Finite Group Databasebeta

← #29580866

Subgroups — #29580866

Order 15616.

566 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 105 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#21455850C212yesmaximal
2#21446651C211yesmaximal, characteristic
3#21444248C212yesmaximal, complemented
4#21443648C211yesmaximal, characteristic, complemented
5C61×#2399C211yesmaximal, characteristic, complemented
6#16840312C2211yescharacteristic
7#16840080—22no—
8#16840047—22no—
9#16839962—22no—
10#16839901—22no—
11C4×#11758201—24no—
12#16839815C2212yes—
13#16839786C2211yescharacteristic
14C2×#14519705—22no—
15Q8⋊#92418—24no—
16C2×C4×Q8×C61C2212yes—
17C61×#398—22no—
18C61×#395—22no—
19C4×C4⋊C4×C61—22no—
20C61×#376C2211yescharacteristic
21C2×#11758222—21no—
22C2×#11758201—24no—
23C2×#11758194—21no—
24#14519777D411yescharacteristic
25#14519774D411yescharacteristic
26#14519765—21no—
27#14519751—22no—
28C4×D244—22no—
29#14519708—24no—
30C4⋊#92418—22no—
31C4⋊#92418—22no—
32#14519705—44no—
33#14519701—24no—
34#14519697—24no—
35C22⋊#92418—22no—
36C4×#92418D414yes—
37C22×Q8×C61D412yes—
38C4×Q8×C61—22no—
39C4×Q8×C61—22no—
40C4×Q8×C61—22no—
41C4×Q8×C61—22no—
42C2×C4⋊C4×C61—24no—
43C2×C4⋊C4×C61—22no—
44C2×C4⋊C4×C61D412yes—
45C2×C42×C61—22no—
46C2×C42×C61D412yes—
47C2×C42×C61C2311yescharacteristic
48#95×C61—22no—
49#95×C61—22no—
50#95×C61—22no—
51C2×Q8×C61—22no—
52C2×Q8×C61—22no—
53C2×Q8×C61—44no—
54C22×C4×C61—22no—
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61C2×D412yes—
58C22×C4×C61C2×D411yescharacteristic
59C2×C22⋊D61—41no—
60#11758222—42no—
61#11758222—42no—
62C2×D244—22no—
63C2×D244—21no—
64C2×C4×D61—41no—
65C2×Dic122—21no—
66C4⋊C4×C61—22no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—48no—
69C4⋊C4×C61—42no—
70C4⋊C4×C61—42no—
71C42×C61—22no—
72C42×C61—42no—
73C42×C61—42no—
74C42×C61SD1614yescomplemented
75C42×C61C2×D411yescharacteristic
76C42×C61C2×D412yes—
77C42×C61#4111yescharacteristic
78#11758201—48no—
79#11758198—42no—
80C4⋊Dic61—22no—
81#11758194—42no—
82C2×#92418—24no—
83C2×#92418—41no—
84C23×C61#12011yescharacteristic
85C22×D61—41no—
86Q8×C61—24no—
87Q8×C61—24no—
88Q8×C61—42no—
89Q8×C61—42no—
90C2×C4×C61—24no—
91C2×C4×C61—24no—
92C2×C4×C61—42no—
93C2×C4×C61—44no—
94C2×C4×C61—42no—
95C2×C4×C61—42no—
96C2×C4×C61—44no—
97C2×C4×C61—42no—
98C2×C4×C61#12012yes—
99C2×C4×C61#12112yes—
100C2×C4×C61#12111yescharacteristic
101C2×C4×C61#12711yescharacteristic
102C2×C4×C61C2×SD1612yes—
103C2×C4×C61#13712yes—
104C22⋊D61—82no—
105C2×Dic61—41no—
106D244—44no—
107D244—42no—
108C4×D61—81no—
109C4×D61—81no—
110Dic122—42no—
111#92418—44no—
112#92418—81no—
113#10014—611nomaximal, Sylow
114C22×C61—22no—
115C22×C61#39211yescharacteristic
116C22×C61#44812yes—
117C2×D61—82no—
118C2×D61—81no—
119C4×C61—22no—
120C4×C61—22no—
121C4×C61—44no—
122C4×C61—44no—
123C4×C61—82no—
124C4×C61#46112yes—
125C4×C61#49211yescharacteristic
126C4×C61#49711yescharacteristic
127Dic61—81no—
128#3103—612no—
129#2399D6111yescharacteristic, complemented
130#1963—611no—
131#1487—612no—
132#1274—611no—
133C2×C61—22no—
134C2×C61#201012yes—
135C2×C61#201311yescharacteristic
136D61—161no—
137#532—611no—
138C2×C4×Q8C2×D6112yes—
139#481—1222no—
140#475—1222no—
141#460—1222no—
142#448—1222no—
143C4×SD16—1224no—
144#431—612no—
145#423—611no—
146C2×#104—1222no—
147#398—22no—
148#395—22no—
149C4×C4⋊C4—22no—
150#376C2×D6111yescharacteristic
151Q8⋊C8—1224no—
152C61#1001411yescharacteristic, complemented, Sylow
153C2×#41—1221no—
154C22×Q8C22⋊D6112yes—
155C2×SD16—1224no—
156C2×M16—1221no—
157C4⋊Q8—611no—
158#127—611no—
159#124—1221no—
160#121—1222no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—1222no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D6112yes—
169C2×C42—22no—
170C2×C42C22⋊D6112yes—
171C2×C42C22×D6111yescharacteristic
172#106—1224no—
173C4⋊C8—1222no—
174C4⋊C8—1222no—
175#104—2444no—
176#103—1224no—
177#102—1224no—
178C22⋊C8—1222no—
179C4×C8—614no—
180#95—22no—
181#95—22no—
182#95—22no—
183#41—2442no—
184#41—2442no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—44no—
188C2×Q8—1221no—
189C2×D4—1222no—
190C2×D4—1221no—
191C2×D4—2441no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—2441no—
196C22×C4C2×C22⋊D6112yes—
197C22×C4C2×C22⋊D6111yescharacteristic
198SD16—2448no—
199M16—2442no—
200C2×C8—1224no—
201C2×C8—2441no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—1222no—
208C22⋊C4—2442no—
209C42—22no—
210C42—42no—
211C42—42no—
212C42#1175820114yescomplemented
213C42D4×D6111yescharacteristic
214C42D4×D6112yes—
215C42#1175822411yescharacteristic
216C23—2441no—
217C23#1451974611yescharacteristic
218Q8—24no—
219Q8—24no—
220Q8—42no—
221Q8—42no—
222Q8—2442no—
223D4—4882no—
224D4—2444no—
225D4—2442no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—4881no—
235C2×C4—4881no—
236C2×C4—2441no—
237C2×C4#1451974512yes—
238C2×C4#1451974811yescharacteristic
239C2×C4#1451974912yes—
240C2×C4#1451977311yescharacteristic
241C2×C4C2×#1175820112yes—
242C2×C4#1451980812yes—
243C8—4881no—
244C8—2444no—
245C22—22no—
246C22—4882no—
247C22—4881no—
248C22#1683967011yescharacteristic
249C22#1683990112yes—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—4881no—
256C4#1683996412yes—
257C4#1684013611yescharacteristic
258C4#1684015411yescharacteristic
259C2—22no—
260C2—9761no—
261C2#2144688412yes—
262C2#2144690211yescharacteristic
subgroupG/H
centerC2×C4#14519773
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2399C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#2399D61
hypocenterC61#10014
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
3. #21444248C2semidirect
4. #21443648C2semidirect
5. C61×#2399C2semidirect
74. C42×C61SD16semidirect
129. #2399D61semidirect
152. C61#10014semidirect
212. C42#11758201semidirect

Maximal subgroups

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

subgroupindex
#214558502
#214466512
#214442482
#214436482
C61×#23992
#1001461