22⋊D6111yescharacteristic
277C42C23×D6111yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—1228no—
282C23—1224no—
283C23—4888no—
284C23—48832no—
285C23—2444no—
286C23—24432no—
287C23—2444no—
288C23#1451974511yescharacteristic
289Q8—44no—
290D4—44no—
291D4—48no—
292D4—48no—
293D4—82no—
294D4—618no—
295D4—1228no—
296D4—1224no—
297D4—24432no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—42no—
301C2×C4—1224no—
302C2×C4—2448no—
303C2×C4—2444no—
304C2×C4—2448no—
305C2×C4#1451974511yescharacteristic
306C2×C4#1451974512yes—
307C2×C4C2×D4×D6112yes—
308C2×C4C22×C22⋊D6111yescharacteristic
309C8—42no—
310C8—2444no—
311C22—22no—
312C22—44no—
313C22—48no—
314C22—88no—
315C22—1228no—
316C22—48832no—
317C22—24416no—
318C22—2444no—
319C22—2444no—
320C22#1683993912yes—
321C22C2×#1451974511yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—82no—
326C4—2444no—
327C4AGL(1,ℤ/8)×D6114yes—
328C2—22no—
329C2—84no—
330C2—2448no—
331C2#2145504412yes—
332C2C2×#1683993911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3138C2
socleC22×C61C2×#120
prime radicalC2×#16840397C2
hypercenter#3138D61
hypocenterC61#18529
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #21455914C2semidirect
3. #21444995C2semidirect
4. #21444533C2semidirect
5. #21444263C2semidirect
6. C61×#3138C2semidirect
15. #16840039C22semidirect
16. #16839964C22semidirect
19. #16839819C22semidirect
20. #16839506C22semidirect
21. #16839491C22semidirect
22. C61×#533C22semidirect
25. C61×#486C22semidirect
27. C4×M16×C61C22semidirect
48. C4⋊#92418C23semidirect
56. C4⋊Q8×C61C23semidirect
67. C4×C8×C61C23semidirect
163. #3138D61semidirect
177. #533C2×D61semidirect
183. #486C2×D61semidirect
191. C4×M16C2×D61semidirect
194. C61#18529semidirect
206. C4⋊Q8C22×D61semidirect
234. C4×C8C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168403972
#214559142
#214449952
#214445332
#214442632
C61×#31382
#1852961