Finite Group Databasebeta

← #30598772

Subgroups — #30598772

Order 15616.

1222 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1222 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#21461647C212yesmaximal
3C22×#14519765C211yesmaximal, characteristic
4C2×#14519836—24no—
5#16840703C2214yes—
6#16840640—28no—
7C23×Dic122C411yescharacteristic, complemented
8C23×D244C2211yescharacteristic
9C2×#14519765—28no—
10C2×#14519765—24no—
11C22×#11758198C412yescomplemented
12C22×#11758198C2212yes—
13C22×C42×C61C411yescharacteristic, complemented
14C2×C22⋊#6923—24no—
15C2×C22⋊#6923—28no—
16#14519836—48no—
17C24×D61C2×C411yescharacteristic
18C24×D61C2311yescharacteristic
19C22×Dic122—22no—
20C22×Dic122—24no—
21C22×Dic122—44no—
22C22×D244—24no—
23C22×D244—44no—
24C22×D244D414yes—
25C23×Dic61C2×C412yes—
26#14519765—232no—
27C2×#11758198—24no—
28C2×#11758198—24no—
29C2×#11758198—24no—
30C2×#11758198—24no—
31C2×#11758198—24no—
32C2×#11758198D418yes—
33C23×C4×C61C2×C411yescharacteristic
34C23×C4×C61C2×C412yes—
35C2×C42×C61—24no—
36C2×C42×C61—22no—
37C24×C61C22×C411yescharacteristic
38C23×D61—22no—
39C23×D61—42no—
40C23×D61—42no—
41C23×D61—44no—
42C23×D61—44no—
43C23×D61C22⋊C412yes—
44C23×D61C2×D414yes—
45C23×D61C2×D412yes—
46C22×#6923—44no—
47C22×C4×C61—22no—
48C22×C4×C61—24no—
49C22×C4×C61—22no—
50C22×C4×C61—22no—
51C22×C4×C61—44no—
52C22×C4×C61C22⋊C412yes—
53C22×C4×C61C22⋊C414yes—
54C22×Dic61—22no—
55C22×Dic61—22no—
56C22×Dic61—44no—
57C22×Dic61C22⋊C414yes—
58C2×D244—28no—
59C2×D244—28no—
60C2×D244—44no—
61C2×D244—416no—
62C2×Dic122—28no—
63C2×Dic122—416no—
64C2×Dic122—44no—
65C2×Dic122—44no—
66C22⋊#6923—48no—
67C22⋊#6923—416no—
68C42×C61—28no—
69#11758198—216no—
70#11758198—216no—
71C23×C61—22no—
72C23×C61—24no—
73C23×C61C2×C22⋊C411yescharacteristic
74C23×C61C2×C22⋊C412yes—
75C22×D61—24no—
76C22×D61—28no—
77C22×D61—22no—
78C22×D61—24no—
79C22×D61—44no—
80C22×D61—44no—
81C22×D61—44no—
82C22×D61—44no—
83C22×D61—44no—
84C22×D61—44no—
85C22×D61—44no—
86C22×D61—88no—
87C22×D61—88no—
88C22×D61#12014yes—
89C2×#6923—84no—
90C2×#6923—88no—
91C2×C4×C61—28no—
92C2×C4×C61—22no—
93C2×C4×C61—22no—
94C2×C4×C61—24no—
95C2×C4×C61—44no—
96C2×C4×C61—44no—
97C2×C4×C61—44no—
98C2×C4×C61#9914yes—
99C2×Dic61—24no—
100C2×Dic61—44no—
101C2×Dic61—44no—
102C2×Dic61—44no—
103D244—416no—
104Dic122—416no—
105#8544—611nomaximal, Sylow
106C22×C61—28no—
107C22×C61—22no—
108C22×C61—22no—
109C22×C61—22no—
110C22×C61#37911yescharacteristic
111C22×C61C2×#9912yes—
112C22×C61#41014yes—
113C2×D61—24no—
114C2×D61—28no—
115C2×D61—42no—
116C2×D61—44no—
117C2×D61—84no—
118C2×D61—88no—
119C2×D61—88no—
120C2×D61—84no—
121#6923—164no—
122C4×C61—24no—
123C4×C61—44no—
124Dic61—44no—
125C22×#124—611no—
126#1819—614no—
127#1781—612no—
128C2×C61—22no—
129C2×C61—24no—
130C2×C61#178111yescharacteristic
131C2×C61#178212yes—
132D61—44no—
133D61—82no—
134C23×Q8—611no—
135C23×D4—611no—
136C2×#124—1228no—
137C2×#124—1224no—
138C22×C22⋊C4—612no—
139C22×C22⋊C4—612no—
140C22×C42#692311yescharacteristic, complemented
141C2×#99—1224no—
142#379—614no—
143#342—1228no—
144C61#854411yescharacteristic, complemented, Sylow
145C25—611no—
146C25—611no—
147C22×Q8—1224no—
148C22×Q8—1222no—
149C22×Q8—2444no—
150C22×D4—614no—
151C22×D4—1224no—
152C22×D4—2444no—
153C23×C4—612no—
154C23×C4C2×#692311yescharacteristic
155C23×C4C2×#692312yes—
156#124—12232no—
157C2×C22⋊C4—618no—
158C2×C22⋊C4—1224no—
159C2×C22⋊C4—1224no—
160C2×C22⋊C4—1224no—
161C2×C22⋊C4—1224no—
162C2×C22⋊C4—1224no—
163C2×C22⋊C4—1228no—
164C2×C22⋊C4—1224no—
165C2×C42—24no—
166C2×C42—22no—
167#99—2448no—
168C24—614no—
169C24—612no—
170C24—612no—
171C24—1222no—
172C24—2442no—
173C24—2442no—
174C24—2444no—
175C24—2444no—
176C24C22×#692311yescharacteristic
177C2×Q8—1228no—
178C2×Q8—24416no—
179C2×Q8—2444no—
180C2×Q8—2444no—
181C2×D4—1228no—
182C2×D4—1228no—
183C2×D4—2444no—
184C2×D4—24416no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—44no—
190C22×C4—614no—
191C22×C4—1222no—
192C22×C4—1222no—
193C22×C4—2444no—
194C22×C4—2444no—
195C22×C4C22⋊#692312yes—
196C22×C4C22⋊#692314yes—
197C22⋊C4—12216no—
198C22⋊C4—12216no—
199C22⋊C4—2448no—
200C22⋊C4—24416no—
201C42—28no—
202C23—22no—
203C23—24no—
204C23—614no—
205C23—1224no—
206C23—1228no—
207C23—1224no—
208C23—1222no—
209C23—4888no—
210C23—4888no—
211C23—2444no—
212C23—2444no—
213C23—2444no—
214C23—2444no—
215C23—2444no—
216C23—2444no—
217C23—2444no—
218C23C2×C22⋊#692311yescharacteristic
219C23C2×C22⋊#692312yes—
220Q8—24416no—
221D4—24416no—
222C2×C4—28no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—44no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—1224no—
230C2×C4—4884no—
231C2×C4—4888no—
232C2×C4—2444no—
233C2×C4—2444no—
234C2×C4—2444no—
235C2×C4#1451983614yes—
236C22—28no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—1224no—
241C22—1228no—
242C22—4884no—
243C22—4888no—
244C22—4884no—
245C22—4888no—
246C22—2442no—
247C22—2444no—
248C22#1684070311yescharacteristic
249C22C2×#1451983612yes—
250C22#1684074114yes—
251C4—24no—
252C4—44no—
253C4—9764no—
254C4—2444no—
255C2—22no—
256C2—24no—
257C2—4882no—
258C2—2444no—
259C2#2146164711yescharacteristic
260C2#2146164812yes—
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C42×C61C4
socleC22×C61#379
prime radicalC23×D244C22
cosocleC24×D61C23
hypercenterC22×C42#6923
hypocenterC61#8544
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. #21461714C2semidirect
7. C23×Dic122C4semidirect
11. C22×#11758198C4semidirect
13. C22×C42×C61C4semidirect
140. C22×C42#6923semidirect
144. C61#8544semidirect

Maximal subgroups

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

subgroupindex
#214617142
#214616472
C22×#145197652
#854461