Finite Group Databasebeta

← #29582292

Subgroups — #29582292

Order 15616.

574 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 103 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#21456940C212yesmaximal, complemented
2#21456265C211yesmaximal, characteristic
3#21444378C212yesmaximal, complemented
4#21443664C211yesmaximal, characteristic, complemented
5C61×#1960C211yesmaximal, characteristic, complemented
6C2×#14519799—22no—
7#16840312C2212yes—
8#16840165—22no—
9#16840156—22no—
10#16840151—22no—
11#16840110—22no—
12#16840075—24no—
13#16840039—24no—
14#16839815C2212yes—
15#16839495—24no—
16C2×C4⋊Q8×C61C2211yescharacteristic
17C2×#103×C61—22no—
18C2×#97×C61C2211yescharacteristic
19C61×#382C2211yescharacteristic
20C2×#11758222—22no—
21#14519799—44no—
22#14519799—44no—
23C2×Dic244—22no—
24C2×#11758190—24no—
25C2×D488—22no—
26#14519777D412yes—
27#14519774D412yes—
28#14519765—22no—
29#14519751—22no—
30#14519751—22no—
31C4×D244—22no—
32C4×D244—22no—
33#14519710—28no—
34C4⋊#92418—22no—
35#14519697—28no—
36C22⋊#92418D414yes—
37C4×#92418—22no—
38C22×Q8×C61—21no—
39C22×C8×C61D412yes—
40C4⋊Q8×C61—22no—
41C4⋊Q8×C61—21no—
42C4⋊Q8×C61D412yes—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61—21no—
45C2×C42×C61C2311yescharacteristic
46#103×C61—44no—
47#97×C61—22no—
48#95×C61—22no—
49C2×Q8×C61—22no—
50C2×Q8×C61—41no—
51C2×Q8×C61—42no—
52C2×Q8×C61—42no—
53C22×C4×C61—41no—
54C22×C4×C61—41no—
55C22×C4×C61C2×D412yes—
56C22×C4×C61C2×D411yescharacteristic
57C2×C22⋊D61—42no—
58#11758222—44no—
59#11758222—44no—
60C2×D244—22no—
61C2×D244—22no—
62C2×D244—22no—
63C2×C4×D61—42no—
64C2×Dic122—22no—
65C2×C8×C61—22no—
66C2×C8×C61—22no—
67C2×C8×C61D814yes—
68C4⋊C4×C61—24no—
69C4⋊C4×C61—42no—
70C4⋊C4×C61—42no—
71C42×C61C2×D411yescharacteristic
72C42×C61C2×D412yes—
73C42×C61#4111yescharacteristic
74#11758198—42no—
75#11758198—42no—
76C4⋊Dic61—22no—
77C4⋊Dic61—22no—
78C2×#92418—24no—
79Dic244—44no—
80D488—44no—
81#11758190—48no—
82C23×C61#12711yescharacteristic
83C22×D61—42no—
84Q8×C61—44no—
85Q8×C61—82no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—81no—
89C2×C4×C61—81no—
90C2×C4×C61—82no—
91C2×C4×C61—81no—
92C2×C4×C61#12012yes—
93C2×C4×C61#12011yescharacteristic
94C2×C4×C61#12112yes—
95C2×C4×C61#12111yescharacteristic
96C2×C4×C61C2×D812yes—
97C2×C4×C61AGL(1,ℤ/8)12yes—
98C22⋊D61—82no—
99C22⋊D61—82no—
100C2×Dic61—42no—
101D244—44no—
102D244—44no—
103D244—82no—
104C4×D61—84no—
105Dic122—44no—
106C8×C61—24no—
107#92418—44no—
108#10225—611nomaximal, Sylow
109C22×C61—22no—
110C22×C61#39211yescharacteristic
111C22×C61#49311yescharacteristic
112C22×C61#49611yescharacteristic
113C2×D61—82no—
114C2×D61—82no—
115C2×D61—82no—
116C4×C61—22no—
117C4×C61—22no—
118C4×C61—82no—
119C4×C61—161no—
120C4×C61#44712yes—
121C4×C61#45912yes—
122Dic61—82no—
123#3226—612no—
124#3150—611no—
125#1960D6111yescharacteristic, complemented
126#1517—612no—
127#1281—611no—
128C2×C61—21no—
129C2×C61—21no—
130C2×C61#200912yes—
131C2×C61#201111yescharacteristic
132D61—162no—
133C2×#135—1222no—
134#532—612no—
135C2×C4⋊Q8C2×D6111yescharacteristic
136#500—1222no—
137#497—1222no—
138#496—1222no—
139#486—1222no—
140#480—1224no—
141#474—1224no—
142#431—612no—
143C2×#103—22no—
144C2×#97C2×D6111yescharacteristic
145#382C2×D6111yescharacteristic
146#327—1224no—
147C61#1022511yescharacteristic, complemented, Sylow
148C2×#41—1222no—
149C22×Q8—21no—
150#135—2444no—
151#135—2444no—
152C2×Q16—1222no—
153C2×SD16—1224no—
154C2×D8—1222no—
155C22×C8C22⋊D6112yes—
156C4⋊Q8—21no—
157C4⋊Q8—22no—
158C4⋊Q8—612no—
159C4⋊Q8D24412yes—
160#127—612no—
161#124—1222no—
162#121—1222no—
163#121—1222no—
164C4×D4—1222no—
165C4×D4—1222no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C42C22×D6111yescharacteristic
169#107—1228no—
170C4⋊C8—1222no—
171#103—44no—
172#102—1228no—
173C22⋊C8—614no—
174#97—22no—
175C4×C8—1222no—
176#95—22no—
177#41—2444no—
178#41—2444no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—1222no—
184C2×D4—1222no—
185C2×D4—1222no—
186C2×D4—1222no—
187C2×D4—2442no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—2442no—
191C22×C4D4×D6112yes—
192C22×C4C2×C22⋊D6111yescharacteristic
193Q16—2444no—
194SD16—2448no—
195D8—2444no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—1224no—
199C2×C8#1175819914yes—
200C4⋊C4—24no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—1222no—
204C4⋊C4—1222no—
205C22⋊C4—2442no—
206C22⋊C4—2442no—
207C42C2×D24411yescharacteristic
208C42#1175822211yescharacteristic
209C42D4×D6112yes—
210C23—2442no—
211C23#1451977311yescharacteristic
212Q8—44no—
213Q8—82no—
214Q8—2444no—
215D4—4882no—
216D4—4882no—
217D4—4882no—
218D4—2444no—
219D4—2444no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4—81no—
226C2×C4—4884no—
227C2×C4—2442no—
228C2×C4#1451974511yescharacteristic
229C2×C4#1451974712yes—
230C2×C4#1451975012yes—
231C2×C4#1451975111yescharacteristic
232C2×C4C2×#1175819912yes—
233C2×C4#1451980212yes—
234C8—24no—
235C8—2444no—
236C22—22no—
237C22—4882no—
238C22—4882no—
239C22—4882no—
240C22#1683967111yescharacteristic
241C22#1684013811yescharacteristic
242C22#1684015211yescharacteristic
243C4—22no—
244C4—22no—
245C4—82no—
246C4—161no—
247C4—4882no—
248C4#1683989512yes—
249C4#1683995712yes—
250C2—21no—
251C2—21no—
252C2—9762no—
253C2#2144688112yes—
254C2#2144688911yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1960C2
socleC22×C61#392
prime radical#21456265C2
hypercenter#1960D61
hypocenterC61#10225
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
1. #21456940C2semidirect
3. #21444378C2semidirect
4. #21443664C2semidirect
5. C61×#1960C2semidirect
125. #1960D61semidirect
147. C61#10225semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214562652
#214443782
#214436642
C61×#19602
#1022561