Finite Group Databasebeta

← #30598144

Subgroups — #30598144

Order 15616.

810 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 132 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#21461710C212yesmaximal
2C2×#16840608C214yesmaximal
3C22×#14519768C211yesmaximal, characteristic
4C2×C22⋊#92420C2214yes—
5#16840628—24no—
6#16840608—28no—
7C2×#14519768—28no—
8C2×#14519768—24no—
9C22×C4⋊Dic61C412yes—
10C22×C4⋊Dic61C411yescharacteristic
11C22×#11758196C2212yes—
12C22×C4×Dic61C2211yescharacteristic
13C22×C4⋊C4×C61C411yescharacteristic
14C22×#92420—24no—
15C22⋊#92420—28no—
16C23×Dic61C2×C411yescharacteristic
17C23×Dic61C2×C412yes—
18C23×Dic61C2311yescharacteristic
19#14519768—232no—
20C2×C4⋊Dic61—24no—
21C2×C4⋊Dic61—22no—
22C2×C4⋊Dic61—24no—
23C2×C4⋊Dic61—24no—
24C2×C4⋊Dic61—24no—
25C2×#11758196—24no—
26C2×#11758196—24no—
27C2×#11758196D418yes—
28C2×C4×Dic61—24no—
29C2×C4×Dic61D414yes—
30C23×C4×C61C2×C412yes—
31C23×C4×C61C2×C411yescharacteristic
32C2×C4⋊C4×C61—24no—
33C2×C4⋊C4×C61—22no—
34C24×C61C22×C411yescharacteristic
35C22×C4×C61—22no—
36C22×C4×C61—24no—
37C22×C4×C61—22no—
38C22×C4×C61—22no—
39C22×C4×C61—44no—
40C22×C4×C61C22⋊C412yes—
41C22×C4×C61C22⋊C414yes—
42C22×Dic61—22no—
43C22×Dic61—22no—
44C22×Dic61—22no—
45C22×Dic61—24no—
46C22×Dic61—28no—
47C22×Dic61—42no—
48C22×Dic61C22⋊C412yes—
49C22×Dic61C22⋊C414yes—
50C22×Dic61C2×D412yes—
51C22×Dic61C2×D414yes—
52C2×#92420—48no—
53C2×#92420—44no—
54C4⋊C4×C61—28no—
55C4⋊Dic61—216no—
56C4⋊Dic61—28no—
57#11758196—216no—
58C4×Dic61—28no—
59C23×C61—22no—
60C23×C61—22no—
61C23×C61C2×C22⋊C412yes—
62C23×C61C2×C22⋊C411yescharacteristic
63C23×C61#13114yes—
64C2×C4×C61—28no—
65C2×C4×C61—22no—
66C2×C4×C61—22no—
67C2×C4×C61—24no—
68C2×C4×C61—44no—
69C2×C4×C61—44no—
70C2×C4×C61—44no—
71C2×C4×C61#10114yes—
72C2×Dic61—24no—
73C2×Dic61—22no—
74C2×Dic61—24no—
75C2×Dic61—28no—
76C2×Dic61—24no—
77C2×Dic61—44no—
78C2×Dic61—44no—
79C2×Dic61—44no—
80C2×Dic61—44no—
81C2×Dic61#12014yes—
82#92420—84no—
83#8306—611nomaximal, Sylow
84C22×C61—28no—
85C22×C61—22no—
86C22×C61—22no—
87C22×C61#37911yescharacteristic
88C22×C61#40814yes—
89C22×C61#40812yes—
90C22×C61#41012yes—
91C22×C61C2×#10112yes—
92C4×C61—24no—
93C4×C61—44no—
94Dic61—24no—
95Dic61—42no—
96Dic61—44no—
97C22×#125—611no—
98#1817—612no—
99C2×#324—614no—
100C2×C61—24no—
101C2×C61#147014yes—
102C2×C61#178011yescharacteristic
103C2×C61#178211yescharacteristic
104C2×C61#178411yescharacteristic
105C2×#125—1224no—
106C2×#125—1228no—
107C22×C4⋊C4—612no—
108C22×C4⋊C4—612no—
109C22×C4⋊C4—611no—
110C22×C4⋊C4#692311yescharacteristic
111C22×C42—611no—
112C2×C22⋊C8—614no—
113#336—1224no—
114#324—1228no—
115C61#830611yescharacteristic, complemented, Sylow
116C23×C4—611no—
117C23×C4—612no—
118C23×C4—611no—
119C23×C4C2×#692312yes—
120C23×C4C2×#692311yescharacteristic
121C22×C8—1224no—
122#125—12232no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—618no—
126C2×C4⋊C4—1224no—
127C2×C4⋊C4—1224no—
128C2×C4⋊C4—1224no—
129C2×C4⋊C4—1224no—
130C2×C4⋊C4—1224no—
131C2×C4⋊C4—1224no—
132C2×C4⋊C4—1222no—
133C2×C42—614no—
134C2×C42—1224no—
135C22⋊C8—1228no—
136C24C22×#692311yescharacteristic
137C22×C4—22no—
138C22×C4—24no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—44no—
142C22×C4—612no—
143C22×C4—614no—
144C22×C4—614no—
145C22×C4—612no—
146C22×C4—1222no—
147C22×C4—1222no—
148C22×C4—1222no—
149C22×C4—1228no—
150C22×C4—1224no—
151C22×C4—2442no—
152C22×C4C22⋊#692312yes—
153C22×C4C22⋊#692314yes—
154C2×C8—2444no—
155C2×C8—2448no—
156C4⋊C4—28no—
157C4⋊C4—12216no—
158C4⋊C4—1228no—
159C4⋊C4—12216no—
160C42—1228no—
161C23—22no—
162C23—22no—
163C23C2×C22⋊#692312yes—
164C23C2×C22⋊#692311yescharacteristic
165C23#1451986514yes—
166C2×C4—28no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—44no—
171C2×C4—44no—
172C2×C4—44no—
173C2×C4—614no—
174C2×C4—1224no—
175C2×C4—1228no—
176C2×C4—1224no—
177C2×C4—1222no—
178C2×C4—1224no—
179C2×C4—2444no—
180C2×C4—2444no—
181C2×C4—2444no—
182C2×C4—2444no—
183C2×C4#1451983814yes—
184C8—4884no—
185C22—28no—
186C22—22no—
187C22—22no—
188C22#1684070311yescharacteristic
189C22#1684073714yes—
190C22#1684073712yes—
191C22#1684074212yes—
192C22C2×#1451983812yes—
193C4—24no—
194C4—44no—
195C4—1224no—
196C4—2442no—
197C4—2444no—
198C2—24no—
199C2#2146109314yes—
200C2#2146164411yescharacteristic
201C2#2146164911yescharacteristic
202C2#2146165311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutator, prime radicalC24×C61C22×C4
FrattiniC24C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
cosocleC23×Dic61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8306
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
115. C61#8306semidirect

Maximal subgroups

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

subgroupindex
#214617102
C2×#168406082
C22×#145197682
#830661