Finite Group Databasebeta

← #30606318

Subgroups — #30606318

Order 15616.

746 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 152 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
1C22×#14519838C212yesmaximal
2#21460916C214yesmaximal, complemented
3#21448403C211yesmaximal, characteristic
4C22×#11758217—22no—
5C2×#14519838—22no—
6C2×#14519838—24no—
7C2×#14519838C418yescomplemented
8C2×#14519838C2214yes—
9C23×Dic122C2211yescharacteristic
10C2×#14519736C411yescharacteristic
11#16839668C412yes—
12#16839668C2212yes—
13#16839667—22no—
14#16839623—22no—
15C2×#117×C61C411yescharacteristic
16C2×#11758217—28no—
17C2×#11758217—24no—
18C2×#11758217—44no—
19C2×#11758217—42no—
20C22×#92420—42no—
21#14519838—28no—
22#14519838—24no—
23#14519838C2×C418yes—
24C22×Dic122—42no—
25C22×Dic122—42no—
26C22×Dic122C2×C411yescharacteristic
27C22×Dic122C2×C414yes—
28C22×Dic122D412yes—
29C22×Dic122D412yes—
30C22×Dic122Q812yes—
31C22×Dic122C2311yescharacteristic
32C23×Dic61D412yes—
33#14519736—42no—
34C2×C4⋊Dic61C2×C412yes—
35C2×C22⋊Dic61—21no—
36C2×C4×Dic61—21no—
37#14519684—24no—
38#14519684—24no—
39C23×C4×C61C2×C411yescharacteristic
40#117×C61—42no—
41C2×C4⋊C4×C61—21no—
42C2×C22⋊C4×C61—21no—
43C2×C42×C61C2×C412yes—
44C24×C61C22⋊C411yescharacteristic
45C22×C4×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61C4212yes—
49C22×C4×C61C22⋊C411yescharacteristic
50C22×C4×C61C4⋊C412yes—
51C22×C4×C61C22×C411yescharacteristic
52C22×Dic61—22no—
53C22×Dic61—24no—
54C22×Dic61—22no—
55C22×Dic61—22no—
56C22×Dic61C22⋊C412yes—
57C22×Dic61C22⋊C414yes—
58C22×Dic61C4⋊C414yes—
59C22×Dic61C2×D412yes—
60C2×Dic122—41no—
61C2×Dic122—41no—
62C2×Dic122—44no—
63C2×Dic122—44no—
64C2×Dic122—42no—
65C2×Dic122—42no—
66C2×Dic122—44no—
67C2×Dic122—44no—
68C2×Dic122—41no—
69C2×Dic122—41no—
70C2×Dic122C4212yes—
71C2×Dic122C22⋊C414yes—
72C2×Dic122C4⋊C412yes—
73C2×Dic122C4⋊C414yes—
74C2×Dic122C22×C412yes—
75C2×Dic122C2×D411yescharacteristic
76C2×Dic122C2×Q811yescharacteristic
77#11758217—28no—
78#11758217—48no—
79#11758217—44no—
80C2×#92420—48no—
81C2×#92420—44no—
82C4⋊C4×C61—42no—
83C22⋊C4×C61—42no—
84C42×C61—42no—
85C22⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4×Dic61—42no—
88C23×C61—42no—
89C23×C61#9512yes—
90C23×C61#9912yes—
91C23×C61#9912yes—
92C23×C61C2×C22⋊C411yescharacteristic
93C2×C4×C61—24no—
94C2×C4×C61—24no—
95C2×C4×C61—21no—
96C2×C4×C61—21no—
97C2×C4×C61—21no—
98C2×C4×C61—21no—
99C2×C4×C61—44no—
100C2×C4×C61—42no—
101C2×C4×C61#9512yes—
102C2×C4×C61C2×C4211yescharacteristic
103C2×C4×C61C2×C4⋊C411yescharacteristic
104C2×Dic61—24no—
105C2×Dic61—24no—
106C2×Dic61—24no—
107C2×Dic61—24no—
108C2×Dic61—22no—
109C2×Dic61—22no—
110C2×Dic61—22no—
111C2×Dic61—22no—
112C2×Dic61—44no—
113C2×Dic61—42no—
114C2×Dic61#9514yes—
115C2×Dic61C2×C22⋊C412yes—
116C2×Dic61C2×C4⋊C412yes—
117Dic122—42no—
118Dic122—44no—
119Dic122—42no—
120Dic122—44no—
121Dic122—42no—
122Dic122—41no—
123Dic122—41no—
124#92420—48no—
125#12586—611nomaximal, Sylow
126C22×C61—22no—
127C22×C61—24no—
128C22×C61—42no—
129C22×C61—42no—
130C22×C61#34214yes—
131C22×C61C2×#9511yescharacteristic
132C22×C61C2×#9911yescharacteristic
133C22×C61C2×#9911yescharacteristic
134C4×C61—22no—
135C4×C61—21no—
136C4×C61—21no—
137C4×C61—82no—
138Dic61—22no—
139Dic61—24no—
140Dic61—22no—
141Dic61—82no—
142C22×#101—612no—
143#2310—611no—
144#1394—614no—
145C2×C61—21no—
146C2×C61—24no—
147C2×C61—21no—
148C2×C61C2×#34211yescharacteristic
149C2×C61#213312yes—
150C23×Q8—611no—
151C22×M16—1222no—
152C2×#117—611no—
153C2×#117#692311yescharacteristic
154C2×#101—618no—
155C2×#101—614no—
156C2×#101—1224no—
157C2×#101—1222no—
158#391—612no—
159#391—612no—
160#391—1222no—
161#381—1222no—
162C61#1258611yescharacteristic, complemented, Sylow
163C22×Q8—614no—
164C22×Q8—612no—
165C22×Q8—612no—
166C22×Q8—612no—
167C22×Q8—611no—
168C22×Q8—611no—
169C22×Q8—2442no—
170C22×Q8—2442no—
171C23×C4—612no—
172C23×C4C2×#692311yescharacteristic
173C2×M16—1228no—
174C2×M16—1224no—
175C2×M16—2444no—
176C2×M16—2442no—
177C22×C8—2442no—
178#117—42no—
179#117—2442no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—612no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—1221no—
184C2×C42—1221no—
185C2×C42C2×#692312yes—
186#101—618no—
187#101—1228no—
188#101—1224no—
189#95—1224no—
190#95—1224no—
191C24C22⋊#692311yescharacteristic
192C2×Q8—614no—
193C2×Q8—614no—
194C2×Q8—612no—
195C2×Q8—612no—
196C2×Q8—611no—
197C2×Q8—612no—
198C2×Q8—611no—
199C2×Q8—2441no—
200C2×Q8—2441no—
201C2×Q8—2442no—
202C2×Q8—2442no—
203C2×Q8—2441no—
204C2×Q8—2441no—
205C2×Q8—2444no—
206C2×Q8—2444no—
207C2×Q8—2444no—
208C2×Q8—2444no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—614no—
213C22×C4—614no—
214C22×C4—612no—
215C22×C4—612no—
216C22×C4—1224no—
217C22×C4—1222no—
218C22×C4—1222no—
219C22×C4—1222no—
220C22×C4C4×#692312yes—
221C22×C4C4⋊#692312yes—
222C22×C4C22⋊#692311yescharacteristic
223C22×C4C22×#692311yescharacteristic
224M16—1228no—
225M16—2444no—
226M16—2448no—
227C2×C8—2444no—
228C2×C8—2448no—
229C4⋊C4—42no—
230C4⋊C4—2442no—
231C22⋊C4—42no—
232C22⋊C4—2442no—
233C42—42no—
234C42—2442no—
235C23—42no—
236C23(C2×#6923)⋊C412yes—
237C23#1451983512yes—
238C23#1451983612yes—
239C23C2×C22⋊#692311yescharacteristic
240Q8—2444no—
241Q8—2444no—
242Q8—2442no—
243Q8—2442no—
244Q8—2441no—
245Q8—2442no—
246Q8—2441no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—614no—
256C2×C4—612no—
257C2×C4—612no—
258C2×C4—1222no—
259C2×C4—1222no—
260C2×C4—1222no—
261C2×C4—1222no—
262C2×C4—1224no—
263C2×C4—1224no—
264C2×C4—1224no—
265C2×C4—1224no—
266C2×C4—2444no—
267C2×C4—2442no—
268C2×C4(C2×#6923)⋊C412yes—
269C2×C4C2×C4×#692311yescharacteristic
270C2×C4C2×C4⋊#692311yescharacteristic
271C8—2448no—
272C22—22no—
273C22—24no—
274C22—42no—
275C22—42no—
276C22#1684064014yes—
277C22C2×(C2×#6923)⋊C411yescharacteristic
278C22C2×#1451983511yescharacteristic
279C22C2×#1451983611yescharacteristic
280C4—22no—
281C4—21no—
282C4—21no—
283C4—82no—
284C4—1222no—
285C4—1224no—
286C4—1222no—
287C4—4882no—
288C2—21no—
289C2—24no—
290C2—21no—
291C2C2×#1684064011yescharacteristic
292C2#2146211612yes—
subgroupG/H
centerC22C2×#14519836
commutatorC2×C4×C61C2×C42
FrattiniC22×C4C22×#6923
FittingC2×#117×C61C4
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC22×Dic122C23
hypercenterC2×#117#6923
hypocenterC61#12586
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #21460916C2semidirect
7. C2×#14519838C4semidirect
162. C61#12586semidirect

Maximal subgroups

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

subgroupindex
C22×#145198382
#214609162
#214484032
#1258661