Finite Group Databasebeta

← #30604190

Subgroups — #30604190

Order 15616.

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

normal subgroups only

This group has 412 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#21461081C212yesmaximal
2#21460761C212yesmaximal
3#21460750C212yesmaximal
4C4×#14519777C211yesmaximal, characteristic
5#16840783—22no—
6#16840608—22no—
7#16840607C2214yes—
8C2×#14519777C411yescharacteristic
9C2×C4×Dic122C411yescharacteristic
10C2×C4×Dic122C2211yescharacteristic
11#16839668C411yescharacteristic
12#16839668C2211yescharacteristic
13#16839660—22no—
14#16839632C2211yescharacteristic
15C4×C4⋊Dic61—22no—
16C43×C61C411yescharacteristic
17C4⋊#92420—42no—
18C22⋊#92420—24no—
19#14519832—42no—
20C22×Dic122C2×C411yescharacteristic
21C22×Dic122C2×C411yescharacteristic
22#14519777—22no—
23#14519777—22no—
24C4×Dic122—41no—
25C4×Dic122—41no—
26C4×Dic122—41no—
27C4×Dic122—41no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61—22no—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×C4⋊Dic61C2311yescharacteristic
32C2×#11758196—21no—
33C2×#11758196D412yes—
34C2×C4×Dic61—21no—
35C2×C4×Dic61D412yes—
36#14519684—22no—
37#14519684—22no—
38C2×C42×C61—21no—
39C2×C42×C61—21no—
40C2×C42×C61C2×C411yescharacteristic
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43C22×C4×C61—21no—
44C22×C4×C61—21no—
45C22×C4×C61C22⋊C411yescharacteristic
46C22×C4×C61C22⋊C411yescharacteristic
47C22×C4×C61C22×C411yescharacteristic
48C22×Dic61—21no—
49C22×Dic61—21no—
50C22×Dic61C22⋊C412yes—
51C22×Dic61C2×D412yes—
52C2×Dic122—21no—
53C2×Dic122—21no—
54C2×Dic122—21no—
55C2×Dic122—41no—
56C2×Dic122—41no—
57C2×Dic122—41no—
58C2×Dic122—41no—
59C2×Dic122M1612yes—
60C2×#92420—44no—
61C42×C61—24no—
62C42×C61—21no—
63C42×C61—21no—
64C42×C61—21no—
65C42×C61—42no—
66C42×C61—41no—
67C42×C61M1612yes—
68C4⋊Dic61—28no—
69C4⋊Dic61—41no—
70C4⋊Dic61—44no—
71C4⋊Dic61—41no—
72#11758196—42no—
73#11758196—42no—
74C4×Dic61—22no—
75C4×Dic61—22no—
76C4×Dic61—41no—
77C4×Dic61—41no—
78C23×C61C2×C22⋊C411yescharacteristic
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—22no—
85C2×C4×C61—22no—
86C2×C4×C61—21no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—41no—
90C2×C4×C61#9911yescharacteristic
91C2×C4×C61#9911yescharacteristic
92C2×C4×C61C2×M1611yescharacteristic
93C2×C4×C61C2×M1611yescharacteristic
94C2×Dic61—22no—
95C2×Dic61—22no—
96C2×Dic61—22no—
97C2×Dic61—22no—
98C2×Dic61—22no—
99C2×Dic61—41no—
100C2×Dic61—41no—
101C2×Dic61—42no—
102C2×Dic61—42no—
103C2×Dic61—41no—
104Dic122—42no—
105Dic122—42no—
106Dic122—42no—
107Dic122—42no—
108#92420—84no—
109#11532—611nomaximal, Sylow
110C22×C61—21no—
111C22×C61—21no—
112C22×C61#40711yescharacteristic
113C22×C61C2×#9911yescharacteristic
114C22×C61#41311yescharacteristic
115C4×C61—24no—
116C4×C61—21no—
117C4×C61—21no—
118C4×C61—21no—
119C4×C61—42no—
120C4×C61—41no—
121C4×C61#35612yes—
122Dic61—42no—
123Dic61—42no—
124Dic61—42no—
125Dic61—81no—
126C4×C4⋊Q8—611no—
127#1467—612no—
128#1325—612no—
129#1319—612no—
130C2×C61—21no—
131C2×C61—21no—
132C2×C61#146511yescharacteristic
133C2×C61#212611yescharacteristic
134C2×C61C2×#35611yescharacteristic
135C2×C4⋊Q8—611no—
136C2×C4×Q8—611no—
137C2×C4×Q8—611no—
138#432—1222no—
139#391—611no—
140#391—611no—
141#389—1222no—
142#384—611no—
143C4×C4⋊C4—1222no—
144C43#692311yescharacteristic
145#324—614no—
146#324—1222no—
147C61#1153211yescharacteristic, complemented, Sylow
148C22×Q8—611no—
149C22×Q8—611no—
150C4⋊Q8—1222no—
151C4⋊Q8—1222no—
152C4×Q8—2441no—
153C4×Q8—2441no—
154C4×Q8—2441no—
155C4×Q8—2441no—
156C2×C4⋊C4—612no—
157C2×C4⋊C4—611no—
158C2×C4⋊C4—611no—
159C2×C4⋊C4—1222no—
160C2×C4⋊C4—1222no—
161C2×C4⋊C4—1221no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—612no—
165C2×C42—1221no—
166C2×C42C2×#692311yescharacteristic
167C2×C42C2×#692311yescharacteristic
168C2×C42C2×#692311yescharacteristic
169C4⋊C8—2442no—
170C22⋊C8—1224no—
171#97—2442no—
172#95—1222no—
173#95—1222no—
174C2×Q8—612no—
175C2×Q8—1221no—
176C2×Q8—1221no—
177C2×Q8—1221no—
178C2×Q8—2441no—
179C2×Q8—2441no—
180C2×Q8—2441no—
181C2×Q8—2441no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—612no—
185C22×C4—612no—
186C22×C4—1221no—
187C22×C4—1221no—
188C22×C4C22⋊#692311yescharacteristic
189C22×C4C22⋊#692311yescharacteristic
190C22×C4C22×#692311yescharacteristic
191C2×C8—2444no—
192C4⋊C4—1228no—
193C4⋊C4—2442no—
194C4⋊C4—2441no—
195C4⋊C4—2441no—
196C4⋊C4—2442no—
197C4⋊C4—2444no—
198C42—24no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—41no—
203C42—42no—
204C42—1222no—
205C42—1222no—
206C42—2441no—
207C42—2441no—
208C42#1175821312yes—
209C23C2×C22⋊#692311yescharacteristic
210Q8—2442no—
211Q8—2442no—
212Q8—2442no—
213Q8—2442no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1222no—
226C2×C4—1222no—
227C2×C4—1222no—
228C2×C4—1222no—
229C2×C4—1222no—
230C2×C4—2442no—
231C2×C4—2441no—
232C2×C4—2441no—
233C2×C4—2441no—
234C2×C4—2442no—
235C2×C4#1451983611yescharacteristic
236C2×C4#1451983611yescharacteristic
237C2×C4D61⋊M1611yescharacteristic
238C2×C4C2×#1175821311yescharacteristic
239C8—4884no—
240C22—21no—
241C22—21no—
242C22#1684073311yescharacteristic
243C22C2×#1451983611yescharacteristic
244C22#1684074811yescharacteristic
245C4—24no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—41no—
251C4—4881no—
252C4—2442no—
253C4—2442no—
254C4—2442no—
255C4#1684067312yes—
256C2—21no—
257C2—21no—
258C2#2146107611yescharacteristic
259C2#2146209211yescharacteristic
260C2C2×#1684067311yescharacteristic
subgroupG/H
centerC22C2×#14519836
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC43×C61C4
socleC22×C61C2×#99
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4⋊Dic61C23
hypercenterC43#6923
hypocenterC61#11532
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
147. C61#11532semidirect

Maximal subgroups

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

subgroupindex
#214610812
#214607612
#214607502
C4×#145197772
#1153261