Finite Group Databasebeta

← #16844147

Subgroups — #16844147

Order 3968.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×#92497C211yesmaximal, characteristic
2#14521071C211yesmaximal, characteristic, complemented
3C4×#92469C211yesmaximal, characteristic, complemented
4#14520781C211yesmaximal, characteristic, complemented
5#14520779C211yesmaximal, characteristic, complemented
6D4⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
7C31×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#92497C411yescharacteristic
9C4×D4×C31C2211yescharacteristic
10D4×Dic31C411yescharacteristic
11C4×C22⋊D31—21no—
12C2×C4×Dic31—21no—
13C2×#92469—21no—
14C2×#92469—21no—
15#11758469C411yescharacteristic
16#11758464C411yescharacteristic
17(C22⋊D31)⋊C4—21no—
18#11758454—21no—
19C42×D31C2211yescharacteristic
20C4×Dic62C2211yescharacteristic
21C31×C4⋊C8C2211yescharacteristic
22C4×C8×C31C2211yescharacteristic
23#11758403—21no—
24Dic31⋊C8—21no—
25C4⋊(C31⋊C8)C2211yescharacteristic
26C4×C31⋊C8C2211yescharacteristic
27C2×D4×C31C2×C411yescharacteristic
28C22×C4×C31—21no—
29C2×C22⋊D31—21no—
30C22×Dic31—21no—
31C22×Dic31—21no—
32#92497—22no—
33#92497—22no—
34C2×C4×D31C2×C411yescharacteristic
35C2×C4×D31D411yescharacteristic
36C2×C4×D31D411yescharacteristic
37C2×Dic62C2×C411yescharacteristic
38C2×C8×C31—21no—
39C2×C8×C31—21no—
40C4⋊C4×C31C2×C411yescharacteristic
41C22⋊C4×C31—21no—
42C42×C31C2311yescharacteristic
43C22⋊Dic31—21no—
44(C2×D31)⋊C4—21no—
45C4⋊Dic31C2×C411yescharacteristic
46Dic31⋊C4—21no—
47Dic31⋊C4—21no—
48C4×Dic31—21no—
49C4×Dic31—21no—
50C4×Dic31—21no—
51C4×Dic31C2×C411yescharacteristic
52C4×Dic31D411yescharacteristic
53C4×Dic31D411yescharacteristic
54C2×C31⋊C8—21no—
55C2×C31⋊C8—21no—
56#92469—24no—
57#92469—42no—
58C23×C31—21no—
59C22×D31C22⋊C411yescharacteristic
60D4×C31—21no—
61D4×C31M1612yescomplemented
62C2×C4×C31—21no—
63C2×C4×C31—41no—
64C2×C4×C31—41no—
65C2×C4×C31C22×C411yescharacteristic
66C2×C4×C31C2×D411yescharacteristic
67C2×C4×C31C2×D411yescharacteristic
68C22⋊D31—24no—
69C2×Dic31—24no—
70C2×Dic31—21no—
71C2×Dic31—21no—
72C2×Dic31—42no—
73C2×Dic31C22⋊C411yescharacteristic
74C2×Dic31C22⋊C411yescharacteristic
75C2×Dic31C22⋊C411yescharacteristic
76C4×D31—21no—
77C4×D31—21no—
78C4×D31—21no—
79C4×D31—21no—
80C4×D31—22no—
81Dic62—21no—
82Dic62M1612yescomplemented
83C8×C31—22no—
84C8×C31—41no—
85C31⋊C8—22no—
86C31⋊C8—41no—
87#1489—311nomaximal, Sylow
88C22×C31—22no—
89C22×C31—41no—
90C22×C31—41no—
91C22×C31C2×C22⋊C411yescharacteristic
92C2×D31—21no—
93C2×D31—21no—
94C2×D31—21no—
95C4×C31—21no—
96C4×C31—41no—
97C4×C31C2×M1611yescharacteristic
98C4×C31C2×M1611yescharacteristic
99C4×C31AGL(1,ℤ/8)11yescharacteristic
100C4×C31#13711yescharacteristic
101Dic31—22no—
102Dic31—21no—
103Dic31—21no—
104Dic31—41no—
105Dic31#10412yes—
106C4×#41—311no—
107#423—311no—
108C4×M16—311no—
109Q8⋊C8—311no—
110Q8⋊C8—311no—
111D4⋊C8—311no—
112D4⋊C8D3111yescharacteristic, complemented
113C2×C31—42no—
114C2×C31#40711yescharacteristic
115C2×C31#41711yescharacteristic
116C2×C31C2×#10411yescharacteristic
117D31—41no—
118C2×#41—311no—
119C2×M16—621no—
120C2×M16—621no—
121C4×Q8—311no—
122C4×Q8—311no—
123C4×D4—311no—
124C4×D4—621no—
125C4×D4—621no—
126C4×D4C2×D3111yescharacteristic
127#117—311no—
128#117—621no—
129C2×C42—311no—
130C2×C42—621no—
131C4⋊C8—311no—
132C4⋊C8—621no—
133C4⋊C8C2×D3111yescharacteristic
134#97—621no—
135C4×C8—311no—
136C4×C8C2×D3111yescharacteristic
137C31#148911yescharacteristic, complemented, Sylow
138#41—622no—
139#41—622no—
140C2×Q8—311no—
141C2×D4—621no—
142C2×D4C4×D3111yescharacteristic
143C22×C4—21no—
144C22×C4—311no—
145C22×C4—311no—
146C22×C4—311no—
147C22×C4—621no—
148C22×C4—621no—
149M16—624no—
150M16—1242no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—621no—
154C2×C8—621no—
155C4⋊C4—311no—
156C4⋊C4—621no—
157C4⋊C4—621no—
158C4⋊C4C4×D3111yescharacteristic
159C22⋊C4—21no—
160C22⋊C4—621no—
161C22⋊C4—621no—
162C42—311no—
163C42—311no—
164C42—311no—
165C42—621no—
166C42—621no—
167C42—621no—
168C42C22×D3111yescharacteristic
169C23—21no—
170C23—311no—
171Q8—312no—
172Q8—621no—
173D4—21no—
174D4—624no—
175D4#9246912yescomplemented
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—311no—
180C2×C4—311no—
181C2×C4—311no—
182C2×C4—622no—
183C2×C4—624no—
184C2×C4—621no—
185C2×C4—621no—
186C2×C4—621no—
187C2×C4—621no—
188C2×C4—621no—
189C2×C4—621no—
190C2×C4—1242no—
191C2×C4C2×C4×D3111yescharacteristic
192C2×C4D4×D3111yescharacteristic
193C2×C4D4×D3111yescharacteristic
194C8—22no—
195C8—41no—
196C8—622no—
197C8—1241no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—621no—
202C22—621no—
203C22—621no—
204C22C22⋊C4×D3111yescharacteristic
205C4—21no—
206C4—41no—
207C4—312no—
208C4—621no—
209C4—622no—
210C4—621no—
211C4—1241no—
212C4C2×#9246911yescharacteristic
213C4M16×D3111yescharacteristic
214C4#1175848811yescharacteristic
215C4#1175849211yescharacteristic
216C2—42no—
217C2—1241no—
218C2#1452101111yescharacteristic
219C2#1452104711yescharacteristic
220C2#104×D3111yescharacteristic
subgroupG/H
centerC2×C4D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC42C22×D31
FittingC31×D4⋊C8C2
socleC22×C31C2×C22⋊C4
prime radicalC2×#92497C4
cosocleC42×C31C23
hypercenterD4⋊C8D31
hypocenterC31#1489
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #14521071C2semidirect
3. C4×#92469C2semidirect
4. #14520781C2semidirect
5. #14520779C2semidirect
6. D4⋊(C31⋊C8)C2semidirect
7. C31×D4⋊C8C2semidirect
61. D4×C31M16semidirect
82. Dic62M16semidirect
112. D4⋊C8D31semidirect
137. C31#1489semidirect
175. D4#92469semidirect

Maximal subgroups

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

subgroupindex
C4×#924972
#145210712
C4×#924692
#145207812
#145207792
D4⋊(C31⋊C8)2
C31×D4⋊C82
#148931