Finite Group Databasebeta

← #16846008

Subgroups — #16846008

Order 3968.

408 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×C4×C22⋊D31C211yesmaximal, characteristic
2C2×C22⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
3C2×(C2×D31)⋊C8C212yesmaximal, complemented
4#14520803C212yesmaximal, complemented
5C22×M16×C31C211yesmaximal, characteristic, complemented
6C23×C4×C31C2211yescharacteristic
7C22×C22⋊D31C411yescharacteristic
8C22×C4×D31C2211yescharacteristic
9C2×M16×C31—22no—
10C2×M16×C31—24no—
11C22×C8×C31C2212yes—
12C2×C22⋊Dic31C411yescharacteristic
13C4×C22⋊D31—22no—
14C4×C22⋊D31—22no—
15C2×(C2×D31)⋊C4C411yescharacteristic
16C2×Dic31⋊C4C411yescharacteristic
17C2×C4×Dic31C2211yescharacteristic
18C22×C31⋊C8C2212yes—
19C22⋊(C31⋊C8)—22no—
20(C2×D31)⋊C8—24no—
21C24×C31C2×C411yescharacteristic
22C23×D31C2×C411yescharacteristic
23C22×C4×C31—22no—
24C22×C4×C31—22no—
25C22×C4×C31C2×C411yescharacteristic
26C22×C4×C31D412yes—
27C22×C4×C31D412yes—
28C22×C4×C31C2311yescharacteristic
29C2×C22⋊D31—22no—
30C2×C22⋊D31—22no—
31C2×C22⋊D31—22no—
32C22×Dic31C2×C411yescharacteristic
33C22×Dic31C2×C411yescharacteristic
34C22×Dic31C2×C411yescharacteristic
35C2×C4×D31—22no—
36C2×C4×D31—21no—
37C2×C4×D31—21no—
38C2×C4×D31—22no—
39M16×C31—28no—
40C2×C8×C31—22no—
41C2×C8×C31—22no—
42C2×C8×C31D414yescomplemented
43C22⋊Dic31—22no—
44(C2×D31)⋊C4—22no—
45Dic31⋊C4—22no—
46C4×Dic31—21no—
47C4×Dic31—21no—
48C2×C31⋊C8—22no—
49C2×C31⋊C8—22no—
50C2×C31⋊C8—22no—
51C23×C31—22no—
52C23×C31—21no—
53C23×C31—21no—
54C23×C31—22no—
55C23×C31C22⋊C412yes—
56C23×C31C22×C411yescharacteristic
57C22×D31—21no—
58C22×D31—22no—
59C22×D31—21no—
60C22×D31—22no—
61C22×D31M1612yescomplemented
62C2×C4×C31—22no—
63C2×C4×C31—22no—
64C2×C4×C31—24no—
65C2×C4×C31—22no—
66C2×C4×C31C22⋊C412yes—
67C2×C4×C31C2×D411yescharacteristic
68C2×C4×C31C2×D412yes—
69C2×C4×C31C2×D411yescharacteristic
70C2×C4×C31#4112yes—
71C22⋊D31—28no—
72C2×Dic31—21no—
73C2×Dic31—21no—
74C2×Dic31—21no—
75C2×Dic31—21no—
76C2×Dic31—22no—
77C2×Dic31—22no—
78C2×Dic31M1612yescomplemented
79C4×D31—41no—
80C4×D31—42no—
81C4×D31—41no—
82C8×C31—24no—
83C31⋊C8—42no—
84#1870—311nomaximal, Sylow
85C22×C31—22no—
86C22×C31—22no—
87C22×C31—22no—
88C22×C31—42no—
89C22×C31—42no—
90C22×C31C2×C22⋊C411yescharacteristic
91C22×C31C4×D412yes—
92C22×C31C2×M1611yescharacteristic
93C22×C31C2×M1611yescharacteristic
94C22×C31#13112yes—
95C2×D31—22no—
96C2×D31—24no—
97C2×D31—41no—
98C2×D31—41no—
99C2×D31—42no—
100C4×C31—22no—
101C4×C31#12011yescharacteristic
102C4×C31#12112yes—
103C4×C31#12311yescharacteristic
104Dic31—22no—
105Dic31—41no—
106Dic31—41no—
107C22×M16D3111yescharacteristic, complemented
108C2×C4×D4—311no—
109C2×C22⋊C8—311no—
110C2×C22⋊C8—312no—
111#336—312no—
112C2×C31—42no—
113C2×C31#38611yescharacteristic
114C2×C31#40711yescharacteristic
115C2×C31#40811yescharacteristic
116C2×C31#43512yes—
117C2×C31#43612yes—
118D31—42no—
119C22×D4—311no—
120C23×C4—311no—
121C23×C4C2×D3111yescharacteristic
122C2×M16—22no—
123C2×M16—24no—
124C22×C8—312no—
125C22×C8C2×D3112yes—
126C4×D4—622no—
127C4×D4—622no—
128C2×C4⋊C4—311no—
129C2×C22⋊C4—311no—
130C2×C22⋊C4—311no—
131C2×C42—311no—
132C22⋊C8—622no—
133C22⋊C8—624no—
134C31#187011yescharacteristic, complemented, Sylow
135C24—311no—
136C24C4×D3111yescharacteristic
137C2×D4—622no—
138C2×D4—622no—
139C2×D4—622no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—311no—
143C22×C4—311no—
144C22×C4—311no—
145C22×C4—621no—
146C22×C4—622no—
147C22×C4—622no—
148C22×C4—621no—
149C22×C4C4×D3111yescharacteristic
150C22×C4D12412yes—
151C22×C4C22⋊D3112yes—
152C22×C4C22×D3111yescharacteristic
153M16—28no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—622no—
157C2×C8—622no—
158C2×C8—622no—
159C2×C8C22⋊D3114yescomplemented
160C4⋊C4—622no—
161C22⋊C4—622no—
162C22⋊C4—622no—
163C42—621no—
164C42—621no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—22no—
169C23—312no—
170C23—621no—
171C23—622no—
172C23—621no—
173C23—622no—
174C23(C2×D31)⋊C412yes—
175C23C2×C4×D3111yescharacteristic
176D4—628no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—312no—
182C2×C4—621no—
183C2×C4—621no—
184C2×C4—622no—
185C2×C4—621no—
186C2×C4—621no—
187C2×C4—622no—
188C2×C4—1241no—
189C2×C4—1242no—
190C2×C4—1241no—
191C2×C4(C2×D31)⋊C412yes—
192C2×C4C2×D12411yescharacteristic
193C2×C4#9249512yes—
194C2×C4C2×C22⋊D3111yescharacteristic
195C2×C4C2×C22⋊D3112yes—
196C8—24no—
197C8—1242no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—624no—
204C22—622no—
205C22—1242no—
206C22—1241no—
207C22—1241no—
208C22M16×D3111yescharacteristic
209C22M16×D3111yescharacteristic
210C22#1175848412yes—
211C22C2×(C2×D31)⋊C411yescharacteristic
212C22C4×C22⋊D3112yes—
213C4—22no—
214C4—622no—
215C4—1241no—
216C4—1241no—
217C4#1175850611yescharacteristic
218C4#1175850712yes—
219C4#1175853011yescharacteristic
220C2—42no—
221C2—1242no—
222C2#1452092811yescharacteristic
223C2#1452101211yescharacteristic
224C2#1452101711yescharacteristic
225C2#1452111812yes—
226C2#1452112512yes—
subgroupG/H
center, FrattiniC22×C4C22×D31
commutator, socleC23×C31C22×C4
FittingC22×M16×C31C2
prime radicalC22×C22⋊D31C4
cosocleC22×C4×C31C23
hypercenterC22×M16D31
hypocenterC31#1870
LCSG ▷ C23×C31 ▷ C31
UCSC1 ◁ C22×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C31⋊C8)C2semidirect
3. C2×(C2×D31)⋊C8C2semidirect
4. #14520803C2semidirect
5. C22×M16×C31C2semidirect
42. C2×C8×C31D4semidirect
61. C22×D31M16semidirect
78. C2×Dic31M16semidirect
107. C22×M16D31semidirect
134. C31#1870semidirect
159. C2×C8C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D312
C2×C22⋊(C31⋊C8)2
C2×(C2×D31)⋊C82
#145208032
C22×M16×C312
#187031