Finite Group Databasebeta

← #16844156

Subgroups — #16844156

Order 3968.

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

normal subgroups only

This group has 252 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×Q8×D31C211yesmaximal, characteristic
2#14521105C211yesmaximal, characteristic
3C4×#92469C211yesmaximal, characteristic
4#14520781C211yesmaximal, characteristic, complemented
5Q8⋊(C31⋊C8)C211yesmaximal, characteristic, complemented
6#14520779C211yesmaximal, characteristic, complemented
7C31×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D31C411yescharacteristic
9C4×Q8×C31C2211yescharacteristic
10Q8×Dic31C411yescharacteristic
11C2×#92469—21no—
12C4⋊C4×D31—21no—
13C4⋊C4×D31C411yescharacteristic
14#11758464—21no—
15#11758464C411yescharacteristic
16C42×D31—21no—
17C42×D31C2211yescharacteristic
18C4×Dic62—21no—
19C4×Dic62C2211yescharacteristic
20C31×C4⋊C8C2211yescharacteristic
21C4×C8×C31C2211yescharacteristic
22(C2×D31)⋊C8—21no—
23#11758403—21no—
24#11758403—21no—
25C4⋊(C31⋊C8)C2211yescharacteristic
26C4×C31⋊C8C2211yescharacteristic
27C2×Q8×C31C2×C411yescharacteristic
28Q8×D31—22no—
29Q8×D31—22no—
30C2×C4×D31—21no—
31C2×C4×D31—21no—
32C2×C4×D31C2×C411yescharacteristic
33C2×C4×D31D411yescharacteristic
34C2×C4×D31D411yescharacteristic
35C2×Dic62—21no—
36C2×Dic62C2×C411yescharacteristic
37C2×C8×C31—21no—
38C2×C8×C31—21no—
39C4⋊C4×C31—21no—
40C4⋊C4×C31C2×C411yescharacteristic
41C42×C31—21no—
42C42×C31C2311yescharacteristic
43C4⋊Dic31—21no—
44C4⋊Dic31C2×C411yescharacteristic
45Dic31⋊C4—21no—
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—
57C22×D31C22⋊C411yescharacteristic
58Q8×C31—21no—
59Q8×C31M1612yescomplemented
60C2×C4×C31—21no—
61C2×C4×C31—21no—
62C2×C4×C31C22×C411yescharacteristic
63C2×C4×C31C2×D411yescharacteristic
64C2×C4×C31C2×D411yescharacteristic
65C2×Dic31—21no—
66C2×Dic31—21no—
67C2×Dic31C22⋊C411yescharacteristic
68C2×Dic31C22⋊C411yescharacteristic
69C2×Dic31C22⋊C411yescharacteristic
70C4×D31—24no—
71C4×D31—22no—
72C4×D31—21no—
73C4×D31—21no—
74C4×D31—21no—
75C4×D31—21no—
76C4×D31—42no—
77Dic62—24no—
78Dic62—21no—
79Dic62M1612yescomplemented
80C8×C31—22no—
81C8×C31—41no—
82C31⋊C8—22no—
83C31⋊C8—41no—
84#1491—311nomaximal, Sylow
85C22×C31C2×C22⋊C411yescharacteristic
86C2×D31—21no—
87C2×D31—21no—
88C2×D31#10412yes—
89C4×C31—22no—
90C4×C31—21no—
91C4×C31—41no—
92C4×C31C2×M1611yescharacteristic
93C4×C31C2×M1611yescharacteristic
94C4×C31#13711yescharacteristic
95C4×C31#13711yescharacteristic
96Dic31—21no—
97Dic31—22no—
98Dic31—21no—
99Dic31—21no—
100Dic31—41no—
101C2×C4×Q8—311no—
102#432—311no—
103C4×M16—311no—
104Q8⋊C8—311no—
105Q8⋊C8—311no—
106Q8⋊C8—311no—
107Q8⋊C8D3111yescharacteristic, complemented
108C2×C31#40711yescharacteristic
109C2×C31#41911yescharacteristic
110C2×C31C2×#10411yescharacteristic
111D31—22no—
112C22×Q8—311no—
113C2×M16—621no—
114C4×Q8—311no—
115C4×Q8—311no—
116C4×Q8—311no—
117C4×Q8—621no—
118C4×Q8—621no—
119C4×Q8C2×D3111yescharacteristic
120C2×C4⋊C4—311no—
121C2×C4⋊C4—621no—
122C2×C42—311no—
123C2×C42—621no—
124C4⋊C8—311no—
125C4⋊C8C2×D3111yescharacteristic
126C22⋊C8—621no—
127#97—621no—
128#97—621no—
129C4×C8—311no—
130C4×C8C2×D3111yescharacteristic
131C31#149111yescharacteristic, complemented, Sylow
132C2×Q8—311no—
133C2×Q8—621no—
134C2×Q8—622no—
135C2×Q8—622no—
136C2×Q8C4×D3111yescharacteristic
137C22×C4—311no—
138C22×C4—311no—
139C22×C4—311no—
140C22×C4—621no—
141C22×C4—621no—
142M16—624no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—621no—
146C2×C8—621no—
147C4⋊C4—21no—
148C4⋊C4—311no—
149C4⋊C4—621no—
150C4⋊C4—621no—
151C4⋊C4—621no—
152C4⋊C4—621no—
153C4⋊C4C4×D3111yescharacteristic
154C42—21no—
155C42—311no—
156C42—311no—
157C42—311no—
158C42—621no—
159C42—621no—
160C42—621no—
161C42C22×D3111yescharacteristic
162C23—311no—
163Q8—21no—
164Q8—312no—
165Q8—624no—
166Q8—621no—
167Q8#9246912yescomplemented
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—311no—
171C2×C4—311no—
172C2×C4—311no—
173C2×C4—622no—
174C2×C4—624no—
175C2×C4—621no—
176C2×C4—621no—
177C2×C4—621no—
178C2×C4—621no—
179C2×C4—621no—
180C2×C4—621no—
181C2×C4—1242no—
182C2×C4C2×C4×D3111yescharacteristic
183C2×C4D4×D3111yescharacteristic
184C2×C4D4×D3111yescharacteristic
185C8—22no—
186C8—41no—
187C8—622no—
188C8—1241no—
189C22—312no—
190C22—621no—
191C22—621no—
192C22C22⋊C4×D3111yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—621no—
197C4—622no—
198C4—621no—
199C4—621no—
200C4—1241no—
201C4C2×#9246911yescharacteristic
202C4M16×D3111yescharacteristic
203C4#1175849211yescharacteristic
204C4#1175849511yescharacteristic
205C2—622no—
206C2#1452101111yescharacteristic
207C2#1452105611yescharacteristic
208C2#104×D3111yescharacteristic
subgroupG/H
centerC2×C4D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC42C22×D31
FittingC31×Q8⋊C8C2
socleC22×C31C2×C22⋊C4
prime radicalC22×D31C22⋊C4
cosocleC42×C31C23
hypercenterQ8⋊C8D31
hypocenterC31#1491
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #14520781C2semidirect
5. Q8⋊(C31⋊C8)C2semidirect
6. #14520779C2semidirect
7. C31×Q8⋊C8C2semidirect
59. Q8×C31M16semidirect
79. Dic62M16semidirect
107. Q8⋊C8D31semidirect
131. C31#1491semidirect
167. Q8#92469semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D312
#145211052
C4×#924692
#145207812
Q8⋊(C31⋊C8)2
#145207792
C31×Q8⋊C82
#149131