Finite Group Databasebeta

← #30598797

Subgroups — #30598797

Order 15616.

1154 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 108 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#21461653C212yesmaximal, complemented
2#21461064C214yesmaximal, complemented
3C22×#14519764C211yesmaximal, characteristic
4C2×#14519838—24no—
5#16840732C2214yes—
6#16840605—28no—
7C23×Dic122C2211yescharacteristic
8C2×#14519764—28no—
9C2×#14519764—24no—
10C22×C22⋊Dic61C412yes—
11C22×C22⋊Dic61C2212yes—
12C22×C4×Dic61C411yescharacteristic
13C23×D4×C61C411yescharacteristic
14C2×#11758217—24no—
15#14519838—48no—
16C22⋊#92420—28no—
17C22×Dic122—24no—
18C22×Dic122—44no—
19C22×Dic122D414yes—
20C23×Dic61C2×C412yes—
21C23×Dic61C2×C411yescharacteristic
22C23×Dic61C2311yescharacteristic
23#14519764—232no—
24C2×C22⋊Dic61—24no—
25C2×C22⋊Dic61—24no—
26C2×C22⋊Dic61—24no—
27C2×C22⋊Dic61—24no—
28C2×C22⋊Dic61—24no—
29C2×C22⋊Dic61D418yes—
30C2×C4×Dic61—22no—
31C2×C4×Dic61—24no—
32C25×C61C2×C412yes—
33C22×D4×C61—24no—
34C22×D4×C61—22no—
35C22×D4×C61—44no—
36C23×C4×C61C2×C411yescharacteristic
37C24×C61—22no—
38C24×C61—22no—
39C24×C61—44no—
40C24×C61—44no—
41C24×C61—44no—
42C24×C61C22⋊C414yes—
43C24×C61C22×C411yescharacteristic
44C2×D4×C61—28no—
45C2×D4×C61—416no—
46C2×D4×C61—44no—
47C2×D4×C61—44no—
48C22×C4×C61—22no—
49C22×C4×C61—24no—
50C22×C4×C61C22⋊C412yes—
51C22×Dic61—22no—
52C22×Dic61—22no—
53C22×Dic61—22no—
54C22×Dic61—42no—
55C22×Dic61—42no—
56C22×Dic61—44no—
57C22×Dic61C22⋊C412yes—
58C22×Dic61C22⋊C414yes—
59C22×Dic61C2×D412yes—
60C22×Dic61C2×D414yes—
61C2×Dic122—28no—
62C2×Dic122—28no—
63C2×Dic122—44no—
64C2×Dic122—416no—
65#11758217—48no—
66C2×#92420—44no—
67C22⋊Dic61—216no—
68C22⋊Dic61—216no—
69C4×Dic61—28no—
70C23×C61—22no—
71C23×C61—24no—
72C23×C61—24no—
73C23×C61—28no—
74C23×C61—44no—
75C23×C61—44no—
76C23×C61—44no—
77C23×C61—44no—
78C23×C61—44no—
79C23×C61—44no—
80C23×C61—44no—
81C23×C61—44no—
82C23×C61—816no—
83C23×C61C2×C22⋊C411yescharacteristic
84C23×C61C2×C22⋊C412yes—
85D4×C61—416no—
86C2×C4×C61—28no—
87C2×C4×C61—22no—
88C2×C4×C61—22no—
89C2×C4×C61#9914yes—
90C2×Dic61—24no—
91C2×Dic61—24no—
92C2×Dic61—22no—
93C2×Dic61—24no—
94C2×Dic61—44no—
95C2×Dic61—44no—
96C2×Dic61—44no—
97C2×Dic61—44no—
98C2×Dic61—44no—
99C2×Dic61#12014yes—
100Dic122—416no—
101#92420—84no—
102#8552—611nomaximal, Sylow
103C22×C61—28no—
104C22×C61—22no—
105C22×C61—22no—
106C22×C61—22no—
107C22×C61—44no—
108C22×C61—44no—
109C22×C61—44no—
110C22×C61—88no—
111C22×C61—88no—
112C22×C61—88no—
113C22×C61#37911yescharacteristic
114C22×C61C2×#9912yes—
115C22×C61#41314yes—
116C4×C61—24no—
117Dic61—24no—
118Dic61—42no—
119Dic61—44no—
120C22×#124—611no—
121#1784—612no—
122#1462—614no—
123C2×C61—22no—
124C2×C61—24no—
125C2×C61—84no—
126C2×C61#178111yescharacteristic
127C2×C61#178512yes—
128C23×Q8—611no—
129C23×D4#692311yescharacteristic
130C2×#124—1228no—
131C2×#124—1224no—
132C22×C22⋊C4—612no—
133C22×C22⋊C4—612no—
134C22×C42—611no—
135C2×#101—1224no—
136#407—614no—
137#323—1228no—
138C61#855211yescharacteristic, complemented, Sylow
139C25C2×#692312yes—
140C22×Q8—614no—
141C22×Q8—1224no—
142C22×Q8—2444no—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—44no—
146C23×C4—611no—
147C23×C4—611no—
148C23×C4—612no—
149C23×C4C2×#692311yescharacteristic
150C2×M16—1224no—
151#124—12232no—
152C2×C22⋊C4—618no—
153C2×C22⋊C4—1224no—
154C2×C22⋊C4—1224no—
155C2×C22⋊C4—1224no—
156C2×C22⋊C4—1224no—
157C2×C22⋊C4—1224no—
158C2×C42—1224no—
159C2×C42—1222no—
160#101—2448no—
161C22⋊C8—1228no—
162C24—22no—
163C24—22no—
164C24—44no—
165C24—44no—
166C24—44no—
167C24C22⋊#692314yes—
168C24C22×#692311yescharacteristic
169C2×Q8—1228no—
170C2×Q8—1228no—
171C2×Q8—2444no—
172C2×Q8—24416no—
173C2×D4—28no—
174C2×D4—416no—
175C2×D4—44no—
176C2×D4—44no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—612no—
180C22×C4—614no—
181C22×C4—612no—
182C22×C4—614no—
183C22×C4—1222no—
184C22×C4—1222no—
185C22×C4—1222no—
186C22×C4—2442no—
187C22×C4—2442no—
188C22×C4—2444no—
189C22×C4C22⋊#692312yes—
190M16—2448no—
191C2×C8—2444no—
192C22⋊C4—12216no—
193C22⋊C4—12216no—
194C42—1228no—
195C23—22no—
196C23—24no—
197C23—24no—
198C23—28no—
199C23—44no—
200C23—44no—
201C23—44no—
202C23—44no—
203C23—44no—
204C23—44no—
205C23—44no—
206C23—44no—
207C23—816no—
208C23C2×C22⋊#692311yescharacteristic
209C23C2×C22⋊#692312yes—
210Q8—24416no—
211D4—416no—
212C2×C4—28no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—614no—
216C2×C4—1224no—
217C2×C4—1224no—
218C2×C4—1224no—
219C2×C4—1222no—
220C2×C4—2444no—
221C2×C4—2444no—
222C2×C4—2444no—
223C2×C4—2444no—
224C2×C4—2444no—
225C2×C4#1451983514yes—
226C8—4884no—
227C22—28no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—44no—
232C22—44no—
233C22—44no—
234C22—88no—
235C22—88no—
236C22—88no—
237C22#1684070311yescharacteristic
238C22C2×#1451983512yes—
239C22#1684074914yes—
240C4—24no—
241C4—1224no—
242C4—2442no—
243C4—2444no—
244C2—22no—
245C2—24no—
246C2—84no—
247C2#2146164511yescharacteristic
248C2#2146165612yes—
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
Fitting, prime radicalC23×D4×C61C4
socleC22×C61#379
cosocleC23×Dic61C23
hypercenterC23×D4#6923
hypocenterC61#8552
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461653C2semidirect
2. #21461064C2semidirect
138. C61#8552semidirect

Maximal subgroups

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

subgroupindex
#214616532
#214610642
C22×#145197642
#855261