Finite Group Databasebeta

← #29575506

Subgroups — #29575506

Order 15616.

566 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 135 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#21454843C211yesmaximal, characteristic
2#21448531C211yesmaximal, characteristic
3#21444213C212yesmaximal, complemented
4#21443647C212yesmaximal, complemented
5C61×#1460C211yesmaximal, characteristic
6C2×#14519764C411yescharacteristic
7C2×C4×Dic122C2211yescharacteristic
8D4×#92418—24no—
9#16839815C2212yes—
10C2×C22⋊#92418—22no—
11#16839667C411yescharacteristic
12#16839665C411yescharacteristic
13#16839650—22no—
14#16839635—22no—
15#16839631C411yescharacteristic
16#16839608C2211yescharacteristic
17#16839492—24no—
18C2×C4×D4×C61C2211yescharacteristic
19C61×#431C2212yes—
20C61×#323—22no—
21C22×Dic122C2×C411yescharacteristic
22C22×#92418—24no—
23#14519764—22no—
24#14519764C814yescomplemented
25C4×Dic122—22no—
26C4×Dic122—22no—
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×#11758196—21no—
29C2×#11758196—22no—
30C2×C22⋊Dic61—22no—
31C2×C22⋊Dic61—22no—
32C2×C4×Dic61—21no—
33C2×C4×Dic61C2×C411yescharacteristic
34C2×C4×Dic61C2×C411yescharacteristic
35C4⋊#92418—22no—
36C22⋊#92418—24no—
37C22⋊#92418—22no—
38C4×#92418D414yes—
39#14519684—22no—
40#14519684—21no—
41#14519684—21no—
42C22×D4×C61C2×C411yescharacteristic
43C23×C4×C61D412yes—
44C4×D4×C61—22no—
45C4×D4×C61—22no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C22⋊C4×C61D412yes—
48C2×C42×C61C2311yescharacteristic
49C4⋊C8×C61—22no—
50C22⋊C8×C61—22no—
51C4×C8×C61—22no—
52C24×C61C22⋊C412yes—
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61C2×C812yes—
56C2×D4×C61M1612yes—
57C22×C4×C61—22no—
58C22×C4×C61—22no—
59C22×C4×C61—82no—
60C22×C4×C61C22⋊C412yes—
61C22×C4×C61C22×C411yescharacteristic
62C22×C4×C61C2×D411yescharacteristic
63C22×C4×C61C2×D411yescharacteristic
64C22×Dic61—21no—
65C22×Dic61—21no—
66C22×Dic61—21no—
67C22×Dic61—21no—
68C2×Dic122—22no—
69C2×Dic122—22no—
70C2×Dic122—21no—
71C2×Dic122C2×C812yescomplemented
72C2×C8×C61—42no—
73C4⋊C4×C61—22no—
74C22⋊C4×C61—24no—
75C42×C61C2×D412yes—
76C42×C61#4112yes—
77C22⋊Dic61—28no—
78C22⋊Dic61—82no—
79C4⋊Dic61—22no—
80#11758196—41no—
81#11758196—41no—
82#11758196—82no—
83C4×Dic61—21no—
84C4×Dic61—22no—
85C4×Dic61—42no—
86C4×Dic61C2×C812yescomplemented
87C2×#92418—24no—
88C2×#92418—84no—
89C23×C61—22no—
90C23×C61—22no—
91C23×C61—42no—
92C23×C61—42no—
93C23×C61C22⋊C814yes—
94C23×C61C2×C22⋊C411yescharacteristic
95D4×C61—24no—
96D4×C61—24no—
97C2×C4×C61—22no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—44no—
101C2×C4×C61—42no—
102C2×C4×C61—82no—
103C2×C4×C61C4×D412yes—
104C2×C4×C61#12011yescharacteristic
105C2×C4×C61#12112yes—
106C2×C4×C61#12311yescharacteristic
107C2×C4×C61C22×C811yescharacteristic
108C2×C4×C61C2×M1611yescharacteristic
109C2×C4×C61#13112yes—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—41no—
115C2×Dic61—41no—
116C2×Dic61—42no—
117C2×Dic61—41no—
118C2×Dic61—41no—
119C2×Dic61—41no—
120C2×Dic61—41no—
121Dic122—24no—
122Dic122—42no—
123C8×C61—82no—
124#92418—44no—
125#9099—611nomaximal, Sylow
126C22×C61—21no—
127C22×C61—21no—
128C22×C61—24no—
129C22×C61—44no—
130C22×C61—42no—
131C22×C61—42no—
132C22×C61—82no—
133C22×C61#38611yescharacteristic
134C22×C61C2×C22⋊C811yescharacteristic
135C22×C61#40811yescharacteristic
136C4×C61—22no—
137C4×C61—21no—
138C4×C61—21no—
139C4×C61—42no—
140C4×C61C8×D412yes—
141C4×C61#43512yes—
142Dic61—42no—
143Dic61—42no—
144Dic61—81no—
145Dic61—81no—
146#2972—611no—
147#2326—611no—
148#1479—612no—
149#1460D6111yescharacteristic
150#1274—612no—
151C2×C61—21no—
152C2×C61—21no—
153C2×C61—44no—
154C2×C61#186911yescharacteristic
155C2×C61#190911yescharacteristic
156C2×C61#191111yescharacteristic
157C2×#124—611no—
158C2×C4×Q8—611no—
159C2×C4×D4C2×D6111yescharacteristic
160C8×D4—1224no—
161#431—612no—
162#431C2×D6112yes—
163C2×C22⋊C8—1222no—
164#391—611no—
165#390—611no—
166#388—1222no—
167#385—1222no—
168#384—611no—
169#376—611no—
170Q8⋊C8—1224no—
171#323—22no—
172C61#909911yescharacteristic, complemented, Sylow
173C22×Q8—611no—
174C22×D4C4×D6111yescharacteristic
175C23×C4D24412yes—
176C22×C8—1224no—
177#124—614no—
178#124—1222no—
179C4×Q8—1222no—
180C4×Q8—1222no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4—611no—
184C2×C4⋊C4—1222no—
185C2×C4⋊C4—1221no—
186C2×C4⋊C4C4×D6111yescharacteristic
187C2×C22⋊C4—1222no—
188C2×C22⋊C4—1222no—
189C2×C22⋊C4C22⋊D6112yes—
190C2×C42—611no—
191C2×C42—611no—
192C2×C42—1221no—
193C2×C42C22×D6111yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—1222no—
196C22⋊C8—22no—
197C22⋊C8—1224no—
198C22⋊C8—1222no—
199C4×C8—22no—
200C4×C8—614no—
201#95—1222no—
202#95—1221no—
203#95—1221no—
204C24#1175819812yes—
205C2×Q8—612no—
206C2×Q8—1221no—
207C2×Q8—1222no—
208C2×Q8—1222no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4C8×D6112yes—
212C2×D4#1175818912yes—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—82no—
216C22×C4—1221no—
217C22×C4—1221no—
218C22×C4—1221no—
219C22×C4—1221no—
220C22×C4#1175819812yes—
221C22×C4C2×C4×D6111yescharacteristic
222C22×C4C2×D24411yescharacteristic
223C22×C4C2×C22⋊D6111yescharacteristic
224C2×C8—42no—
225C2×C8—1224no—
226C2×C8—4884no—
227C4⋊C4—22no—
228C4⋊C4—1222no—
229C4⋊C4—4882no—
230C4⋊C4—2441no—
231C4⋊C4—2441no—
232C22⋊C4—24no—
233C22⋊C4—1228no—
234C22⋊C4—4882no—
235C42—612no—
236C42—1221no—
237C42—1222no—
238C42—2442no—
239C42D4×D6112yes—
240C42#1175822412yes—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23#1451968914yes—
246C23C2×#1175819811yescharacteristic
247Q8—1224no—
248Q8—2442no—
249D4—24no—
250D4—24no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—2441no—
260C2×C4—2442no—
261C2×C4—2441no—
262C2×C4—2441no—
263C2×C4—2441no—
264C2×C4—2441no—
265C2×C4—2441no—
266C2×C4—2441no—
267C2×C4—2441no—
268C2×C4#1451973712yes—
269C2×C4#1451974711yescharacteristic
270C2×C4#1451974912yes—
271C2×C4#1451976111yescharacteristic
272C2×C4C2×C8×D6111yescharacteristic
273C2×C4C2×#1175818911yescharacteristic
274C2×C4#1451978512yes—
275C8—82no—
276C8—2444no—
277C22—21no—
278C22—21no—
279C22—24no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—82no—
284C22#1683963911yescharacteristic
285C22C2×#1451968911yescharacteristic
286C22#1683973011yescharacteristic
287C4—22no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—4881no—
292C4—4881no—
293C4—2442no—
294C4—2442no—
295C4#1683982512yes—
296C4#1683983012yes—
297C2—21no—
298C2—21no—
299C2—44no—
300C2#2144610911yescharacteristic
301C2#2144633911yescharacteristic
302C2#2144634911yescharacteristic
subgroupG/H
centerC22#16839639
commutatorC2×C4×C61C22×C8
FrattiniC2×C42C22×D61
FittingC61×#1460C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C42×C61C23
hypercenter#1460D61
hypocenterC61#9099
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #21444213C2semidirect
4. #21443647C2semidirect
24. #14519764C8semidirect
71. C2×Dic122C2×C8semidirect
86. C4×Dic61C2×C8semidirect
172. C61#9099semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214485312
#214442132
#214436472
C61×#14602
#909961