Finite Group Databasebeta

← #16781700

Subgroups — #16781700

Order 3328.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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#13130670C212yesmaximal
2#13130086C214yesmaximal
3C22×#88198C211yesmaximal, characteristic
4C23×Q8×C13C411yescharacteristic
5C23×Dic26C2211yescharacteristic
6C2×#88198—24no—
7C2×#88198—28no—
8C22×#6101C412yes—
9C22×#6101C2212yes—
10C22×C4×Dic13C411yescharacteristic
11#11719169C2214yes—
12C2×#88113—24no—
13#11718286—28no—
14C22×Q8×C13—24no—
15C22×Q8×C13—22no—
16C22×Q8×C13—44no—
17C23×C4×C13C2×C411yescharacteristic
18C23×C4×C13C2×C412yes—
19C23×Dic13C2×C412yes—
20C23×Dic13C2×C411yescharacteristic
21C23×Dic13C2311yescharacteristic
22C22×Dic26—24no—
23C22×Dic26—44no—
24C22×Dic26D414yes—
25C2×#6122—24no—
26#88198—232no—
27C2×#6101—24no—
28C2×#6101—24no—
29C2×#6101—24no—
30C2×#6101—24no—
31C2×#6101—24no—
32C2×#6101D418yes—
33C2×C4×Dic13—22no—
34C2×C4×Dic13—24no—
35C22⋊#1057—28no—
36#88113—48no—
37#8551—131nomaximal, Sylow
38C24×C13C22×C411yescharacteristic
39C2×Q8×C13—28no—
40C2×Q8×C13—416no—
41C2×Q8×C13—44no—
42C2×Q8×C13—44no—
43C22×C4×C13—22no—
44C22×C4×C13—24no—
45C22×C4×C13—22no—
46C22×C4×C13—22no—
47C22×C4×C13—44no—
48C22×C4×C13C22⋊C412yes—
49C22×C4×C13C22⋊C414yes—
50C22×Dic13—22no—
51C22×Dic13—22no—
52C22×Dic13—28no—
53C22×Dic13—22no—
54C22×Dic13—42no—
55C22×Dic13—42no—
56C22×Dic13C22⋊C412yes—
57C22×Dic13C22⋊C414yes—
58C22×Dic13C2×D412yes—
59C22×Dic13C2×D414yes—
60C2×Dic26—28no—
61C2×Dic26—28no—
62C2×Dic26—44no—
63C2×Dic26—416no—
64#6122—48no—
65C2×#1057—44no—
66#6101—216no—
67#6101—216no—
68C4×Dic13—28no—
69C22×C4⋊Q8—131no—
70#1784—132no—
71#1463—134no—
72C23×C13—22no—
73C23×C13—24no—
74C23×C13C2×C22⋊C411yescharacteristic
75C23×C13C2×C22⋊C412yes—
76Q8×C13—416no—
77C2×C4×C13—28no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—24no—
81C2×C4×C13—44no—
82C2×C4×C13—44no—
83C2×C4×C13—44no—
84C2×C4×C13#9914yes—
85C2×Dic13—28no—
86C2×Dic13—28no—
87C2×Dic13—24no—
88C2×Dic13—24no—
89C2×Dic13—24no—
90C2×Dic13—22no—
91C2×Dic13—24no—
92C2×Dic13—44no—
93C2×Dic13—44no—
94C2×Dic13#10118yescomplemented
95C2×Dic13#12014yes—
96Dic26—416no—
97#1057—84no—
98C23×Q8—131no—
99C23×Q8#26111yescharacteristic
100C2×C4⋊Q8—268no—
101C2×C4⋊Q8—264no—
102C22×C4⋊C4—132no—
103C22×C4⋊C4—132no—
104C22×C42—131no—
105C2×#101—264no—
106#407—134no—
107#324—268no—
108C22×C13—28no—
109C22×C13—22no—
110C22×C13—22no—
111C22×C13—22no—
112C22×C13#37911yescharacteristic
113C22×C13C2×#9912yes—
114C22×C13C2×#10114yes—
115C4×C13—24no—
116C4×C13—44no—
117Dic13—24no—
118Dic13—28no—
119Dic13—42no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—134no—
124C22×Q8—264no—
125C22×Q8—524no—
126C23×C4—131no—
127C23×C4—132no—
128C23×C4—131no—
129C23×C4C2×#26111yescharacteristic
130C23×C4C2×#26112yes—
131C2×M16—264no—
132C4⋊Q8—2632no—
133C2×C4⋊C4—138no—
134C2×C4⋊C4—264no—
135C2×C4⋊C4—264no—
136C2×C4⋊C4—264no—
137C2×C4⋊C4—264no—
138C2×C4⋊C4—264no—
139C2×C42—264no—
140C2×C42—262no—
141#101—528no—
142C22⋊C8—268no—
143C2×C13—22no—
144C2×C13—24no—
145C2×C13#178111yescharacteristic
146C2×C13#178412yes—
147C24C22×#26111yescharacteristic
148C2×Q8—28no—
149C2×Q8—416no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—268no—
153C2×Q8—268no—
154C2×Q8—524no—
155C2×Q8—5216no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—44no—
161C22×C4—134no—
162C22×C4—132no—
163C22×C4—134no—
164C22×C4—132no—
165C22×C4—268no—
166C22×C4—262no—
167C22×C4—262no—
168C22×C4—262no—
169C22×C4—522no—
170C22×C4—522no—
171C22×C4C22⋊#26112yes—
172C22×C4C22⋊#26114yes—
173M16—528no—
174C2×C8—524no—
175C4⋊C4—2616no—
176C4⋊C4—2616no—
177C42—268no—
178C13#855111yescharacteristic, complemented, Sylow
179C23—24no—
180C23—22no—
181C23C2×C22⋊#26112yes—
182C23C2×C22⋊#26111yescharacteristic
183Q8—416no—
184Q8—5216no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—24no—
189C2×C4—44no—
190C2×C4—44no—
191C2×C4—44no—
192C2×C4—134no—
193C2×C4—138no—
194C2×C4—268no—
195C2×C4—268no—
196C2×C4—264no—
197C2×C4—264no—
198C2×C4—264no—
199C2×C4—262no—
200C2×C4—264no—
201C2×C4—524no—
202C2×C4—524no—
203C2×C4#8811914yes—
204C8—1044no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22D13⋊#10114yes—
210C22C2×#8811912yes—
211C22#1171917111yescharacteristic
212C4—24no—
213C4—44no—
214C4—264no—
215C4—268no—
216C4—522no—
217C2—24no—
218C2—22no—
219C2#1313066211yescharacteristic
220C2#1313067112yes—
subgroupG/H
centerC22#11719171
commutator, prime radicalC24×C13C22×C4
FrattiniC24C22×#261
FittingC23×Q8×C13C4
socleC22×C13#379
cosocleC23×Dic13C23
hypercenterC23×Q8#261
hypocenterC13#8551
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
94. C2×Dic13#101semidirect
178. C13#8551semidirect

Maximal subgroups

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

subgroupindex
#131306702
#131300862
C22×#881982
#855113