Finite Group Databasebeta

← #29661298

Subgroups — #29661298

Order 15616.

960 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16840510C211yesmaximal, characteristic, complemented
2#21456802C212yesmaximal, complemented
3#21445140C214yesmaximal, complemented
4#21445111C214yesmaximal, complemented
5#21444446C211yesmaximal, characteristic, complemented
6#21444416C212yesmaximal, complemented
7C61×#3142C211yesmaximal, characteristic, complemented
8#16840510—24no—
9#16840510—22no—
10#16840510C2214yes—
11C2×#14519765C2212yes—
12C2×#14519762—22no—
13C2×#14519755—22no—
14C2×#14519752—22no—
15#16840149C2214yescomplemented
16#16840124C2214yescomplemented
17#16839977—24no—
18#16839900—22no—
19#16839895—22no—
20#16839819C2212yes—
21#16839504C2214yescomplemented
22#16839501C2218yescomplemented
23C61×#532C2211yescharacteristic
24C2×#125×C61C2211yescharacteristic
25C61×#491C2212yescomplemented
26C61×#490C2212yescomplemented
27C61×#488—22no—
28C61×#417—22no—
29C4×M16×C61C2211yescharacteristic
30C22×Dic122D412yes—
31C22×D244—22no—
32C22×D244D412yes—
33C22×C4×D61—22no—
34C2×#11758200—24no—
35C2×#11758199—24no—
36#14519765—22no—
37#14519765—22no—
38#14519765D414yes—
39#14519765C2314yes—
40#14519762—28no—
41#14519762—28no—
42#14519755—28no—
43#14519755—28no—
44#14519752—28no—
45#14519752—28no—
46C2×#11758196—22no—
47C2×#11758198—22no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C4⋊#92418—22no—
51#14519700—24no—
52#14519697—24no—
53C22⋊#92418—22no—
54#14519688C2314yes—
55C2×#41×C61—21no—
56C2×M16×C61—21no—
57C4⋊Q8×C61C2311yescharacteristic
58#127×C61C2311yescharacteristic
59#125×C61—22no—
60#125×C61D412yes—
61#125×C61C2312yes—
62#124×C61—21no—
63#121×C61—22no—
64C4×D4×C61—22no—
65C2×C4⋊C4×C61—22no—
66C2×C4⋊C4×C61D412yes—
67C2×C42×C61C2311yescharacteristic
68#103×C61—24no—
69#102×C61—24no—
70#97×C61C2312yes—
71C4×C8×C61—21no—
72C23×D61—22no—
73#41×C61—44no—
74C2×Q8×C61—21no—
75C2×D4×C61—21no—
76C2×D4×C61—22no—
77C2×D4×C61—41no—
78C22×C4×C61—22no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×Dic61—22no—
83C2×D244—28no—
84C2×D244—22no—
85C2×D244—82no—
86C2×D244—84no—
87C2×D244C2×D414yes—
88C2×C4×D61—28no—
89C2×C4×D61—84no—
90C2×Dic122—22no—
91C2×Dic122—82no—
92C2×Dic122C2×D414yes—
93M16×C61—42no—
94C2×C8×C61—22no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61—22no—
99C4⋊C4×C61C2×D414yes—
100C22⋊C4×C61—42no—
101C42×C61C2×D412yes—
102C42×C61C2×D411yescharacteristic
103C42×C61C2411yescharacteristic
104#11758200—84no—
105#11758199—84no—
106#11758198—28no—
107#11758198—24no—
108#11758198—24no—
109#11758198—24no—
110#11758198—24no—
111#11758196—28no—
112C2×#92418—24no—
113C23×C61—41no—
114C23×C61#12011yescharacteristic
115C22×D61—24no—
116C22×D61—44no—
117C22×D61—44no—
118C22×D61—44no—
119Q8×C61—42no—
120D4×C61—44no—
121D4×C61—42no—
122D4×C61—82no—
123C2×C4×C61—22no—
124C2×C4×C61—24no—
125C2×C4×C61—42no—
126C2×C4×C61—44no—
127C2×C4×C61—41no—
128C2×C4×C61—82no—
129C2×C4×C61#12011yescharacteristic
130C2×C4×C61#12012yes—
131C2×C4×C61C22×D412yes—
132C2×C4×C61C22×D411yescharacteristic
133C2×Dic61—24no—
134C2×Dic61—44no—
135D244—44no—
136D244—82no—
137D244—88no—
138C4×D61—88no—
139Dic122—44no—
140Dic122—82no—
141C8×C61—42no—
142#92418—44no—
143#19001—611nomaximal, Sylow
144C22×C61—22no—
145C22×C61—81no—
146C22×C61—82no—
147C22×C61#45412yes—
148C22×C61C2×#12011yescharacteristic
149C2×D61—44no—
150C2×D61—44no—
151C2×D61—44no—
152C2×D61—88no—
153C4×C61—21no—
154C4×C61—21no—
155C4×C61—22no—
156C4×C61—22no—
157C4×C61—44no—
158C4×C61—81no—
159Dic61—44no—
160C2×#562—611no—
161#3213—612no—
162#3142D6111yescharacteristic, complemented
163#1662—614no—
164#1656—614no—
165#1534—611no—
166#1527—612no—
167C2×C61—22no—
168C2×C61—161no—
169C2×C61C2×#45411yescharacteristic
170C2×C61#301912yes—
171D61—84no—
172#562—614no—
173#562—1224no—
174#562—1222no—
175#532C2×D6111yescharacteristic
176C2×#125C2×D6111yescharacteristic
177C2×#124—612no—
178C2×#123—1222no—
179C2×C22⋊Q8—1222no—
180C2×#121—1222no—
181#496—614no—
182#491C2×D6112yescomplemented
183#490C2×D6112yescomplemented
184#489—614no—
185#488—22no—
186#463—1224no—
187#448—1222no—
188#447—1222no—
189#432—612no—
190#417—22no—
191C4×M16C2×D6111yescharacteristic
192#330—614no—
193#329—618no—
194C61#1900111yescharacteristic, complemented, Sylow
195C2×#41—21no—
196C22×Q8—612no—
197C22×D4—612no—
198C22×D4—1222no—
199C23×C4—1222no—
200C2×SD16—1224no—
201C2×D8—1224no—
202C2×M16—21no—
203C4⋊Q8C22×D6111yescharacteristic
204#127C22×D6111yescharacteristic
205#125—22no—
206#125C22⋊D6112yes—
207#125C22×D6112yes—
208#124—21no—
209#124—614no—
210#124—614no—
211#124—1222no—
212#124—1222no—
213#123—1228no—
214#123—1228no—
215C22⋊Q8—1228no—
216C22⋊Q8—1228no—
217#121—22no—
218#121—1228no—
219#121—1228no—
220C4×D4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—1222no—
223C2×C4⋊C4C22⋊D6112yes—
224C2×C22⋊C4—1222no—
225C2×C22⋊C4—1222no—
226C2×C22⋊C4—1222no—
227C2×C42C22×D6111yescharacteristic
228C4⋊C8—1222no—
229#103—24no—
230#103—1224no—
231#102—24no—
232#102—1224no—
233C22⋊C8—1222no—
234#97—614no—
235#97C22×D6112yes—
236C4×C8—21no—
237C24—1222no—
238#41—44no—
239C2×Q8—21no—
240C2×Q8—614no—
241C2×Q8—1222no—
242C2×Q8—4882no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—41no—
246C2×D4—614no—
247C2×D4—1228no—
248C2×D4—1222no—
249C2×D4—4884no—
250C2×D4—4882no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—1228no—
254C22×C4—1222no—
255C22×C4—4884no—
256C22×C4D4×D6112yes—
257C22×C4C2×C22⋊D6111yescharacteristic
258SD16—4884no—
259D8—4884no—
260M16—42no—
261C2×C8—22no—
262C2×C8—1224no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—1228no—
268C4⋊C4C2×C22⋊D6114yes—
269C22⋊C4—42no—
270C22⋊C4—1228no—
271C22⋊C4—1224no—
272C22⋊C4—1224no—
273C22⋊C4—1224no—
274C22⋊C4—1224no—
275C42D4×D6112yes—
276C42C2×C22⋊D6111yescharacteristic
277C42C23×D6111yescharacteristic
278C23—41no—
279C23—1224no—
280C23—2444no—
281C23—2444no—
282C23—2444no—
283C23#1451974511yescharacteristic
284Q8—42no—
285Q8—4882no—
286Q8—2444no—
287D4—44no—
288D4—42no—
289D4—82no—
290D4—4882no—
291D4—4888no—
292D4—2444no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—41no—
298C2×C4—82no—
299C2×C4—1224no—
300C2×C4—4888no—
301C2×C4—2444no—
302C2×C4#1451974511yescharacteristic
303C2×C4#1451974512yes—
304C2×C4C2×D4×D6112yes—
305C2×C4C22×C22⋊D6111yescharacteristic
306C8—42no—
307C8—2444no—
308C22—22no—
309C22—81no—
310C22—82no—
311C22—4888no—
312C22—2444no—
313C22—2444no—
314C22—2444no—
315C22#1683993412yes—
316C22C2×#1451974511yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—81no—
323C4—2444no—
324C2—22no—
325C2—161no—
326C2—4884no—
327C2C2×#1683993411yescharacteristic
328C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3142C2
socleC22×C61C2×#120
hypercenter#3142D61
hypocenterC61#19001
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. C2×#16840510C2semidirect
2. #21456802C2semidirect
3. #21445140C2semidirect
4. #21445111C2semidirect
5. #21444446C2semidirect
6. #21444416C2semidirect
7. C61×#3142C2semidirect
15. #16840149C22semidirect
16. #16840124C22semidirect
21. #16839504C22semidirect
22. #16839501C22semidirect
25. C61×#491C22semidirect
26. C61×#490C22semidirect
162. #3142D61semidirect
182. #491C2×D61semidirect
183. #490C2×D61semidirect
194. C61#19001semidirect

Maximal subgroups

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

subgroupindex
C2×#168405102
#214568022
#214451402
#214451112
#214444462
#214444162
C61×#31422
#1900161