Finite Group Databasebeta

← #30604116

Subgroups — #30604116

Order 15616.

578 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 78 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#21461068C212yesmaximal
2#21460762C212yesmaximal, complemented
3#21460761C212yesmaximal, complemented
4#384×D61C211yesmaximal, characteristic
5#16840733—22no—
6#16840608—22no—
7#16840607C2214yes—
8C2×C4⋊C4×D61—22no—
9C2×C4⋊C4×D61C411yescharacteristic
10C2×C4⋊C4×D61C2211yescharacteristic
11C2×C42×D61C411yescharacteristic
12#16839632C2211yescharacteristic
13#16839631—22no—
14#16839631C411yescharacteristic
15#16839631C2211yescharacteristic
16C61×#384C411yescharacteristic
17C2×#11758213—42no—
18#14519834—42no—
19C22⋊#92420—24no—
20C22×C4×D61—21no—
21C22×C4×D61—21no—
22C22×C4×D61C2×C411yescharacteristic
23C22×C4×D61D412yes—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2311yescharacteristic
27C2×#11758196—22no—
28C2×#11758196—21no—
29C2×#11758196—22no—
30C2×#11758196D412yes—
31C4⋊C4×D61—41no—
32C4⋊C4×D61—41no—
33C4⋊C4×D61—41no—
34C4⋊C4×D61—41no—
35C4⋊C4×D61—48no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61C2×C411yescharacteristic
38C42×D61—22no—
39C42×D61—22no—
40C2×C4⋊C4×C61—22no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C42×C61C2×C411yescharacteristic
44C23×D61C22⋊C411yescharacteristic
45C22×C4×C61—21no—
46C22×C4×C61—21no—
47C22×C4×C61C22⋊C412yes—
48C22×C4×C61C22×C411yescharacteristic
49C22×Dic61—21no—
50C22×Dic61—21no—
51C22×Dic61C22⋊C411yescharacteristic
52C22×Dic61C2×D412yes—
53C2×C4×D61—24no—
54C2×C4×D61—24no—
55C2×C4×D61—24no—
56C2×C4×D61—21no—
57C2×C4×D61—21no—
58C2×C4×D61—21no—
59C2×C4×D61—21no—
60C2×C4×D61—22no—
61C2×C4×D61—22no—
62C2×C4×D61—22no—
63C2×C4×D61—44no—
64C2×C4×D61—44no—
65C2×C4×D61—41no—
66C2×C4×D61—41no—
67C2×#92420—44no—
68#11758213—84no—
69C4⋊C4×C61—41no—
70C4⋊C4×C61—44no—
71C4⋊C4×C61—41no—
72C42×C61—21no—
73C42×C61M1612yescomplemented
74C4⋊Dic61—41no—
75C4⋊Dic61—44no—
76C4⋊Dic61—41no—
77#11758196—42no—
78#11758196—44no—
79#11758196—44no—
80#11758196—42no—
81C4×Dic61—24no—
82C4×Dic61—21no—
83C4×Dic61M1612yescomplemented
84C23×C61C2×C22⋊C411yescharacteristic
85C22×D61—21no—
86C22×D61—21no—
87C22×D61—22no—
88C22×D61—22no—
89C22×D61#9911yescharacteristic
90C22×D61#9911yescharacteristic
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—21no—
95C2×C4×C61—21no—
96C2×C4×C61—22no—
97C2×C4×C61—42no—
98C2×C4×C61—42no—
99C2×C4×C61—41no—
100C2×C4×C61C2×M1611yescharacteristic
101C2×C4×C61C2×M1611yescharacteristic
102C2×Dic61—22no—
103C2×Dic61—22no—
104C2×Dic61—21no—
105C2×Dic61—22no—
106C2×Dic61—21no—
107C2×Dic61—22no—
108C2×Dic61—42no—
109C2×Dic61—42no—
110C2×Dic61—41no—
111C2×Dic61#10112yes—
112C4×D61—216no—
113C4×D61—22no—
114C4×D61—22no—
115C4×D61—22no—
116C4×D61—22no—
117C4×D61—44no—
118C4×D61—82no—
119C4×D61—82no—
120C4×D61—82no—
121#92420—84no—
122#11512—611nomaximal, Sylow
123C22×C61—21no—
124C22×C61—21no—
125C22×C61#40711yescharacteristic
126C22×C61C2×#9911yescharacteristic
127C22×C61C2×#10111yescharacteristic
128C2×D61—24no—
129C2×D61—24no—
130C2×D61—21no—
131C2×D61—21no—
132C2×D61—21no—
133C2×D61—21no—
134C2×D61#35312yes—
135C2×D61#35612yes—
136C4×C61—24no—
137C4×C61—21no—
138C4×C61—21no—
139C4×C61—42no—
140C4×C61—81no—
141Dic61—24no—
142Dic61—22no—
143Dic61—42no—
144Dic61—81no—
145C2×#384—611no—
146#1463—612no—
147#1325—612no—
148#1325—612no—
149C2×C61—21no—
150C2×C61—21no—
151C2×C61#146311yescharacteristic
152C2×C61C2×#35311yescharacteristic
153C2×C61C2×#35611yescharacteristic
154D61—22no—
155D61—22no—
156C22×C4⋊C4—611no—
157C22×C4⋊C4—611no—
158C22×C4⋊C4—1222no—
159C22×C42—611no—
160#407—1222no—
161#384—611no—
162#384—611no—
163#384—611no—
164#384—1222no—
165#384#692311yescharacteristic
166#324—614no—
167#324—1222no—
168C61#1151211yescharacteristic, complemented, Sylow
169C23×C4—611no—
170C23×C4—612no—
171C23×C4—1221no—
172C23×C4—1221no—
173C2×M16—2442no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—612no—
176C2×C4⋊C4—611no—
177C2×C4⋊C4—611no—
178C2×C4⋊C4—1221no—
179C2×C4⋊C4—1222no—
180C2×C4⋊C4—1222no—
181C2×C4⋊C4—1222no—
182C2×C4⋊C4—2441no—
183C2×C4⋊C4—2441no—
184C2×C4⋊C4—2441no—
185C2×C4⋊C4—2441no—
186C2×C4⋊C4—2448no—
187C2×C4⋊C4C2×#692311yescharacteristic
188C2×C4⋊C4C2×#692311yescharacteristic
189C2×C42—611no—
190C2×C42—1222no—
191C2×C42—1221no—
192C2×C42—1222no—
193C2×C42C2×#692311yescharacteristic
194C22⋊C8—1224no—
195C22⋊C8—2442no—
196C24—611no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—611no—
200C22×C4—612no—
201C22×C4—1222no—
202C22×C4—1221no—
203C22×C4—1221no—
204C22×C4—1224no—
205C22×C4—1224no—
206C22×C4—1224no—
207C22×C4—1222no—
208C22×C4—1222no—
209C22×C4—1221no—
210C22×C4—1221no—
211C22×C4—1221no—
212C22×C4—1221no—
213C22×C4—2441no—
214C22×C4—2441no—
215C22×C4—2444no—
216C22×C4—2444no—
217C22×C4C22⋊#692312yes—
218C22×C4C22×#692311yescharacteristic
219M16—4884no—
220C2×C8—2444no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—44no—
224C4⋊C4—2442no—
225C4⋊C4—2441no—
226C4⋊C4—2441no—
227C4⋊C4—2444no—
228C4⋊C4—2444no—
229C4⋊C4—2444no—
230C4⋊C4—2442no—
231C42—21no—
232C42—612no—
233C42—1224no—
234C42—1221no—
235C42#1175821312yescomplemented
236C23—611no—
237C23—611no—
238C23—1222no—
239C23—1222no—
240C23—1221no—
241C23—1221no—
242C23C2×C22⋊#692311yescharacteristic
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—612no—
253C2×C4—12216no—
254C2×C4—1222no—
255C2×C4—1222no—
256C2×C4—1222no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1221no—
262C2×C4—1221no—
263C2×C4—1222no—
264C2×C4—4882no—
265C2×C4—4882no—
266C2×C4—4882no—
267C2×C4—2444no—
268C2×C4—2442no—
269C2×C4—2441no—
270C2×C4—2442no—
271C2×C4D61⋊M1611yescharacteristic
272C2×C4C2×#1175821311yescharacteristic
273C8—4884no—
274C22—21no—
275C22—21no—
276C22—612no—
277C22—612no—
278C22—1224no—
279C22—1224no—
280C22—1221no—
281C22—1221no—
282C22—1221no—
283C22—1221no—
284C22#1684073311yescharacteristic
285C22D61⋊#9911yescharacteristic
286C22D61⋊#10111yescharacteristic
287C4—24no—
288C4—21no—
289C4—21no—
290C4—42no—
291C4—81no—
292C4—1224no—
293C4—1222no—
294C4—4881no—
295C4—2442no—
296C2—21no—
297C2—21no—
298C2—1222no—
299C2—1222no—
300C2#2146106811yescharacteristic
301C2D61⋊#35311yescharacteristic
302C2D61⋊#35611yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC22×C61C2×#99
prime radicalC23×D61C22⋊C4
cosocleC2×C4⋊Dic61C23
hypercenter#384#6923
hypocenterC61#11512
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #21460762C2semidirect
3. #21460761C2semidirect
73. C42×C61M16semidirect
83. C4×Dic61M16semidirect
168. C61#11512semidirect
235. C42#11758213semidirect

Maximal subgroups

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

subgroupindex
#214610682
#214607622
#214607612
#384×D612
#1151261