Finite Group Databasebeta

← #16855078

Subgroups — #16855078

Order 3968.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#11758564C211yesmaximal, characteristic, complemented
2#14521849C212yesmaximal, complemented
3#14521623C211yesmaximal, characteristic, complemented
4#14521234C214yesmaximal, complemented
5#14521224C214yesmaximal, complemented
6#14521065C212yesmaximal, complemented
7C2×C31×#101C211yesmaximal, characteristic, complemented
8C22×Q8×C31C2211yescharacteristic
9#11758564—22no—
10#11758564—24no—
11#11758564C2214yes—
12C2×#92499—22no—
13C2×Q8×D31—22no—
14C2×#92495—22no—
15C2×#92495C2212yes—
16C2×#92495C2211yescharacteristic
17C2×M16×C31C2212yes—
18#11758524C2212yescomplemented
19#11758523—22no—
20#11758521C2212yescomplemented
21#11758506—22no—
22#11758503C2211yescharacteristic
23#11758486—22no—
24#11758486C2214yescomplemented
25#11758485—22no—
26#11758485C2214yescomplemented
27#11758481—24no—
28C31×#101C2214yescomplemented
29#11758413C2218yescomplemented
30C2×Q8×C31—22no—
31C2×Q8×C31—22no—
32C2×Q8×C31D412yes—
33C2×Q8×C31C2312yes—
34C22×C4×C31D412yes—
35C22×C4×C31C2311yescharacteristic
36C2×C22⋊D31—22no—
37C2×C22⋊D31—21no—
38#92499—28no—
39#92499—28no—
40Q8×D31—28no—
41Q8×D31—28no—
42#92495—28no—
43#92495—28no—
44#92495—22no—
45#92495—22no—
46#92495—22no—
47#92495—22no—
48#92495D414yes—
49#92495C2314yes—
50C2×D124—22no—
51C2×D124D412yes—
52C2×D124C2311yescharacteristic
53C2×C4×D31—24no—
54C2×C4×D31—22no—
55C2×C4×D31—21no—
56C2×Dic62—22no—
57C2×Dic62D412yes—
58C2×Dic62C2311yescharacteristic
59M16×C31—22no—
60M16×C31C2314yes—
61C2×C8×C31—22no—
62C22⋊Dic31—21no—
63(C2×D31)⋊C4—24no—
64C4⋊Dic31—21no—
65Dic31⋊C4—24no—
66C4×Dic31C2312yes—
67Dic124—24no—
68D248—24no—
69#92470—24no—
70#92470—24no—
71C23×C31C2×D411yescharacteristic
72C22×D31—22no—
73C22×D31—21no—
74Q8×C31—24no—
75Q8×C31—24no—
76C2×C4×C31—22no—
77C2×C4×C31—22no—
78C2×C4×C31C2×D414yes—
79C2×C4×C31C2×D411yescharacteristic
80C2×C4×C31C2×D412yes—
81C2×C4×C31C2×D412yes—
82C2×C4×C31C2411yescharacteristic
83C22⋊D31—24no—
84C22⋊D31—24no—
85C22⋊D31—41no—
86C22⋊D31—41no—
87C2×Dic31—22no—
88C2×Dic31—22no—
89C2×Dic31—21no—
90D124—28no—
91D124—22no—
92D124—22no—
93D124C2×D414yes—
94C4×D31—216no—
95C4×D31—24no—
96C4×D31—24no—
97C4×D31—42no—
98Dic62—28no—
99Dic62—22no—
100Dic62—22no—
101Dic62C2×D414yes—
102C8×C31—24no—
103#3020—311nomaximal, Sylow
104C22×C31—41no—
105C22×C31#12011yescharacteristic
106C22×C31#12011yescharacteristic
107C22×C31C22×D411yescharacteristic
108C2×D31—24no—
109C2×D31—44no—
110C2×D31—42no—
111C2×D31—41no—
112C4×C31—24no—
113C4×C31#12012yes—
114C4×C31C22×D412yes—
115Dic31—24no—
116Dic31—42no—
117Dic31—41no—
118C2×2−1+4—311no—
119#575—312no—
120#536—311no—
121#456—314no—
122#454—314no—
123#421—312no—
124C2×#101D3111yescharacteristic, complemented
125C2×C31—21no—
126C2×C31—21no—
127C2×C31—21no—
128C2×C31C2×#12011yescharacteristic
129D31—44no—
130D31—81no—
1312−1+4—314no—
1322−1+4—624no—
1332−1+4—622no—
134C2×#41—312no—
135C2×#41—311no—
136C2×#41—622no—
137C2×#41—622no—
138C22×Q8—622no—
139C22×Q8C2×D3111yescharacteristic
140#137—314no—
141#137—622no—
142AGL(1,ℤ/8)—314no—
143AGL(1,ℤ/8)—622no—
144#135—624no—
145C2×M16C2×D3112yes—
146C4⋊Q8—312no—
147#124—312no—
148#123—622no—
149C22⋊Q8—622no—
150#117—311no—
151#104—318no—
152#101C2×D3114yescomplemented
153C31#302011yescharacteristic, complemented, Sylow
154#41—314no—
155#41—314no—
156#41—628no—
157#41—628no—
158#41—628no—
159#41—622no—
160#41—628no—
161#41—622no—
162#41—622no—
163#41—622no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—312no—
167C2×Q8—311no—
168C2×Q8—622no—
169C2×Q8—628no—
170C2×Q8—628no—
171C2×Q8D12412yes—
172C2×Q8C22×D3112yes—
173C2×D4—312no—
174C2×D4—311no—
175C2×D4—622no—
176C2×D4—622no—
177C2×D4—621no—
178C22×C4—624no—
179C22×C4—622no—
180C22×C4—621no—
181C22×C4D12412yes—
182C22×C4C22×D3111yescharacteristic
183Q16—624no—
184SD16—624no—
185SD16—624no—
186D8—624no—
187M16—22no—
188M16C22×D3114yes—
189C2×C8—22no—
190C4⋊C4—624no—
191C4⋊C4—621no—
192C22⋊C4—621no—
193C22⋊C4—624no—
194C42—312no—
195C23—622no—
196C23—621no—
197C23C2×D12411yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—314no—
201Q8—628no—
202Q8—622no—
203Q8—622no—
204D4—314no—
205D4—628no—
206D4—624no—
207D4—624no—
208D4—622no—
209D4—622no—
210D4—1241no—
211D4—1241no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—6216no—
215C2×C4—622no—
216C2×C4—624no—
217C2×C4—624no—
218C2×C4—622no—
219C2×C4—621no—
220C2×C4—1242no—
221C2×C4C2×D12414yes—
222C2×C4C2×D12411yescharacteristic
223C2×C4D4×D3112yes—
224C2×C4D4×D3112yes—
225C2×C4C23×D3111yescharacteristic
226C8—24no—
227C22—41no—
228C22—624no—
229C22—1244no—
230C22—1242no—
231C22—1241no—
232C22#1175845911yescharacteristic
233C22#1175845911yescharacteristic
234C22C22×D12411yescharacteristic
235C4—24no—
236C4—624no—
237C4—1242no—
238C4—1241no—
239C4#1175845912yes—
240C4C2×D4×D3112yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—1244no—
245C2—2481no—
246C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#101C2
socleC2×C31C2×#120
hypercenterC2×#101D31
hypocenterC31#3020
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#11758564C2semidirect
2. #14521849C2semidirect
3. #14521623C2semidirect
4. #14521234C2semidirect
5. #14521224C2semidirect
6. #14521065C2semidirect
7. C2×C31×#101C2semidirect
18. #11758524C22semidirect
20. #11758521C22semidirect
24. #11758486C22semidirect
26. #11758485C22semidirect
28. C31×#101C22semidirect
29. #11758413C22semidirect
124. C2×#101D31semidirect
152. #101C2×D31semidirect
153. C31#3020semidirect

Maximal subgroups

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

subgroupindex
C2×#117585642
#145218492
#145216232
#145212342
#145212242
#145210652
C2×C31×#1012
#302031