Finite Group Databasebeta

← #30595576

Subgroups — #30595576

Order 15616.

658 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 102 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#21461546C212yesmaximal
2#21460640C212yesmaximal, complemented
3#21460639C212yesmaximal, complemented
4#391×D61C211yesmaximal, characteristic
5C2×#14519834—22no—
6C2×#14519832—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—28no—
19#14519832—44no—
20C22×Dic122—21no—
21C22×Dic122C2×C411yescharacteristic
22C2×Q8×D61—22no—
23C2×Q8×D61—22no—
24C2×Q8×D61—42no—
25C2×Q8×D61—42no—
26C22×C4×D61—21no—
27C22×C4×D61—21no—
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61D412yes—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C4⋊C4×D61—42no—
33C2×C4×Dic61D412yes—
34C2×C4×Dic61C2311yescharacteristic
35C42×D61—42no—
36#14519684—22no—
37#14519684—22no—
38#14519684—22no—
39C22×Q8×C61C2×C411yescharacteristic
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C42×C61C2×C411yescharacteristic
42#95×C61—22no—
43C23×D61C22⋊C411yescharacteristic
44C2×Q8×C61—21no—
45C2×Q8×C61—41no—
46C2×Q8×C61—41no—
47C2×Q8×C61M1612yes—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C412yes—
51C22×C4×C61C22×C411yescharacteristic
52C22×Dic61—21no—
53C22×Dic61—21no—
54C22×Dic61C22⋊C411yescharacteristic
55C22×Dic61C2×D412yes—
56Q8×D61—48no—
57Q8×D61—48no—
58C2×C4×D61—22no—
59C2×C4×D61—21no—
60C2×C4×D61—21no—
61C2×C4×D61—22no—
62C2×C4×D61—24no—
63C2×C4×D61—24no—
64C2×C4×D61—24no—
65C2×C4×D61—42no—
66C2×C4×D61—42no—
67C2×C4×D61—42no—
68C2×C4×D61—42no—
69C2×C4×D61—44no—
70C2×Dic122—24no—
71C2×Dic122—21no—
72C2×Dic122—42no—
73C2×Dic122—42no—
74C2×Dic122—41no—
75C2×Dic122—41no—
76C2×Dic122M1612yes—
77C2×#92420—28no—
78C2×#92420—44no—
79C2×#92420—44no—
80C4⋊C4×C61—41no—
81C42×C61—41no—
82C4⋊Dic61—41no—
83#11758196—42no—
84C4×Dic61—24no—
85C4×Dic61—41no—
86C23×C61C2×C22⋊C411yescharacteristic
87C22×D61—21no—
88C22×D61—21no—
89C22×D61—22no—
90C22×D61#10012yes—
91C22×D61#10414yes—
92Q8×C61—42no—
93Q8×C61—42no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—22no—
97C2×C4×C61—22no—
98C2×C4×C61—22no—
99C2×C4×C61—22no—
100C2×C4×C61—41no—
101C2×C4×C61—41no—
102C2×C4×C61—41no—
103C2×C4×C61—42no—
104C2×C4×C61C2×M1611yescharacteristic
105C2×C4×C61C2×M1611yescharacteristic
106C2×Dic61—21no—
107C2×Dic61—22no—
108C2×Dic61—21no—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—41no—
112C2×Dic61—41no—
113C2×Dic61—41no—
114C2×Dic61—42no—
115C2×Dic61#9912yes—
116C2×Dic61#13714yes—
117C4×D61—24no—
118C4×D61—24no—
119C4×D61—48no—
120C4×D61—48no—
121C4×D61—84no—
122Dic122—48no—
123Dic122—42no—
124Dic122—42no—
125#92420—48no—
126#7629—611nomaximal, Sylow
127C22×C61#40711yescharacteristic
128C22×C61C2×#9911yescharacteristic
129C22×C61C2×#10011yescharacteristic
130C22×C61#41912yes—
131C22×C61C2×#10412yes—
132C2×D61—22no—
133C2×D61—22no—
134C2×D61—24no—
135C2×D61—24no—
136C2×D61#32914yes—
137C4×C61—21no—
138C4×C61—21no—
139C4×C61—42no—
140C4×C61—42no—
141C4×C61—81no—
142Dic61—22no—
143Dic61—24no—
144Dic61—42no—
145Dic61—81no—
146C2×#391—611no—
147#1749—612no—
148#1272—612no—
149#1272—612no—
150C2×C61#146211yescharacteristic
151C2×C61#149112yes—
152C2×C61#150812yes—
153C2×C61C2×#32911yescharacteristic
154C2×C61#155811yescharacteristic
155D61—24no—
156C23×Q8—611no—
157C22×C4⋊C4—611no—
158C22×C42—611no—
159C2×C22⋊C8—1222no—
160C2×#97—1222no—
161#391—611no—
162#391—611no—
163#391—611no—
164#391—1222no—
165#391#692311yescharacteristic
166C2×#95—1222no—
167#336—614no—
168C61#762911yescharacteristic, complemented, Sylow
169C22×Q8—611no—
170C22×Q8—1221no—
171C22×Q8—1222no—
172C22×Q8—1222no—
173C22×Q8—2442no—
174C22×Q8—2442no—
175C22×Q8C2×#692311yescharacteristic
176C23×C4—611no—
177C23×C4—612no—
178C23×C4—1221no—
179C23×C4—1221no—
180C22×C8—1224no—
181C2×C4⋊C4—611no—
182C2×C4⋊C4—1221no—
183C2×C4⋊C4—2442no—
184C2×C4⋊C4C2×#692311yescharacteristic
185C2×C42—612no—
186C2×C42—611no—
187C2×C42—2442no—
188C2×C42C2×#692311yescharacteristic
189C22⋊C8—1228no—
190#97—2444no—
191#95—22no—
192#95—1222no—
193#95—1222no—
194#95—1222no—
195C24—611no—
196C2×Q8—21no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—612no—
200C2×Q8—1224no—
201C2×Q8—1221no—
202C2×Q8—2441no—
203C2×Q8—2441no—
204C2×Q8—2442no—
205C2×Q8—2442no—
206C2×Q8—2448no—
207C2×Q8—2448no—
208C2×Q8#1175821312yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—612no—
212C22×C4—611no—
213C22×C4—1222no—
214C22×C4—1224no—
215C22×C4—1222no—
216C22×C4—1224no—
217C22×C4—1224no—
218C22×C4—1221no—
219C22×C4—1221no—
220C22×C4—1221no—
221C22×C4—1221no—
222C22×C4—2442no—
223C22×C4—2442no—
224C22×C4—2444no—
225C22×C4—2442no—
226C22×C4—2442no—
227C22×C4C22⋊#692312yes—
228C22×C4C22×#692311yescharacteristic
229C2×C8—1228no—
230C2×C8—2444no—
231C2×C8—2444no—
232C4⋊C4—41no—
233C4⋊C4—2441no—
234C4⋊C4—2442no—
235C42—41no—
236C42—1224no—
237C42—2441no—
238C23—612no—
239C23—614no—
240C23—1221no—
241C23—1221no—
242C23—1222no—
243C23C2×C22⋊#692311yescharacteristic
244Q8—42no—
245Q8—42no—
246Q8—2442no—
247Q8—2442no—
248Q8—2448no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—612no—
260C2×C4—614no—
261C2×C4—1224no—
262C2×C4—1224no—
263C2×C4—1221no—
264C2×C4—1222no—
265C2×C4—1221no—
266C2×C4—1222no—
267C2×C4—1222no—
268C2×C4—4884no—
269C2×C4—2448no—
270C2×C4—2448no—
271C2×C4—2441no—
272C2×C4—2442no—
273C2×C4—2441no—
274C2×C4—2441no—
275C2×C4D61⋊M1611yescharacteristic
276C2×C4C2×#1175821311yescharacteristic
277C8—2448no—
278C22—614no—
279C22—1222no—
280C22—1222no—
281C22—1224no—
282C22—1224no—
283C22#1684073311yescharacteristic
284C22D61⋊#9911yescharacteristic
285C22D61⋊#10011yescharacteristic
286C22#1684075512yes—
287C22D61⋊#10412yes—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—42no—
292C4—81no—
293C4—1222no—
294C4—1224no—
295C4—4881no—
296C4—2442no—
297C2—1224no—
298C2#2146106511yescharacteristic
299C2#2146116212yes—
300C2#2146121712yes—
301C2D61⋊#32911yescharacteristic
302C2#2146135311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#391C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#391#6923
hypocenterC61#7629
LCSG ▷ C22×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
168. C61#7629semidirect

Maximal subgroups

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

subgroupindex
#214615462
#214606402
#214606392
#391×D612
#762961