Finite Group Databasebeta

← #30599352

Subgroups — #30599352

Order 15616.

538 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 72 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#21461714C212yesmaximal
2#21460718C214yesmaximal
3#21448416C211yesmaximal, characteristic
4#16840703C2214yes—
5#16840640—22no—
6#16840640—22no—
7C22×#11758198C2211yescharacteristic
8#16839654—22no—
9#16839652—22no—
10#16839622C411yescharacteristic, complemented
11#16839617C2212yes—
12C22⋊C4×Dic61C412yescomplemented
13C61×#380C411yescharacteristic, complemented
14C2×C22⋊#6923—24no—
15C2×C22⋊#6923—24no—
16C2×C22⋊#6923—24no—
17C24×D61C2311yescharacteristic
18C23×Dic61C2×C411yescharacteristic
19C2×C4⋊Dic61—21no—
20C2×C22⋊Dic61C2×C412yes—
21C2×#11758198—24no—
22C2×#11758198—22no—
23C2×#11758198—22no—
24C2×#11758198D412yes—
25C2×#11758198D412yes—
26C2×C4×Dic61—22no—
27#14519684—21no—
28#14519684—22no—
29C23×C4×C61C2×C411yescharacteristic
30C2×C4⋊C4×C61—21no—
31C2×C22⋊C4×C61C2×C412yes—
32#95×C61—21no—
33C24×C61C22×C411yescharacteristic
34C23×D61—41no—
35C23×D61—41no—
36C23×D61—81no—
37C23×D61—81no—
38C23×D61C2×D412yes—
39C23×D61#4114yes—
40C22×#6923—44no—
41C22×C4×C61—22no—
42C22×C4×C61—22no—
43C22×C4×C61—81no—
44C22×C4×C61C22⋊C411yescharacteristic
45C22×C4×C61C22⋊C411yescharacteristic
46C22×Dic61—22no—
47C22×Dic61—22no—
48C22×Dic61—81no—
49C22×Dic61C22⋊C412yes—
50C22⋊#6923—44no—
51C22⋊#6923—44no—
52C22⋊#6923—84no—
53C22⋊#6923—84no—
54C4⋊C4×C61—81no—
55C22⋊C4×C61—42no—
56C22⋊Dic61—42no—
57#11758198—24no—
58#11758198—24no—
59#11758198—82no—
60#11758198—82no—
61C4⋊Dic61—81no—
62C4×Dic61—82no—
63C23×C61—21no—
64C23×C61—21no—
65C23×C61—41no—
66C23×C61—41no—
67C23×C61C2×C22⋊C411yescharacteristic
68C23×C61#11712yes—
69C22×D61—22no—
70C22×D61—22no—
71C22×D61—42no—
72C22×D61—42no—
73C22×D61—44no—
74C22×D61—44no—
75C22×D61—44no—
76C22×D61—84no—
77C22×D61—84no—
78C22×D61#12314yes—
79C2×#6923—84no—
80C2×#6923—84no—
81C2×#6923—84no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—41no—
85C2×C4×C61—41no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61—82no—
90C2×Dic61—22no—
91C2×Dic61—42no—
92C2×Dic61—42no—
93C2×Dic61—42no—
94C2×Dic61—42no—
95C2×Dic61—82no—
96#8750—611nomaximal, Sylow
97C22×C61—21no—
98C22×C61—21no—
99C22×C61—22no—
100C22×C61—22no—
101C22×C61—22no—
102C22×C61—42no—
103C22×C61—42no—
104C22×C61#38111yescharacteristic
105C22×C61#41011yescharacteristic
106C22×C61#41011yescharacteristic
107C2×D61—24no—
108C2×D61—24no—
109C2×D61—24no—
110C2×D61—82no—
111C2×D61—82no—
112C2×D61—88no—
113#6923—164no—
114C4×C61—41no—
115C4×C61—41no—
116C4×C61—82no—
117Dic61—42no—
118Dic61—82no—
119#2312—611no—
120#1819—612no—
121#1304—614no—
122C2×C61—21no—
123C2×C61—21no—
124C2×C61—42no—
125C2×C61#182011yescharacteristic
126C2×C61#182811yescharacteristic
127C2×C61#182811yescharacteristic
128D61—44no—
129C22×C22⋊C4—611no—
130#388—1222no—
131#388—1222no—
132#380—611no—
133#380#692311yescharacteristic, complemented
134#379—612no—
135#379—614no—
136C4×C22⋊C4—612no—
137#342—1222no—
138#342—1222no—
139C61#875011yescharacteristic, complemented, Sylow
140C25—611no—
141C23×C4—611no—
142C23×C4C2×#692311yescharacteristic
143C2×C4⋊C4—21no—
144C2×C4⋊C4—1221no—
145C2×C22⋊C4—612no—
146C2×C22⋊C4—612no—
147C2×C22⋊C4—612no—
148C2×C22⋊C4—1224no—
149C2×C22⋊C4—1222no—
150C2×C22⋊C4—1222no—
151C2×C22⋊C4—1224no—
152C2×C22⋊C4—1224no—
153C2×C22⋊C4—1224no—
154C2×C22⋊C4C2×#692312yes—
155C2×C42—1222no—
156#95—21no—
157#95—1222no—
158#95—1221no—
159C24—612no—
160C24—614no—
161C24—4881no—
162C24—4881no—
163C24—2441no—
164C24—2441no—
165C24C22×#692311yescharacteristic
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—81no—
169C22×C4—612no—
170C22×C4—1222no—
171C22×C4—1222no—
172C22×C4—4881no—
173C22×C4—2444no—
174C22×C4C22⋊#692311yescharacteristic
175C22×C4C22⋊#692311yescharacteristic
176C4⋊C4—81no—
177C4⋊C4—4881no—
178C22⋊C4—42no—
179C22⋊C4—1224no—
180C22⋊C4—1224no—
181C22⋊C4—4882no—
182C22⋊C4—4882no—
183C22⋊C4—4884no—
184C22⋊C4—4884no—
185C22⋊C4—2444no—
186C22⋊C4—2444no—
187C22⋊C4—2442no—
188C42—4882no—
189C23—21no—
190C23—21no—
191C23—41no—
192C23—41no—
193C23—614no—
194C23—1222no—
195C23—1222no—
196C23—4884no—
197C23—4884no—
198C23—2442no—
199C23—2442no—
200C23—2444no—
201C23—2444no—
202C23—2444no—
203C23C2×C22⋊#692311yescharacteristic
204C23#1451985612yes—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—1222no—
214C2×C4—4884no—
215C2×C4—4884no—
216C2×C4—4884no—
217C2×C4—4882no—
218C2×C4—2442no—
219C2×C4—2442no—
220C2×C4—2442no—
221C2×C4—2442no—
222C22—21no—
223C22—21no—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—1224no—
230C22—1224no—
231C22—1224no—
232C22—4882no—
233C22—4882no—
234C22—4888no—
235C22#1684070511yescharacteristic
236C22#1684074211yescharacteristic
237C22#1684074311yescharacteristic
238C4—41no—
239C4—41no—
240C4—82no—
241C4—9764no—
242C4—4882no—
243C4—2442no—
244C2—21no—
245C2—21no—
246C2—42no—
247C2—2444no—
248C2#2146171711yescharacteristic
249C2#2146173011yescharacteristic
250C2#2146173111yescharacteristic
subgroupG/H
centerC22#16840705
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC61×#380C4
socleC22×C61#381
prime radical, cosocleC24×D61C23
hypercenter#380#6923
hypocenterC61#8750
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ #380

Complements

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

NH ≅ G/Ntype
10. #16839622C4semidirect
12. C22⋊C4×Dic61C4semidirect
13. C61×#380C4semidirect
133. #380#6923semidirect
139. C61#8750semidirect

Maximal subgroups

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

subgroupindex
#214617142
#214607182
#214484162
#875061