Finite Group Databasebeta

← #29661300

Subgroups — #29661300

Order 15616.

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

normal subgroups only

This group has 960 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
1#21456806C212yesmaximal, complemented
2#21456222C211yesmaximal, characteristic, complemented
3#21445142C214yesmaximal, complemented
4#21445112C214yesmaximal, complemented
5#21444446C211yesmaximal, characteristic, complemented
6#21444417C212yesmaximal, complemented
7C2×C61×#562C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840152C2214yescomplemented
10#16840132C2212yescomplemented
11#16840128C2212yescomplemented
12#16840121C2214yescomplemented
13#16840119—22no—
14#16839974—24no—
15#16839901—22no—
16#16839896—22no—
17#16839819C2212yes—
18#16839761—22no—
19C4×#11758194C2211yescharacteristic
20#16839504C2214yescomplemented
21#16839502C2218yescomplemented
22C61×#562—22no—
23C61×#562—24no—
24C61×#562C2214yes—
25C2×#125×C61C2211yescharacteristic
26C2×#124×C61C2212yes—
27C2×#123×C61—22no—
28C2×C22⋊Q8×C61—22no—
29C2×#121×C61—22no—
30C2×#11758222—21no—
31C2×#11758201—24no—
32C2×#11758199—24no—
33C2×#11758194—21no—
34#14519777C2311yescharacteristic
35#14519774C2311yescharacteristic
36#14519765—21no—
37#14519751—22no—
38C4×D244—22no—
39C4⋊#92418—22no—
40#14519701—24no—
41#14519700—24no—
42#14519698—24no—
43#14519697—24no—
44C22⋊#92418—22no—
45#14519688C2312yes—
46#14519688C2314yes—
47C4×#92418—21no—
48C22×Q8×C61D412yes—
49C22×D4×C61—22no—
50C22×D4×C61D412yes—
51C23×C4×C61—22no—
52#125×C61—22no—
53#125×C61D412yes—
54#125×C61C2312yes—
55#124×C61—22no—
56#124×C61—22no—
57#124×C61D414yes—
58#124×C61C2314yes—
59#123×C61—28no—
60#123×C61—28no—
61C22⋊Q8×C61—28no—
62C22⋊Q8×C61—28no—
63#121×C61—28no—
64#121×C61—28no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61—22no—
67C2×C4⋊C4×C61D412yes—
68C2×C22⋊C4×C61—22no—
69C2×C22⋊C4×C61—22no—
70C2×C22⋊C4×C61—22no—
71C2×C42×C61C2311yescharacteristic
72C24×C61—22no—
73C2×Q8×C61—22no—
74C2×Q8×C61—82no—
75C2×Q8×C61C2×D414yes—
76C2×D4×C61—28no—
77C2×D4×C61—22no—
78C2×D4×C61—82no—
79C2×D4×C61—84no—
80C2×D4×C61C2×D414yes—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61—28no—
84C22×C4×C61—84no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D412yes—
87C2×C22⋊D61—41no—
88#11758222—44no—
89C2×D244—21no—
90C2×D244—22no—
91C2×C4×D61—41no—
92C2×Dic122—21no—
93C4⋊C4×C61—28no—
94C4⋊C4×C61—24no—
95C4⋊C4×C61—22no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61C2×D414yes—
98C22⋊C4×C61—28no—
99C22⋊C4×C61—24no—
100C22⋊C4×C61—24no—
101C22⋊C4×C61—24no—
102C22⋊C4×C61—24no—
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D412yes—
105C42×C61C2411yescharacteristic
106#11758201—84no—
107#11758199—84no—
108#11758198—42no—
109C4⋊Dic61—22no—
110#11758194—42no—
111C2×#92418—22no—
112C2×#92418—24no—
113C23×C61—24no—
114C23×C61—44no—
115C23×C61—44no—
116C23×C61—44no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119Q8×C61—44no—
120Q8×C61—82no—
121D4×C61—44no—
122D4×C61—82no—
123D4×C61—88no—
124C2×C4×C61—22no—
125C2×C4×C61—24no—
126C2×C4×C61—24no—
127C2×C4×C61—42no—
128C2×C4×C61—44no—
129C2×C4×C61—44no—
130C2×C4×C61—88no—
131C2×C4×C61#12012yes—
132C2×C4×C61#12011yescharacteristic
133C2×C4×C61C22×D412yes—
134C2×C4×C61C22×D411yescharacteristic
135C22⋊D61—82no—
136C2×Dic61—41no—
137D244—44no—
138D244—42no—
139C4×D61—82no—
140Dic122—42no—
141#92418—42no—
142#92418—44no—
143#19001—611nomaximal, Sylow
144C22×C61—22no—
145C22×C61—44no—
146C22×C61—44no—
147C22×C61—44no—
148C22×C61—88no—
149C22×C61#45412yes—
150C22×C61C2×#12011yescharacteristic
151C2×D61—81no—
152C2×D61—82no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—22no—
156C4×C61—22no—
157C4×C61—44no—
158C4×C61—44no—
159Dic61—81no—
160C2×#562D6111yescharacteristic, complemented
161#3213—612no—
162#3142—611no—
163#1662—614no—
164#1656—614no—
165#1534—611no—
166#1527—612no—
167C2×C61—22no—
168C2×C61—84no—
169C2×C61C2×#45411yescharacteristic
170C2×C61#301912yes—
171D61—161no—
172#562—22no—
173#562—24no—
174#562C2×D6114yes—
175#532—611no—
176C2×#125C2×D6111yescharacteristic
177C2×#124C2×D6112yes—
178C2×#123—22no—
179C2×C22⋊Q8—22no—
180C2×#121—22no—
181#496—614no—
182#491—612no—
183#490—612no—
184#489—614no—
185#488—1222no—
186#463—1224no—
187#448—1222no—
188#447—1222no—
189#432—612no—
190#417—1222no—
191C4×M16—611no—
192#330—614no—
193#329—618no—
194C61#1900111yescharacteristic, complemented, Sylow
195C2×#41—1221no—
196C22×Q8C22⋊D6112yes—
197C22×D4—22no—
198C22×D4C22⋊D6112yes—
199C23×C4—22no—
200C2×SD16—1224no—
201C2×D8—1224no—
202C2×M16—1221no—
203C4⋊Q8—611no—
204#127—611no—
205#125—22no—
206#125C22⋊D6112yes—
207#125C22×D6112yes—
208#124—22no—
209#124—22no—
210#124—1221no—
211#124C22⋊D6114yes—
212#124C22×D6114yes—
213#123—28no—
214#123—28no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217#121—28no—
218#121—28no—
219#121—1222no—
220C4×D4—1222no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊D6112yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42C22×D6111yescharacteristic
228C4⋊C8—1222no—
229#103—1224no—
230#103—1224no—
231#102—1224no—
232#102—1224no—
233C22⋊C8—1222no—
234#97—612no—
235#97—614no—
236C4×C8—1221no—
237C24—22no—
238#41—2444no—
239C2×Q8—22no—
240C2×Q8—82no—
241C2×Q8—1221no—
242C2×Q8C2×C22⋊D6114yes—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—82no—
246C2×D4—84no—
247C2×D4—1221no—
248C2×D4—1222no—
249C2×D4—2441no—
250C2×D4C2×C22⋊D6114yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—84no—
255C22×C4—2441no—
256C22×C4C2×C22⋊D6111yescharacteristic
257C22×C4C2×C22⋊D6112yes—
258SD16—4884no—
259D8—4884no—
260M16—2442no—
261C2×C8—1222no—
262C2×C8—1224no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—1222no—
268C4⋊C4C2×C22⋊D6114yes—
269C22⋊C4—28no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—2442no—
275C42C2×C22⋊D6111yescharacteristic
276C42C2×C22⋊D6112yes—
277C42C23×D6111yescharacteristic
278C23—24no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—2441no—
283C23#1451974611yescharacteristic
284Q8—44no—
285Q8—82no—
286Q8—2442no—
287D4—44no—
288D4—82no—
289D4—88no—
290D4—4882no—
291D4—2444no—
292D4—2442no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—88no—
300C2×C4—4882no—
301C2×C4—2441no—
302C2×C4#1451974611yescharacteristic
303C2×C4#1451974612yes—
304C2×C4C22×C22⋊D6111yescharacteristic
305C2×C4C22×C22⋊D6112yes—
306C8—2442no—
307C8—2444no—
308C22—22no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—88no—
313C22—4881no—
314C22—4882no—
315C22#1683993512yes—
316C22C2×#1451974611yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—44no—
323C4—4881no—
324C2—22no—
325C2—84no—
326C2—9761no—
327C2C2×#1683993511yescharacteristic
328C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#562C2
socleC22×C61C2×#120
hypercenterC2×#562D61
hypocenterC61#19001
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #21456806C2semidirect
2. #21456222C2semidirect
3. #21445142C2semidirect
4. #21445112C2semidirect
5. #21444446C2semidirect
6. #21444417C2semidirect
7. C2×C61×#562C2semidirect
9. #16840152C22semidirect
10. #16840132C22semidirect
11. #16840128C22semidirect
12. #16840121C22semidirect
20. #16839504C22semidirect
21. #16839502C22semidirect
160. C2×#562D61semidirect
194. C61#19001semidirect

Maximal subgroups

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

subgroupindex
#214568062
#214562222
#214451422
#214451122
#214444462
#214444172
C2×C61×#5622
#1900161