Finite Group Databasebeta

← #16854956

Subgroups — #16854956

Order 3968.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#92495C211yesmaximal, characteristic, complemented
2C2×#11758486C211yesmaximal, characteristic, complemented
3C2×#11758485C211yesmaximal, characteristic, complemented
4#14521560C211yesmaximal, characteristic, complemented
5#14521213C214yesmaximal, complemented
6#14521196C214yesmaximal, complemented
7#14521059C211yesmaximal, characteristic, complemented
8#14521055C211yesmaximal, characteristic, complemented
9C31×#407C211yesmaximal, characteristic, complemented
10C23×C4×C31C2211yescharacteristic
11C22×C22⋊D31—21no—
12C2×#92495—22no—
13C2×#92495—24no—
14C2×#92495—24no—
15C2×#92495C2214yes—
16C2×#92495C2211yescharacteristic
17C22×D124C2211yescharacteristic
18C22×C4×D31—21no—
19C22×Dic62C2211yescharacteristic
20C2×M16×C31C2211yescharacteristic
21C2×M16×C31C2211yescharacteristic
22#11758530—21no—
23#11758507C2212yescomplemented
24#11758506—21no—
25C4×C22⋊D31—22no—
26#11758503C2211yescharacteristic
27#11758501C2212yescomplemented
28#11758486—24no—
29#11758485—24no—
30C2×Dic124C2212yescomplemented
31C2×D248C2212yescomplemented
32C2×#92470C2212yescomplemented
33C2×#92470C2212yescomplemented
34C31×C22⋊C8C2214yescomplemented
35#11758409C2214yescomplemented
36#11758404C2214yescomplemented
37C24×C31D411yescharacteristic
38C23×D31—21no—
39C22×C4×C31—22no—
40C22×C4×C31—22no—
41C22×C4×C31D412yes—
42C22×C4×C31D411yescharacteristic
43C22×C4×C31C2312yes—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—28no—
46C2×C22⋊D31—24no—
47C2×C22⋊D31—22no—
48C2×C22⋊D31—21no—
49C22×Dic31—21no—
50#92495—216no—
51#92495—28no—
52#92495—22no—
53#92495—22no—
54#92495—28no—
55C2×D124—21no—
56C2×D124—22no—
57C2×D124D414yescomplemented
58C2×D124C2311yescharacteristic
59C2×D124C2312yes—
60C2×C4×D31—28no—
61C2×C4×D31—24no—
62C2×C4×D31—21no—
63C2×Dic62—21no—
64C2×Dic62—22no—
65C2×Dic62D414yescomplemented
66C2×Dic62C2311yescharacteristic
67C2×Dic62C2312yes—
68M16×C31—22no—
69M16×C31—22no—
70C2×C8×C31C2312yes—
71C2×C8×C31C2312yes—
72C22⋊Dic31—22no—
73C22⋊Dic31—21no—
74(C2×D31)⋊C4—22no—
75C4⋊Dic31C2312yes—
76Dic31⋊C4—22no—
77C4×Dic31—21no—
78Dic124—24no—
79D248—24no—
80#92470—24no—
81#92470—24no—
82C23×C31—22no—
83C23×C31—42no—
84C23×C31C2×D412yes—
85C23×C31C2×D411yescharacteristic
86C22×D31—24no—
87C22×D31—22no—
88C22×D31—21no—
89C22×D31—44no—
90C2×C4×C31—22no—
91C2×C4×C31—24no—
92C2×C4×C31—22no—
93C2×C4×C31—22no—
94C2×C4×C31C2×D412yes—
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2×D412yes—
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2411yescharacteristic
99C22⋊D31—216no—
100C22⋊D31—44no—
101C22⋊D31—42no—
102C2×Dic31—24no—
103C2×Dic31—22no—
104C2×Dic31—22no—
105C2×Dic31—21no—
106D124—24no—
107D124—22no—
108D124—21no—
109D124—21no—
110D124C2×D414yes—
111C4×D31—216no—
112C4×D31—42no—
113Dic62—24no—
114Dic62—22no—
115Dic62—21no—
116Dic62—21no—
117Dic62C2×D414yes—
118C8×C31—22no—
119C8×C31—22no—
120#3005—311nomaximal, Sylow
121C22×C31—22no—
122C22×C31—22no—
123C22×C31—22no—
124C22×C31—22no—
125C22×C31—44no—
126C22×C31#12012yes—
127C22×C31C22×D411yescharacteristic
128C2×D31—24no—
129C2×D31—48no—
130C2×D31—44no—
131C2×D31—41no—
132C2×D31—42no—
133C4×C31—22no—
134C4×C31#12012yes—
135C4×C31C22×D411yescharacteristic
136C4×C31C22×D411yescharacteristic
137Dic31—24no—
138Dic31—41no—
139Dic31—42no—
140C22×#41—311no—
141C2×#137—311no—
142C2×AGL(1,ℤ/8)—311no—
143#525—311no—
144#452—314no—
145#449—314no—
146#419—311no—
147#418—311no—
148#407D3111yescharacteristic, complemented
149C2×C31—22no—
150C2×C31—42no—
151C2×C31C2×#12011yescharacteristic
152C2×C31#57512yes—
153D31—44no—
154D31—81no—
155C2×#41—311no—
156C2×#41—314no—
157C2×#41—624no—
158C2×#41—624no—
159C2×#41—622no—
160C22×Q8—311no—
161C22×D4—311no—
162C22×D4—621no—
163C23×C4—621no—
164C23×C4C2×D3111yescharacteristic
165#137—624no—
166AGL(1,ℤ/8)—624no—
167C2×Q16—312no—
168C2×SD16—312no—
169C2×SD16—312no—
170C2×D8—312no—
171C2×M16C2×D3111yescharacteristic
172C2×M16C2×D3111yescharacteristic
173#123—621no—
174C22⋊Q8—312no—
175#121—312no—
176#120—621no—
177C4×D4—622no—
178#117—311no—
179#103—314no—
180#102—314no—
181C22⋊C8C2×D3114yescomplemented
182C31#300511yescharacteristic, complemented, Sylow
183C24—621no—
184C24D12411yescharacteristic
185#41—628no—
186#41—628no—
187#41—6216no—
188#41—622no—
189#41—622no—
190C2×Q8—314no—
191C2×Q8—312no—
192C2×Q8—311no—
193C2×Q8—621no—
194C2×Q8—622no—
195C2×D4—314no—
196C2×D4—312no—
197C2×D4—311no—
198C2×D4—628no—
199C2×D4—624no—
200C2×D4—622no—
201C2×D4—621no—
202C2×D4—621no—
203C2×D4—622no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—628no—
207C22×C4—621no—
208C22×C4—624no—
209C22×C4—621no—
210C22×C4D12412yes—
211C22×C4D12411yescharacteristic
212C22×C4C22×D3112yes—
213C22×C4C22×D3111yescharacteristic
214Q16—624no—
215SD16—624no—
216SD16—624no—
217D8—624no—
218M16—22no—
219M16—22no—
220C2×C8C22×D3112yes—
221C2×C8C22×D3112yes—
222C4⋊C4—312no—
223C4⋊C4—622no—
224C22⋊C4—621no—
225C22⋊C4—622no—
226C22⋊C4—622no—
227C42—621no—
228C23—22no—
229C23—42no—
230C23—624no—
231C23—622no—
232C23—621no—
233C23—1244no—
234C23C2×D12412yes—
235C23C2×D12411yescharacteristic
236Q8—314no—
237Q8—624no—
238Q8—622no—
239Q8—621no—
240Q8—621no—
241D4—314no—
242D4—6216no—
243D4—624no—
244D4—622no—
245D4—621no—
246D4—621no—
247D4—1244no—
248D4—1242no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—6216no—
254C2×C4—624no—
255C2×C4—622no—
256C2×C4—622no—
257C2×C4—621no—
258C2×C4—1242no—
259C2×C4C2×D12412yes—
260C2×C4C2×D12411yescharacteristic
261C2×C4D4×D3112yes—
262C2×C4D4×D3112yes—
263C2×C4C23×D3111yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—624no—
272C22—1248no—
273C22—1244no—
274C22—1241no—
275C22—1242no—
276C22#1175845912yes—
277C22C22×D12411yescharacteristic
278C4—22no—
279C4—624no—
280C4—1241no—
281C4—1242no—
282C4#1175845912yes—
283C4C2×D4×D3111yescharacteristic
284C4C2×D4×D3111yescharacteristic
285C2—22no—
286C2—42no—
287C2—1244no—
288C2—2481no—
289C2C2×#1175845911yescharacteristic
290C2#1452184912yes—
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#407C2
socleC22×C31C22×D4
hypercenter#407D31
hypocenterC31#3005
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#92495C2semidirect
2. C2×#11758486C2semidirect
3. C2×#11758485C2semidirect
4. #14521560C2semidirect
5. #14521213C2semidirect
6. #14521196C2semidirect
7. #14521059C2semidirect
8. #14521055C2semidirect
9. C31×#407C2semidirect
23. #11758507C22semidirect
27. #11758501C22semidirect
30. C2×Dic124C22semidirect
31. C2×D248C22semidirect
32. C2×#92470C22semidirect
33. C2×#92470C22semidirect
34. C31×C22⋊C8C22semidirect
35. #11758409C22semidirect
36. #11758404C22semidirect
57. C2×D124D4semidirect
65. C2×Dic62D4semidirect
148. #407D31semidirect
181. C22⋊C8C2×D31semidirect
182. C31#3005semidirect

Maximal subgroups

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

subgroupindex
C22×#924952
C2×#117584862
C2×#117584852
#145215602
#145212132
#145211962
#145210592
#145210552
C31×#4072
#300531