Finite Group Databasebeta

← #30595456

Subgroups — #30595456

Order 15616.

630 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 110 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#21461711C212yesmaximal
2#21460640C212yesmaximal, complemented
3#21460639C212yesmaximal, complemented
4#391×D61C211yesmaximal, characteristic
5C2×#14519834—22no—
6#16840628—22no—
7#16840628C2214yes—
8C22×Q8×D61C411yescharacteristic
9C2×C4⋊C4×D61C411yescharacteristic
10C2×C42×D61C2211yescharacteristic
11#16839669C2211yescharacteristic
12#16839668C411yescharacteristic
13#16839667C2211yescharacteristic
14#16839666—22no—
15#95×D61—22no—
16C61×#391C411yescharacteristic
17C22×#92420—24no—
18#14519834—44no—
19C22×Dic122—21no—
20C22×Dic122C2×C411yescharacteristic
21C2×Q8×D61—22no—
22C2×Q8×D61—42no—
23C2×Q8×D61—42no—
24C2×Q8×D61C814yescomplemented
25C22×C4×D61—21no—
26C22×C4×D61—21no—
27C22×C4×D61C2×C411yescharacteristic
28C22×C4×D61D412yes—
29C2×C4⋊Dic61C2×C411yescharacteristic
30C2×#11758196—21no—
31C4⋊C4×D61—42no—
32C2×C4×Dic61D412yes—
33C2×C4×Dic61C2311yescharacteristic
34C42×D61—42no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38C22×Q8×C61C2×C411yescharacteristic
39C2×C4⋊C4×C61C2×C411yescharacteristic
40C2×C42×C61C2×C411yescharacteristic
41#95×C61—22no—
42C23×D61C22⋊C411yescharacteristic
43C2×Q8×C61—21no—
44C2×Q8×C61—41no—
45C2×Q8×C61—41no—
46C2×Q8×C61C2×C812yes—
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61C22⋊C412yes—
50C22×C4×C61C22×C411yescharacteristic
51C22×Dic61—21no—
52C22×Dic61—21no—
53C22×Dic61C22⋊C411yescharacteristic
54C22×Dic61C2×D412yes—
55Q8×D61—48no—
56Q8×D61—48no—
57C2×C4×D61—22no—
58C2×C4×D61—22no—
59C2×C4×D61—24no—
60C2×C4×D61—24no—
61C2×C4×D61—24no—
62C2×C4×D61—42no—
63C2×C4×D61—42no—
64C2×C4×D61—42no—
65C2×C4×D61—44no—
66C2×C4×D61—42no—
67C2×C4×D61C2×C812yes—
68C2×C4×D61M1612yes—
69C2×Dic122—21no—
70C2×Dic122—24no—
71C2×Dic122—42no—
72C2×Dic122—42no—
73C2×Dic122—41no—
74C2×Dic122—41no—
75C2×Dic122C2×C812yes—
76C2×#92420—44no—
77C2×#92420—44no—
78C2×#92420—44no—
79C4⋊C4×C61—41no—
80C42×C61—41no—
81C4⋊Dic61—41no—
82#11758196—42no—
83C4×Dic61—42no—
84C4×Dic61—41no—
85C23×C61C2×C22⋊C411yescharacteristic
86C22×D61—22no—
87C22×D61—22no—
88C22×D61C22⋊C812yes—
89C22×D61#9912yes—
90C22×D61#10012yes—
91Q8×C61—42no—
92Q8×C61—42no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—22no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—41no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61—42no—
103C2×C4×C61C22×C811yescharacteristic
104C2×C4×C61C2×M1611yescharacteristic
105C2×Dic61—22no—
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—41no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—42no—
114C2×Dic61C22⋊C812yes—
115C2×Dic61#13714yes—
116C4×D61—24no—
117C4×D61—24no—
118C4×D61—48no—
119C4×D61—48no—
120C4×D61—84no—
121Dic122—48no—
122Dic122—42no—
123Dic122—42no—
124#92420—84no—
125#7595—611nomaximal, Sylow
126C22×C61C2×C22⋊C811yescharacteristic
127C22×C61C2×#9911yescharacteristic
128C22×C61C2×#10011yescharacteristic
129C22×C61#41912yes—
130C22×C61#42112yes—
131C2×D61—22no—
132C2×D61—24no—
133C2×D61—24no—
134C2×D61—22no—
135C2×D61#32314yes—
136C4×C61—21no—
137C4×C61—21no—
138C4×C61—42no—
139C4×C61—42no—
140C4×C61—81no—
141Dic61—22no—
142Dic61—24no—
143Dic61—42no—
144Dic61—81no—
145C2×#391—611no—
146#1817—612no—
147#1272—612no—
148#1272—612no—
149C2×C61C2×#32311yescharacteristic
150C2×C61#147912yes—
151C2×C61#151112yes—
152C2×C61#153311yescharacteristic
153C2×C61#155811yescharacteristic
154D61—24no—
155C23×Q8—611no—
156C22×C4⋊C4—611no—
157C22×C42—611no—
158C2×C22⋊C8—1222no—
159#391—611no—
160#391—611no—
161#391—611no—
162#391—1222no—
163#391#692311yescharacteristic
164C2×#95—1222no—
165#336—614no—
166#336—1222no—
167C61#759511yescharacteristic, complemented, Sylow
168C22×Q8—614no—
169C22×Q8—611no—
170C22×Q8—1221no—
171C22×Q8—1222no—
172C22×Q8—2442no—
173C22×Q8—2442no—
174C22×Q8C2×#692311yescharacteristic
175C23×C4—611no—
176C23×C4—612no—
177C23×C4—1221no—
178C23×C4—1221no—
179C22×C8—1224no—
180C2×C4⋊C4—611no—
181C2×C4⋊C4—1221no—
182C2×C4⋊C4—2442no—
183C2×C4⋊C4C2×#692311yescharacteristic
184C2×C42—612no—
185C2×C42—611no—
186C2×C42—2442no—
187C2×C42C2×#692311yescharacteristic
188C22⋊C8—2444no—
189#95—22no—
190#95—1222no—
191#95—1222no—
192#95—1222no—
193C24—611no—
194C2×Q8—21no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8—612no—
198C2×Q8—1224no—
199C2×Q8—1221no—
200C2×Q8—2441no—
201C2×Q8—2441no—
202C2×Q8—2442no—
203C2×Q8—2442no—
204C2×Q8—2448no—
205C2×Q8—2448no—
206C2×Q8D61⋊C812yes—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—612no—
210C22×C4—611no—
211C22×C4—612no—
212C22×C4—612no—
213C22×C4—1224no—
214C22×C4—1222no—
215C22×C4—1222no—
216C22×C4—1224no—
217C22×C4—1224no—
218C22×C4—1221no—
219C22×C4—1221no—
220C22×C4—2442no—
221C22×C4—2442no—
222C22×C4—2444no—
223C22×C4—2442no—
224C22×C4—2442no—
225C22×C4C22⋊#692312yes—
226C22×C4C22×#692311yescharacteristic
227C2×C8—2444no—
228C2×C8—2444no—
229C2×C8—2444no—
230C4⋊C4—41no—
231C4⋊C4—2441no—
232C4⋊C4—2442no—
233C42—41no—
234C42—2441no—
235C42—2442no—
236C23—612no—
237C23—612no—
238C23—612no—
239C23—1222no—
240C23—1222no—
241C23C2×C22⋊#692311yescharacteristic
242Q8—42no—
243Q8—42no—
244Q8—2442no—
245Q8—2442no—
246Q8—2448no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—612no—
258C2×C4—614no—
259C2×C4—1224no—
260C2×C4—1224no—
261C2×C4—1222no—
262C2×C4—1221no—
263C2×C4—1221no—
264C2×C4—1222no—
265C2×C4—1222no—
266C2×C4—4884no—
267C2×C4—2448no—
268C2×C4—2448no—
269C2×C4—2441no—
270C2×C4—2442no—
271C2×C4—2441no—
272C2×C4—2441no—
273C2×C4C2×D61⋊C811yescharacteristic
274C2×C4D61⋊M1611yescharacteristic
275C8—4884no—
276C22—614no—
277C22—1222no—
278C22—1222no—
279C22—1224no—
280C22—1224no—
281C22D61⋊(C22⋊C8)11yescharacteristic
282C22D61⋊#9911yescharacteristic
283C22D61⋊#10011yescharacteristic
284C22#1684075512yes—
285C22#1684075912yes—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—1222no—
292C4—1224no—
293C4—4881no—
294C4—2442no—
295C2—1224no—
296C2D61⋊#32311yescharacteristic
297C2#2146112212yes—
298C2#2146122812yes—
299C2#2146129911yescharacteristic
300C2#2146135311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC2×C4×C61C22×C8
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#7595
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #21460640C2semidirect
3. #21460639C2semidirect
24. C2×Q8×D61C8semidirect
167. C61#7595semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214606402
#214606392
#391×D612
#759561