Finite Group Databasebeta

← #16855071

Subgroups — #16855071

Order 3968.

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

normal subgroups only

This group has 762 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×#11758561C211yesmaximal, characteristic, complemented
2#14521849C212yesmaximal, complemented
3#14521616C211yesmaximal, characteristic, complemented
4#14521229C214yesmaximal, complemented
5#14521219C214yesmaximal, complemented
6#14521065C212yesmaximal, complemented
7C2×C31×#100C211yesmaximal, characteristic, complemented
8C22×D4×C31C2211yescharacteristic
9C22×C22⋊D31—22no—
10#11758561—22no—
11#11758561—24no—
12#11758561C2214yes—
13C2×#92497—22no—
14C2×D4×D31—22no—
15C2×#92495C2212yes—
16C2×#92495C2211yescharacteristic, complemented
17C2×M16×C31C2212yescomplemented
18#11758530—22no—
19#11758517C2212yescomplemented
20#11758515—22no—
21#11758513C2212yescomplemented
22#11758503C2211yescharacteristic, complemented
23#11758486—22no—
24#11758486C2214yescomplemented
25#11758485—22no—
26#11758485C2214yescomplemented
27#11758481—24no—
28C31×#100C2214yescomplemented
29#11758413C2218yescomplemented
30C24×C31D412yes—
31C23×D31—22no—
32C2×D4×C31—22no—
33C2×D4×C31—22no—
34C2×D4×C31D412yes—
35C2×D4×C31C2312yes—
36C22×C4×C31C2311yescharacteristic
37C2×C22⋊D31—24no—
38C2×C22⋊D31—28no—
39C2×C22⋊D31—22no—
40C2×C22⋊D31—28no—
41C2×C22⋊D31—21no—
42C2×C22⋊D31—24no—
43C22×Dic31—22no—
44#92497—28no—
45#92497—28no—
46D4×D31—28no—
47D4×D31—28no—
48#92495—22no—
49#92495—22no—
50#92495—22no—
51#92495—22no—
52#92495D414yescomplemented
53#92495C2314yes—
54C2×D124D412yescomplemented
55C2×D124C2311yescharacteristic, complemented
56C2×C4×D31—22no—
57C2×C4×D31—21no—
58C2×Dic62D412yescomplemented
59C2×Dic62C2311yescharacteristic, complemented
60M16×C31—22no—
61M16×C31C2314yescomplemented
62C2×C8×C31—22no—
63C22⋊Dic31—21no—
64C22⋊Dic31—24no—
65C4⋊Dic31—21no—
66C4×Dic31C2312yescomplemented
67Dic124—24no—
68D248—24no—
69#92470—24no—
70#92470—24no—
71C23×C31—22no—
72C23×C31—22no—
73C23×C31—82no—
74C23×C31C2×D414yes—
75C23×C31C2×D411yescharacteristic
76C22×D31—22no—
77C22×D31—28no—
78C22×D31—21no—
79C22×D31—84no—
80D4×C31—24no—
81D4×C31—24no—
82C2×C4×C31C2×D411yescharacteristic
83C2×C4×C31C2×D412yes—
84C2×C4×C31C2×D412yes—
85C2×C4×C31C2411yescharacteristic
86C22⋊D31—216no—
87C22⋊D31—24no—
88C22⋊D31—24no—
89C22⋊D31—41no—
90C22⋊D31—41no—
91C22⋊D31—84no—
92C2×Dic31—28no—
93C2×Dic31—22no—
94C2×Dic31—22no—
95C2×Dic31—21no—
96D124—22no—
97D124—22no—
98D124C2×D414yes—
99C4×D31—24no—
100C4×D31—24no—
101C4×D31—42no—
102Dic62—22no—
103Dic62—22no—
104Dic62C2×D414yes—
105C8×C31—24no—
106#3019—311nomaximal, Sylow
107C22×C31—24no—
108C22×C31—41no—
109C22×C31—44no—
110C22×C31—44no—
111C22×C31—82no—
112C22×C31#12011yescharacteristic
113C22×C31#12011yescharacteristic
114C22×C31C22×D411yescharacteristic
115C2×D31—24no—
116C2×D31—44no—
117C2×D31—42no—
118C2×D31—41no—
119C2×D31—88no—
120C4×C31#12012yes—
121C4×C31C22×D412yes—
122Dic31—24no—
123Dic31—42no—
124Dic31—41no—
125C2×H(5,𝔽2)—311no—
126#575—312no—
127#535—311no—
128#455—314no—
129#453—314no—
130#421—312no—
131C2×#100D3111yescharacteristic, complemented
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—21no—
135C2×C31—44no—
136C2×C31C2×#12011yescharacteristic
137D31—44no—
138D31—81no—
139H(5,𝔽2)—314no—
140H(5,𝔽2)—624no—
141H(5,𝔽2)—622no—
142C2×#41—312no—
143C2×#41—311no—
144C2×#41—622no—
145C22×D4—622no—
146C22×D4—622no—
147C22×D4C2×D3111yescharacteristic
148#137—314no—
149#137—622no—
150AGL(1,ℤ/8)—314no—
151AGL(1,ℤ/8)—622no—
152#135—624no—
153C2×M16C2×D3112yescomplemented
154#127—312no—
155#124—312no—
156#121—622no—
157#120—622no—
158#117—311no—
159#104—318no—
160#100C2×D3114yescomplemented
161C31#301911yescharacteristic, complemented, Sylow
162C24—622no—
163C24D12412yes—
164#41—314no—
165#41—314no—
166#41—628no—
167#41—628no—
168#41—622no—
169#41—622no—
170#41—622no—
171#41—622no—
172C2×Q8—312no—
173C2×Q8—311no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—312no—
177C2×D4—311no—
178C2×D4—628no—
179C2×D4—628no—
180C2×D4—624no—
181C2×D4—628no—
182C2×D4—622no—
183C2×D4—628no—
184C2×D4—621no—
185C2×D4—624no—
186C2×D4D12412yes—
187C2×D4C22×D3112yes—
188C22×C4—622no—
189C22×C4—622no—
190C22×C4—621no—
191C22×C4C22×D3111yescharacteristic
192Q16—624no—
193SD16—624no—
194SD16—624no—
195D8—624no—
196M16—22no—
197M16C22×D3114yescomplemented
198C2×C8—22no—
199C4⋊C4—621no—
200C22⋊C4—621no—
201C22⋊C4—624no—
202C42—312no—
203C23—22no—
204C23—22no—
205C23—82no—
206C23—628no—
207C23—622no—
208C23—621no—
209C23—2484no—
210C23C2×D12414yes—
211C23C2×D12411yescharacteristic
212Q8—314no—
213Q8—622no—
214Q8—622no—
215D4—24no—
216D4—24no—
217D4—314no—
218D4—6216no—
219D4—624no—
220D4—624no—
221D4—622no—
222D4—622no—
223D4—1241no—
224D4—1241no—
225D4—2484no—
226C2×C4—628no—
227C2×C4—622no—
228C2×C4—624no—
229C2×C4—624no—
230C2×C4—622no—
231C2×C4—621no—
232C2×C4—1242no—
233C2×C4C2×D12411yescharacteristic
234C2×C4D4×D3112yes—
235C2×C4D4×D3112yes—
236C2×C4C23×D3111yescharacteristic
237C8—24no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—82no—
243C22—624no—
244C22—1244no—
245C22—1242no—
246C22—1241no—
247C22—2488no—
248C22#1175845911yescharacteristic
249C22#1175845911yescharacteristic
250C22C22×D12411yescharacteristic
251C4—624no—
252C4—1242no—
253C4—1241no—
254C4#1175845912yes—
255C4C2×D4×D3112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—1244no—
261C2—2481no—
262C2C2×#1175845911yescharacteristic
subgroupG/H
centerC2C2×#11758459
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×#100C2
socleC2×C31C2×#120
hypercenterC2×#100D31
hypocenterC31#3019
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
1. C2×#11758561C2semidirect
2. #14521849C2semidirect
3. #14521616C2semidirect
4. #14521229C2semidirect
5. #14521219C2semidirect
6. #14521065C2semidirect
7. C2×C31×#100C2semidirect
16. C2×#92495C22semidirect
17. C2×M16×C31C22semidirect
19. #11758517C22semidirect
21. #11758513C22semidirect
22. #11758503C22semidirect
24. #11758486C22semidirect
26. #11758485C22semidirect
28. C31×#100C22semidirect
29. #11758413C22semidirect
52. #92495D4semidirect
54. C2×D124D4semidirect
55. C2×D124C23semidirect
58. C2×Dic62D4semidirect
59. C2×Dic62C23semidirect
61. M16×C31C23semidirect
66. C4×Dic31C23semidirect
131. C2×#100D31semidirect
153. C2×M16C2×D31semidirect
160. #100C2×D31semidirect
161. C31#3019semidirect
197. M16C22×D31semidirect

Maximal subgroups

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

subgroupindex
C2×#117585612
#145218492
#145216162
#145212292
#145212192
#145210652
C2×C31×#1002
#301931