Finite Group Databasebeta

← #30601917

Subgroups — #30601917

Order 15616.

478 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 84 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#21461741C211yesmaximal, characteristic, complemented
2#21461250C212yesmaximal, complemented
3#21461248C212yesmaximal, complemented
4#21461033C211yesmaximal, characteristic, complemented
5#21456242C211yesmaximal, characteristic
6#16840780C2212yes—
7#16840706C2212yes—
8#16840636—22no—
9#16840614—24no—
10#16840613—24no—
11#92420⋊C8—22no—
12C2×#14519798—22no—
13#16840313C2211yescharacteristic
14#16840310C2211yescharacteristic
15#16840143—22no—
16#16840141C411yescharacteristic, complemented
17#16840138C411yescharacteristic, complemented
18#16840135C411yescharacteristic, complemented
19C4×C8×D61C2211yescharacteristic
20C61×#492C411yescharacteristic, complemented
21C2×C4⋊#6923—22no—
22#14519844—22no—
23#14519834D414yes—
24C4×#92420—22no—
25(C2×#6923)⋊C4—22no—
26(C2×#6923)⋊C4—22no—
27C2×#11758226—21no—
28C2×#11758224—21no—
29#14519798—48no—
30C2×#11758202—22no—
31C2×#11758190—22no—
32C2×#11758199—22no—
33C2×C8×D61D412yes—
34#14519777C2×C411yescharacteristic
35#14519774C2×C411yescharacteristic
36#14519766—21no—
37#14519764—21no—
38#14519752—22no—
39#14519748—22no—
40D4×Dic61—22no—
41#14519739—22no—
42C42×D61C2311yescharacteristic
43C4×#92418C2×C411yescharacteristic
44C8×Dic61D412yes—
45C2×SD16×C61—22no—
46C4⋊Q8×C61C2×C411yescharacteristic
47#127×C61C2×C411yescharacteristic
48C4×C8×C61C2×C411yescharacteristic
49C22×#6923—42no—
50C2×Q8×C61—21no—
51C2×D4×C61—21no—
52C2×D4×C61—22no—
53C2×C22⋊D61—41no—
54C22×Dic61—41no—
55#11758226—44no—
56#11758224—44no—
57C2×D244—21no—
58C2×D244—22no—
59C2×D244—41no—
60C2×C4×D61—41no—
61C2×C4×D61C2×D412yes—
62C2×C4×D61C2×D411yescharacteristic
63C2×Dic122—21no—
64C2×#92420—24no—
65C4⋊#6923—44no—
66SD16×C61—44no—
67C2×C8×C61C22⋊C412yes—
68C4⋊C4×C61—22no—
69C42×C61C22×C411yescharacteristic
70C22⋊Dic61—42no—
71#11758202—44no—
72#11758199—44no—
73#11758198—42no—
74C4⋊Dic61—22no—
75C4×Dic61C2×D411yescharacteristic
76C4×Dic61#4112yes—
77C2×#92418C22⋊C412yes—
78#11758190—44no—
79C8×D61—24no—
80C23×C61—41no—
81C22×D61—41no—
82C22×D61#12711yescharacteristic
83C2×#6923—84no—
84C2×#6923—82no—
85Q8×C61—42no—
86D4×C61—42no—
87D4×C61—82no—
88C2×C4×C61—41no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C2×C4×C61C4×D412yes—
91C22⋊D61—82no—
92C2×Dic61—41no—
93C2×Dic61—82no—
94C2×Dic61#12112yes—
95C2×Dic61#12411yescharacteristic
96D244—42no—
97D244—82no—
98D244—82no—
99C4×D61—42no—
100C4×D61—82no—
101C4×D61AGL(1,ℤ/8)14yes—
102Dic122—42no—
103#92420—44no—
104C8×C61—22no—
105#92418—22no—
106#9760—611nomaximal, Sylow
107C22×C61—81no—
108C22×C61—82no—
109C22×C61#39011yescharacteristic
110C2×D61—22no—
111C2×D61—81no—
112C2×D61—82no—
113C2×D61#49612yes—
114#6923—162no—
115C4×C61—22no—
116C4×C61—81no—
117C4×C61#41812yes—
118Dic61—22no—
119Dic61—81no—
120Dic61#49012yes—
121#3147—611no—
122#1835—611no—
123#1517—612no—
124#1517—612no—
125#1452—611no—
126C2×C61—161no—
127C2×C61#196111yescharacteristic
128C2×C61#197612yes—
129D61—22no—
130D61—161no—
131C2×#135—1222no—
132#532—611no—
133#532—611no—
134#495—1222no—
135#494—611no—
136#493—611no—
137#492—611no—
138#492#692311yescharacteristic, complemented
139#431—612no—
140C2×C4×C8—611no—
141#382—612no—
142#340—1222no—
143#327—1224no—
144#327—1224no—
145#322—1222no—
146C61#976011yescharacteristic, complemented, Sylow
147C2×#41—1221no—
148C2×#41—1221no—
149#135—2448no—
150C2×Q16—1222no—
151C2×SD16—22no—
152C2×SD16—1222no—
153C2×D8—1222no—
154C22×C8—612no—
155C4⋊Q8—611no—
156C4⋊Q8C2×#692311yescharacteristic
157#127—611no—
158#127C2×#692311yescharacteristic
159#124—1221no—
160#124—1221no—
161#121—1222no—
162#121—1222no—
163C4×D4—1222no—
164C4×D4—1222no—
165C2×C4⋊C4—1222no—
166C2×C42—611no—
167C4⋊C8—1222no—
168C22⋊C8—614no—
169C4×C8—612no—
170C4×C8—611no—
171C4×C8—1222no—
172C4×C8C2×#692311yescharacteristic
173#95—1222no—
174#95—1222no—
175#41—2444no—
176#41—2444no—
177C2×Q8—21no—
178C2×Q8—1221no—
179C2×D4—21no—
180C2×D4—22no—
181C2×D4—1221no—
182C2×D4—1222no—
183C2×D4—2441no—
184C2×D4—2441no—
185C22×C4—611no—
186C22×C4—612no—
187C22×C4—2441no—
188C22×C4—2441no—
189C22×C4—2442no—
190Q16—2444no—
191SD16—44no—
192SD16—2444no—
193D8—2444no—
194C2×C8—612no—
195C2×C8—1224no—
196C2×C8—1224no—
197C2×C8C22⋊#692312yes—
198C4⋊C4—22no—
199C4⋊C4—1222no—
200C4⋊C4—2444no—
201C22⋊C4—2442no—
202C22⋊C4—2442no—
203C42—611no—
204C42—612no—
205C42C22×#692311yescharacteristic
206C23—41no—
207C23—611no—
208C23—2441no—
209Q8—42no—
210Q8—2442no—
211D4—42no—
212D4—82no—
213D4—4882no—
214D4—4882no—
215D4—4882no—
216D4—2442no—
217C2×C4—41no—
218C2×C4—611no—
219C2×C4—614no—
220C2×C4—612no—
221C2×C4—4882no—
222C2×C4—4882no—
223C2×C4—4882no—
224C2×C4—4884no—
225C2×C4—2442no—
226C2×C4—2441no—
227C2×C4C2×C22⋊#692311yescharacteristic
228C2×C4D4×#692312yes—
229C8—22no—
230C8—1222no—
231C8—2444no—
232C22—81no—
233C22—82no—
234C22—612no—
235C22—1222no—
236C22—4881no—
237C22—4882no—
238C22#1684071411yescharacteristic
239C4—22no—
240C4—81no—
241C4—612no—
242C4—1222no—
243C4—9762no—
244C4—4881no—
245C4#1684075412yes—
246C2—161no—
247C2—1222no—
248C2—9761no—
249C2#2146195111yescharacteristic
250C2#2146197712yes—
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#492C4
socleC22×C61#390
prime radical#21456242C2
cosocleC42×D61C23
hypercenter#492#6923
hypocenterC61#9760
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #21461741C2semidirect
2. #21461250C2semidirect
3. #21461248C2semidirect
4. #21461033C2semidirect
16. #16840141C4semidirect
17. #16840138C4semidirect
18. #16840135C4semidirect
20. C61×#492C4semidirect
138. #492#6923semidirect
146. C61#9760semidirect

Maximal subgroups

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

subgroupindex
#214617412
#214612502
#214612482
#214610332
#214562422
#976061