Finite Group Databasebeta

← #16856094

Subgroups — #16856094

Order 3968.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#14521605C211yesmaximal, characteristic, complemented
2#124×D31C211yesmaximal, characteristic
3#14521410C212yesmaximal, complemented
4#14521407C212yesmaximal, complemented
5#14521406C211yesmaximal, characteristic, complemented
6#14521404C211yesmaximal, characteristic, complemented
7#14521403C211yesmaximal, characteristic, complemented
8#14521056C212yesmaximal, complemented
9#14521052C212yesmaximal
10C4×#92469C211yesmaximal, characteristic, complemented
11C31×#488C211yesmaximal, characteristic, complemented
12C2×Q8×D31C2211yescharacteristic
13C2×D4×D31C2211yescharacteristic
14C31×#125C2211yescharacteristic, complemented
15C31×#124C2211yescharacteristic
16#11758524C2211yescharacteristic
17#11758513C2211yescharacteristic
18C2×#92469C2212yes—
19#11758473—22no—
20#11758469C2212yes—
21#11758465C2212yescomplemented
22#11758464—22no—
23#11758462—22no—
24C22⋊C4×D31—22no—
25#11758452C2211yescharacteristic
26C42×D31C2211yescharacteristic
27#11758447C2211yescharacteristic, complemented
28C31×#103C2212yescomplemented
29C31×#102C2212yes—
30C4×C8×C31C2211yescharacteristic, complemented
31#11758423C2212yescomplemented
32#11758420C2212yes—
33#11758409C2212yes—
34#11758404C2212yescomplemented
35#11758403C2212yescomplemented
36#11758398C2212yescomplemented
37#11758397C2212yes—
38C4×C31⋊C8C2211yescharacteristic, complemented
39C23×D31—21no—
40C2×Q8×C31C2311yescharacteristic
41C2×D4×C31C2311yescharacteristic
42C2×C22⋊D31—21no—
43Q8×D31—24no—
44D4×D31—24no—
45C2×D124C2311yescharacteristic
46C2×C4×D31—21no—
47C2×C4×D31D412yes—
48C2×C4×D31C2311yescharacteristic
49C2×Dic62—21no—
50C2×Dic62—22no—
51C2×Dic62C2311yescharacteristic
52C2×C8×C31C2312yes—
53C4⋊C4×C31—22no—
54C4⋊C4×C31C2312yes—
55C22⋊C4×C31—22no—
56C42×C31C2311yescharacteristic
57C22⋊Dic31—22no—
58(C2×D31)⋊C4—22no—
59(C2×D31)⋊C4—22no—
60(C2×D31)⋊C4—22no—
61C4⋊Dic31C2312yes—
62Dic31⋊C4—22no—
63Dic31⋊C4—22no—
64C4×Dic31—22no—
65C4×Dic31D412yes—
66C4×Dic31C2311yescharacteristic
67C2×C31⋊C8C2312yes—
68#92469—24no—
69C23×C31—21no—
70C22×D31—21no—
71C22×D31—44no—
72C22×D31—42no—
73C22×D31C2×D411yescharacteristic
74Q8×C31—22no—
75D4×C31—22no—
76C2×C4×C31—22no—
77C2×C4×C31—21no—
78C2×C4×C31C2×D412yes—
79C2×C4×C31C2411yescharacteristic
80C22⋊D31—42no—
81C2×Dic31—22no—
82C2×Dic31—21no—
83C2×Dic31C2×D412yes—
84C2×Dic31C2×D411yescharacteristic
85D124—22no—
86C4×D31—42no—
87C4×D31—42no—
88C4×D31#4114yes—
89Dic62—22no—
90Dic62—44no—
91Dic62—42no—
92C8×C31—22no—
93C31⋊C8—22no—
94#3140—311nomaximal, Sylow
95C22×C31—41no—
96C22×C31—42no—
97C22×C31C22×D411yescharacteristic
98C2×D31—22no—
99C2×D31—41no—
100C2×D31—42no—
101C2×D31—84no—
102C2×D31#12412yes—
103C4×C31—22no—
104C4×C31—41no—
105C4×C31—42no—
106C4×C31C2×#4112yes—
107Dic31—22no—
108Dic31—41no—
109Dic31—42no—
110Dic31#12412yes—
111#533—311no—
112C2×#124—311no—
113#489—312no—
114#489—312no—
115#488—311no—
116#488—311no—
117#488—311no—
118#488D3111yescharacteristic, complemented
119#419—312no—
120#418—312no—
121C4×M16—311no—
122C2×C31—81no—
123C2×C31C2×#12411yescharacteristic
124C2×C31#57512yes—
125D31—22no—
126D31—81no—
127C22×Q8—311no—
128C22×D4—311no—
129C2×M16—312no—
130C4⋊Q8—312no—
131#125—311no—
132#125C2×D3111yescharacteristic, complemented
133#124—311no—
134#124—311no—
135#124—311no—
136#124—622no—
137#124C2×D3111yescharacteristic
138C22⋊Q8—622no—
139C4×Q8—622no—
140#117—312no—
141C2×C22⋊C4—622no—
142C2×C42—311no—
143#103—312no—
144#103—312no—
145#103—312no—
146#103C2×D3112yescomplemented
147#102—312no—
148#102—312no—
149#102—312no—
150#102C2×D3112yes—
151#97—312no—
152C4×C8—311no—
153C4×C8C2×D3111yescharacteristic, complemented
154C31#314011yescharacteristic, complemented, Sylow
155C24—621no—
156C2×Q8—311no—
157C2×Q8—622no—
158C2×Q8—621no—
159C2×Q8—624no—
160C2×Q8C22×D3111yescharacteristic
161C2×D4—311no—
162C2×D4—624no—
163C2×D4—621no—
164C2×D4C22×D3111yescharacteristic
165C22×C4—311no—
166C22×C4—312no—
167C22×C4—621no—
168M16—624no—
169C2×C8—312no—
170C2×C8C22×D3112yes—
171C4⋊C4—22no—
172C4⋊C4—312no—
173C4⋊C4—622no—
174C4⋊C4—622no—
175C4⋊C4C22×D3112yes—
176C22⋊C4—22no—
177C22⋊C4—622no—
178C22⋊C4—622no—
179C22⋊C4—622no—
180C22⋊C4—622no—
181C42—312no—
182C42—311no—
183C42—622no—
184C42C22×D3111yescharacteristic
185C23—21no—
186C23—311no—
187C23—621no—
188C23—1242no—
189C23—1244no—
190Q8—22no—
191Q8—622no—
192Q8—1242no—
193Q8—1244no—
194D4—22no—
195D4—622no—
196D4—1242no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—311no—
200C2×C4—314no—
201C2×C4—312no—
202C2×C4—622no—
203C2×C4—621no—
204C2×C4—1242no—
205C2×C4—1242no—
206C2×C4D4×D3112yes—
207C2×C4C23×D3111yescharacteristic
208C8—22no—
209C8—622no—
210C22—41no—
211C22—42no—
212C22—312no—
213C22—622no—
214C22—1241no—
215C22—1242no—
216C22—2484no—
217C22C2×D4×D3111yescharacteristic
218C4—22no—
219C4—41no—
220C4—42no—
221C4—312no—
222C4—622no—
223C4—1241no—
224C4—1242no—
225C4#41×D3112yes—
226C2—81no—
227C2—622no—
228C2—2481no—
229C2#124×D3111yescharacteristic
230C2#1452184612yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#488C2
socleC22×C31C22×D4
prime radicalC2×D4×D31C22
hypercenter#488D31
hypocenterC31#3140
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #14521605C2semidirect
3. #14521410C2semidirect
4. #14521407C2semidirect
5. #14521406C2semidirect
6. #14521404C2semidirect
7. #14521403C2semidirect
8. #14521056C2semidirect
10. C4×#92469C2semidirect
11. C31×#488C2semidirect
14. C31×#125C22semidirect
21. #11758465C22semidirect
27. #11758447C22semidirect
28. C31×#103C22semidirect
30. C4×C8×C31C22semidirect
31. #11758423C22semidirect
34. #11758404C22semidirect
35. #11758403C22semidirect
36. #11758398C22semidirect
38. C4×C31⋊C8C22semidirect
118. #488D31semidirect
132. #125C2×D31semidirect
146. #103C2×D31semidirect
153. C4×C8C2×D31semidirect
154. C31#3140semidirect

Maximal subgroups

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

subgroupindex
#145216052
#124×D312
#145214102
#145214072
#145214062
#145214042
#145214032
#145210562
#145210522
C4×#924692
C31×#4882
#314031