Finite Group Databasebeta

← #16855068

Subgroups — #16855068

Order 3968.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#14521847C212yesmaximal, complemented
2#14521616C211yesmaximal, characteristic, complemented
3#14521227C214yesmaximal, complemented
4#14521217C214yesmaximal, complemented
5#14521066C212yesmaximal, complemented
6C2×#11758421C211yesmaximal, characteristic, complemented
7C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
8C31×H(5,𝔽2)—24no—
9C31×H(5,𝔽2)—22no—
10C31×H(5,𝔽2)C2214yes—
11C2×#41×C31—22no—
12C2×#41×C31C2212yes—
13C22×D4×C31—22no—
14C22×D4×C31—22no—
15C22×D4×C31C2211yescharacteristic
16C2×#92495C2211yescharacteristic, complemented
17#11758530—22no—
18#11758528C2214yescomplemented
19#11758527—24no—
20#11758526C2214yescomplemented
21#11758519—22no—
22#11758517C2212yescomplemented
23#11758515—22no—
24#11758513C2212yescomplemented
25#11758509—22no—
26#11758503C2211yescharacteristic, complemented
27C2×#92474C2212yescomplemented
28#11758425C2218yescomplemented
29#11758421C2214yescomplemented
30C24×C31—22no—
31C24×C31D412yes—
32#41×C31—28no—
33#41×C31—28no—
34#41×C31—22no—
35#41×C31—22no—
36#41×C31D414yescomplemented
37#41×C31C2314yes—
38C2×Q8×C31D412yescomplemented
39C2×D4×C31—24no—
40C2×D4×C31—28no—
41C2×D4×C31—28no—
42C2×D4×C31—22no—
43C2×D4×C31—28no—
44C2×D4×C31—28no—
45C2×D4×C31—22no—
46C2×D4×C31—22no—
47C2×D4×C31D412yes—
48C2×D4×C31D412yescomplemented
49C2×D4×C31C2312yes—
50C22×C4×C31—22no—
51C22×C4×C31—22no—
52C22×C4×C31C2311yescharacteristic
53C2×C22⋊D31—21no—
54C2×C22⋊D31—24no—
55#92495—22no—
56#92495—22no—
57C2×D124C2311yescharacteristic, complemented
58C2×C4×D31—21no—
59C2×Dic62C2311yescharacteristic, complemented
60C22⋊Dic31—21no—
61C22⋊Dic31—24no—
62#92482—24no—
63#92481—24no—
64#92480—24no—
65#92479—24no—
66C4⋊Dic31—21no—
67C4×Dic31C2312yescomplemented
68#92474—22no—
69#92474C2314yescomplemented
70C2×C31⋊C8—22no—
71C23×C31—28no—
72C23×C31—22no—
73C23×C31—22no—
74C23×C31—22no—
75C23×C31—82no—
76C23×C31—84no—
77C23×C31C2×D414yes—
78C23×C31C2×D411yescharacteristic
79C22×D31—21no—
80Q8×C31—22no—
81Q8×C31C2×D414yes—
82D4×C31—216no—
83D4×C31—24no—
84D4×C31—24no—
85D4×C31—24no—
86D4×C31—22no—
87D4×C31—24no—
88D4×C31C2×D414yes—
89C2×C4×C31—28no—
90C2×C4×C31—22no—
91C2×C4×C31—24no—
92C2×C4×C31—24no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2×D412yes—
95C2×C4×C31C2×D412yes—
96C2×C4×C31C2411yescharacteristic
97C22⋊D31—41no—
98C22⋊D31—41no—
99C22⋊D31—84no—
100C2×Dic31—22no—
101C2×Dic31—21no—
102D124—22no—
103C4×D31—42no—
104Dic62—22no—
105C31⋊C8—24no—
106#3019—311nomaximal, Sylow
107C22×C31—24no—
108C22×C31—24no—
109C22×C31—41no—
110C22×C31—44no—
111C22×C31—44no—
112C22×C31—44no—
113C22×C31—82no—
114C22×C31—88no—
115C22×C31#12011yescharacteristic
116C22×C31#12011yescharacteristic
117C22×C31C22×D411yescharacteristic
118C2×D31—42no—
119C2×D31—41no—
120C4×C31—24no—
121C4×C31#12012yes—
122C4×C31C22×D412yes—
123Dic31—42no—
124Dic31—41no—
125C2×H(5,𝔽2)D3111yescharacteristic, complemented
126#575—312no—
127#535—311no—
128#455—314no—
129#453—314no—
130#421—312no—
131C2×#100—311no—
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—21no—
135C2×C31—44no—
136C2×C31—44no—
137C2×C31C2×#12011yescharacteristic
138D31—81no—
139H(5,𝔽2)—24no—
140H(5,𝔽2)—22no—
141H(5,𝔽2)C2×D3114yes—
142C2×#41—22no—
143C2×#41—311no—
144C2×#41C2×D3112yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C2×D3111yescharacteristic
148#137—314no—
149#137—622no—
150AGL(1,ℤ/8)—314no—
151AGL(1,ℤ/8)—622no—
152#135—624no—
153C2×M16—312no—
154#127—312no—
155#124—312no—
156#121—622no—
157#120—622no—
158#117—311no—
159#104—318no—
160#100—314no—
161C31#301911yescharacteristic, complemented, Sylow
162C24—22no—
163C24C22⋊D3112yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—622no—
169#41—622no—
170#41C22⋊D3114yescomplemented
171#41C22×D3114yes—
172C2×Q8—311no—
173C2×Q8C22⋊D3112yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—311no—
183C2×D4—621no—
184C2×D4—624no—
185C2×D4C22⋊D3112yes—
186C2×D4C22⋊D3112yescomplemented
187C2×D4C22×D3112yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—621no—
191C22×C4C22×D3111yescharacteristic
192Q16—624no—
193SD16—624no—
194SD16—624no—
195D8—624no—
196M16—314no—
197M16—622no—
198C2×C8—622no—
199C4⋊C4—621no—
200C22⋊C4—621no—
201C22⋊C4—624no—
202C42—312no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—621no—
210C23C2×C22⋊D3114yes—
211C23C2×C22⋊D3111yescharacteristic
212Q8—22no—
213Q8—622no—
214Q8C2×C22⋊D3114yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—622no—
222D4—1241no—
223D4—1241no—
224D4—2484no—
225D4C2×C22⋊D3114yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—622no—
231C2×C4—621no—
232C2×C4—1242no—
233C2×C4C2×C22⋊D3111yescharacteristic
234C2×C4C2×C22⋊D3112yes—
235C2×C4C2×C22⋊D3112yes—
236C2×C4C23×D3111yescharacteristic
237C8—624no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—1242no—
247C22—1241no—
248C22#1175853011yescharacteristic
249C22#1175853011yescharacteristic
250C22C22×C22⋊D3111yescharacteristic
251C4—24no—
252C4—1242no—
253C4—1241no—
254C4#1175853012yes—
255C4C22×C22⋊D3112yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—2481no—
262C2C2×#1175853011yescharacteristic
subgroupG/H
centerC2C2×#11758530
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C31×H(5,𝔽2)C2
socleC2×C31C2×#120
hypercenterC2×H(5,𝔽2)D31
hypocenterC31#3019
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #14521847C2semidirect
2. #14521616C2semidirect
3. #14521227C2semidirect
4. #14521217C2semidirect
5. #14521066C2semidirect
6. C2×#11758421C2semidirect
7. C2×C31×H(5,𝔽2)C2semidirect
16. C2×#92495C22semidirect
18. #11758528C22semidirect
20. #11758526C22semidirect
22. #11758517C22semidirect
24. #11758513C22semidirect
26. #11758503C22semidirect
27. C2×#92474C22semidirect
28. #11758425C22semidirect
29. #11758421C22semidirect
36. #41×C31D4semidirect
38. C2×Q8×C31D4semidirect
48. C2×D4×C31D4semidirect
57. C2×D124C23semidirect
59. C2×Dic62C23semidirect
67. C4×Dic31C23semidirect
69. #92474C23semidirect
125. C2×H(5,𝔽2)D31semidirect
161. C31#3019semidirect
170. #41C22⋊D31semidirect
173. C2×Q8C22⋊D31semidirect
186. C2×D4C22⋊D31semidirect

Maximal subgroups

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

subgroupindex
#145218472
#145216162
#145212272
#145212172
#145210662
C2×#117584212
C2×C31×H(5,𝔽2)2
#301931