Finite Group Databasebeta

← #16856101

Subgroups — #16856101

Order 3968.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521597C211yesmaximal, characteristic
2C4⋊Q8×D31C211yesmaximal, characteristic, complemented
3#14521407C214yesmaximal, complemented
4#14521401C211yesmaximal, characteristic, complemented
5#14521400C211yesmaximal, characteristic, complemented
6#14521395C211yesmaximal, characteristic, complemented
7#14521047C214yesmaximal, complemented
8C4×#92469C211yesmaximal, characteristic, complemented
9C31×#486C211yesmaximal, characteristic, complemented
10C2×Q8×D31—21no—
11C2×#92497C2212yes—
12C31×C4⋊Q8C2211yescharacteristic
13C31×#127C2211yescharacteristic
14#11758521—21no—
15#11758515—21no—
16#11758515—21no—
17#11758513C2212yes—
18D4×Dic31—21no—
19D4×Dic31—21no—
20C2×#92469C2212yes—
21C4⋊C4×D31—21no—
22C4⋊C4×D31C2212yes—
23#11758465—21no—
24#11758465C2212yes—
25C42×D31C2211yescharacteristic
26#11758446C2211yescharacteristic
27#11758446C2211yescharacteristic
28C31×#102C2214yes—
29C4×C8×C31C2211yescharacteristic
30#11758420C2214yes—
31#11758404C2214yes—
32#11758403C2212yes—
33#11758398C2214yes—
34C4×C31⋊C8C2211yescharacteristic
35C2×Q8×C31—21no—
36C2×D4×C31—21no—
37C2×D4×C31—21no—
38C2×D4×C31C2312yes—
39C2×C22⋊D31—22no—
40C22×Dic31—22no—
41Q8×D31—44no—
42#92497—28no—
43C2×C4×D31—22no—
44C2×C4×D31D411yescharacteristic
45C2×C4×D31D411yescharacteristic
46C2×C4×D31C2311yescharacteristic
47C2×Dic62—21no—
48C2×Dic62—22no—
49C2×Dic62C2312yes—
50C2×C8×C31C2312yes—
51C4⋊C4×C31—21no—
52C4⋊C4×C31C2312yes—
53C42×C31C2311yescharacteristic
54C22⋊Dic31—22no—
55C22⋊Dic31—22no—
56C4⋊Dic31—21no—
57C4⋊Dic31—21no—
58C4⋊Dic31—21no—
59C4⋊Dic31C2312yes—
60Dic31⋊C4—22no—
61Dic31⋊C4—22no—
62C4×Dic31D411yescharacteristic
63C4×Dic31D411yescharacteristic
64C4×Dic31C2311yescharacteristic
65C2×C31⋊C8C2312yes—
66#92469—24no—
67C23×C31—22no—
68C22×D31C2×D411yescharacteristic
69Q8×C31—42no—
70D4×C31—24no—
71D4×C31—42no—
72D4×C31—42no—
73C2×C4×C31—22no—
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31C2×D411yescharacteristic
76C2×C4×C31C2411yescharacteristic
77C22⋊D31—44no—
78C2×Dic31—22no—
79C2×Dic31—22no—
80C2×Dic31—44no—
81C2×Dic31C2×D411yescharacteristic
82C2×Dic31C2×D411yescharacteristic
83C2×Dic31C2×D411yescharacteristic
84C4×D31—22no—
85C4×D31—22no—
86C4×D31—44no—
87C4×D31#4114yes—
88Dic62—24no—
89Dic62—44no—
90Dic62—42no—
91C8×C31—22no—
92C31⋊C8—22no—
93#3141—311nomaximal, Sylow
94C22×C31—42no—
95C22×C31—42no—
96C22×C31—42no—
97C22×C31C22×D411yescharacteristic
98C2×D31—21no—
99C2×D31—21no—
100C2×D31#12412yes—
101C4×C31—42no—
102C4×C31AGL(1,ℤ/8)12yes—
103C4×C31#13712yes—
104C4×C31C2×#4112yes—
105Dic31—21no—
106Dic31—21no—
107Dic31—42no—
108Dic31—42no—
109Dic31#12412yes—
110#532—311no—
111C2×C4⋊Q8—311no—
112#489—314no—
113#487—311no—
114#487—311no—
115#486—311no—
116#486D3111yescharacteristic, complemented
117#417—314no—
118C4×M16—311no—
119C2×C31—82no—
120C2×C31C2×#12411yescharacteristic
121C2×C31C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C31C2×#13711yescharacteristic
123D31—22no—
124C2×#41—312no—
125C22×Q8—621no—
126C2×M16—312no—
127C4⋊Q8—312no—
128C4⋊Q8—311no—
129C4⋊Q8—311no—
130C4⋊Q8—621no—
131C4⋊Q8—621no—
132C4⋊Q8C2×D3111yescharacteristic
133#127C2×D3111yescharacteristic
134#124—312no—
135#121—621no—
136#121—621no—
137C4×D4—621no—
138C4×D4—621no—
139C2×C4⋊C4—312no—
140C2×C4⋊C4—621no—
141C2×C42—311no—
142#103—314no—
143#103—314no—
144#102—314no—
145#102C2×D3114yes—
146#97—312no—
147C4×C8—311no—
148C4×C8C2×D3111yescharacteristic
149C31#314111yescharacteristic, complemented, Sylow
150#41—628no—
151C2×Q8—21no—
152C2×Q8—312no—
153C2×Q8—622no—
154C2×Q8—621no—
155C2×Q8—1244no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—622no—
159C2×D4C22×D3112yes—
160C22×C4—311no—
161C22×C4—311no—
162C22×C4—311no—
163C22×C4—622no—
164C22×C4—622no—
165M16—624no—
166C2×C8—312no—
167C2×C8C22×D3112yes—
168C4⋊C4—21no—
169C4⋊C4—312no—
170C4⋊C4—622no—
171C4⋊C4—622no—
172C4⋊C4—621no—
173C4⋊C4—621no—
174C4⋊C4—621no—
175C4⋊C4C22×D3112yes—
176C22⋊C4—622no—
177C22⋊C4—622no—
178C42—311no—
179C42—311no—
180C42—311no—
181C42C22×D3111yescharacteristic
182C23—22no—
183C23—311no—
184Q8—42no—
185Q8—624no—
186Q8—1244no—
187Q8—1242no—
188D4—24no—
189D4—42no—
190D4—42no—
191D4—1244no—
192C2×C4—22no—
193C2×C4—311no—
194C2×C4—314no—
195C2×C4—311no—
196C2×C4—311no—
197C2×C4—622no—
198C2×C4—622no—
199C2×C4—622no—
200C2×C4—622no—
201C2×C4—1244no—
202C2×C4—1244no—
203C2×C4D4×D3111yescharacteristic
204C2×C4D4×D3111yescharacteristic
205C2×C4C23×D3111yescharacteristic
206C8—22no—
207C8—622no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—312no—
212C22—621no—
213C22—621no—
214C22C2×D4×D3111yescharacteristic
215C4—42no—
216C4—312no—
217C4—621no—
218C4—621no—
219C4—1242no—
220C4—1242no—
221C4#1175848812yes—
222C4#1175849212yes—
223C4#41×D3112yes—
224C2—82no—
225C2—622no—
226C2#124×D3111yescharacteristic
227C2C2×#1175848811yescharacteristic
228C2C2×#1175849211yescharacteristic
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#486C2
socleC22×C31C22×D4
prime radical#14521597C2
hypercenter#486D31
hypocenterC31#3141
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D31C2semidirect
3. #14521407C2semidirect
4. #14521401C2semidirect
5. #14521400C2semidirect
6. #14521395C2semidirect
7. #14521047C2semidirect
8. C4×#92469C2semidirect
9. C31×#486C2semidirect
116. #486D31semidirect
149. C31#3141semidirect

Maximal subgroups

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

subgroupindex
#145215972
C4⋊Q8×D312
#145214072
#145214012
#145214002
#145213952
#145210472
C4×#924692
C31×#4862
#314131