Finite Group Databasebeta

← #30598769

Subgroups — #30598769

Order 15616.

1266 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1266 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#21461714C214yesmaximal, complemented
2#21461645C212yesmaximal
3C22×#14519760C211yesmaximal, characteristic
4C2×#14519835—24no—
5#16840703C2214yes—
6#16840640—28no—
7C23×C22⋊D61C2211yescharacteristic
8C2×#14519760—28no—
9C2×#14519760—24no—
10C22×#11758196C412yescomplemented
11C22×C22⋊Dic61C411yescharacteristic, complemented
12C22×#11758198C2212yes—
13C24×C4×C61C411yescharacteristic, complemented
14C2×C22⋊#6923—24no—
15C2×C22⋊#6923—28no—
16#14519835—48no—
17C24×D61C2311yescharacteristic
18C22×C22⋊D61—24no—
19C22×C22⋊D61—44no—
20C22×C22⋊D61D414yes—
21C23×Dic61C2×C412yes—
22C23×Dic61C2×C411yescharacteristic
23#14519760—232no—
24C2×#11758196—24no—
25C2×#11758196—24no—
26C2×#11758196—24no—
27C2×C22⋊Dic61—22no—
28C2×C22⋊Dic61—24no—
29C2×#11758198—24no—
30C2×#11758198—24no—
31C2×#11758198D418yes—
32C25×C61C2×C411yescharacteristic
33C23×C4×C61—22no—
34C23×C4×C61—24no—
35C23×C4×C61—44no—
36C23×C4×C61C2×C412yes—
37C24×C61—24no—
38C24×C61—22no—
39C24×C61—44no—
40C24×C61C22⋊C412yes—
41C24×C61C22×C411yescharacteristic
42C23×D61—42no—
43C23×D61—44no—
44C23×D61C2×D414yes—
45C23×D61C2×D412yes—
46C22×#6923—44no—
47C22×C4×C61—28no—
48C22×C4×C61—22no—
49C22×C4×C61—22no—
50C22×C4×C61—416no—
51C22×C4×C61—44no—
52C22×C4×C61—44no—
53C22×C4×C61—44no—
54C22×C4×C61C22⋊C414yes—
55C2×C22⋊D61—28no—
56C2×C22⋊D61—28no—
57C2×C22⋊D61—44no—
58C2×C22⋊D61—416no—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C22×Dic61—22no—
62C22×Dic61—42no—
63C22×Dic61—44no—
64C22×Dic61C22⋊C412yes—
65C22×Dic61C22⋊C414yes—
66C22⋊#6923—48no—
67C22⋊#6923—416no—
68C22⋊Dic61—28no—
69#11758198—216no—
70#11758196—216no—
71C23×C61—22no—
72C23×C61—24no—
73C23×C61—28no—
74C23×C61—22no—
75C23×C61—22no—
76C23×C61—416no—
77C23×C61—44no—
78C23×C61—44no—
79C23×C61—44no—
80C23×C61#9914yes—
81C23×C61C2×C22⋊C411yescharacteristic
82C23×C61C2×C22⋊C412yes—
83C22×D61—24no—
84C22×D61—28no—
85C22×D61—44no—
86C22×D61—44no—
87C22×D61—44no—
88C22×D61—88no—
89C22×D61#12014yes—
90C2×#6923—84no—
91C2×#6923—88no—
92C2×C4×C61—24no—
93C2×C4×C61—44no—
94C2×C4×C61—416no—
95C2×C4×C61—44no—
96C2×C4×C61—44no—
97C22⋊D61—416no—
98C2×Dic61—22no—
99C2×Dic61—24no—
100C2×Dic61—24no—
101C2×Dic61—44no—
102C2×Dic61—44no—
103C2×Dic61—44no—
104C2×Dic61—44no—
105#8543—611nomaximal, Sylow
106C22×C61—28no—
107C22×C61—22no—
108C22×C61—22no—
109C22×C61—22no—
110C22×C61—24no—
111C22×C61—44no—
112C22×C61—44no—
113C22×C61—416no—
114C22×C61—44no—
115C22×C61#37911yescharacteristic
116C22×C61C2×#9912yes—
117C22×C61#41014yes—
118C2×D61—24no—
119C2×D61—28no—
120C2×D61—84no—
121C2×D61—88no—
122#6923—164no—
123C4×C61—44no—
124Dic61—42no—
125Dic61—44no—
126C22×#123—611no—
127#1819—614no—
128#1781—612no—
129C2×C61—22no—
130C2×C61—24no—
131C2×C61—44no—
132C2×C61#178111yescharacteristic
133C2×C61#178212yes—
134D61—44no—
135C23×D4—611no—
136C24×C4#692311yescharacteristic, complemented
137C2×#123—1228no—
138C2×#123—1224no—
139C22×C4⋊C4—612no—
140C22×C22⋊C4—612no—
141C22×C22⋊C4—611no—
142C2×#99—1224no—
143#379—614no—
144#342—1228no—
145C61#854311yescharacteristic, complemented, Sylow
146C25—611no—
147C25C2×#692311yescharacteristic
148C22×D4—614no—
149C22×D4—1224no—
150C22×D4—2444no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—44no—
154C23×C4—612no—
155C23×C4—611no—
156C23×C4C2×#692312yes—
157#123—12232no—
158C2×C4⋊C4—1224no—
159C2×C4⋊C4—1224no—
160C2×C4⋊C4—1224no—
161C2×C22⋊C4—618no—
162C2×C22⋊C4—1224no—
163C2×C22⋊C4—1224no—
164C2×C22⋊C4—1224no—
165C2×C22⋊C4—1222no—
166C2×C22⋊C4—1228no—
167C2×C22⋊C4—1224no—
168#99—2448no—
169C24—24no—
170C24—22no—
171C24—44no—
172C24—614no—
173C24—612no—
174C24—2442no—
175C24—2444no—
176C24C22⋊#692312yes—
177C24C22×#692311yescharacteristic
178C2×D4—1228no—
179C2×D4—1228no—
180C2×D4—2444no—
181C2×D4—24416no—
182C22×C4—28no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—416no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—612no—
190C22×C4—614no—
191C22×C4—1222no—
192C22×C4—1222no—
193C22×C4—1222no—
194C22×C4—2442no—
195C22×C4—2444no—
196C22×C4—2444no—
197C22×C4C22⋊#692314yes—
198C4⋊C4—12216no—
199C22⋊C4—12216no—
200C22⋊C4—1228no—
201C22⋊C4—2448no—
202C22⋊C4—24416no—
203C23—22no—
204C23—24no—
205C23—28no—
206C23—22no—
207C23—22no—
208C23—44no—
209C23—416no—
210C23—44no—
211C23—44no—
212C23—614no—
213C23—1224no—
214C23—1228no—
215C23—4888no—
216C23—2444no—
217C23—2444no—
218C23—2444no—
219C23#1451983514yes—
220C23C2×C22⋊#692311yescharacteristic
221C23C2×C22⋊#692312yes—
222D4—24416no—
223C2×C4—24no—
224C2×C4—44no—
225C2×C4—416no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—1224no—
229C2×C4—1224no—
230C2×C4—1222no—
231C2×C4—4884no—
232C2×C4—4888no—
233C2×C4—2444no—
234C2×C4—2444no—
235C2×C4—2444no—
236C2×C4—2444no—
237C22—28no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—24no—
242C22—44no—
243C22—44no—
244C22—416no—
245C22—44no—
246C22—1224no—
247C22—1228no—
248C22—4884no—
249C22—4888no—
250C22#1684070311yescharacteristic
251C22C2×#1451983512yes—
252C22#1684074114yes—
253C4—44no—
254C4—9764no—
255C4—2442no—
256C4—2444no—
257C2—22no—
258C2—24no—
259C2—44no—
260C2—2444no—
261C2#2146164511yescharacteristic
262C2#2146164812yes—
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC24×C4×C61C4
socleC22×C61#379
prime radicalC23×C22⋊D61C22
cosocleC24×D61C23
hypercenterC24×C4#6923
hypocenterC61#8543
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #21461714C2semidirect
10. C22×#11758196C4semidirect
11. C22×C22⋊Dic61C4semidirect
13. C24×C4×C61C4semidirect
136. C24×C4#6923semidirect
145. C61#8543semidirect

Maximal subgroups

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

subgroupindex
#214617142
#214616452
C22×#145197602
#854361