Finite Group Databasebeta

← #16854957

Subgroups — #16854957

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×#92499C211yesmaximal, characteristic, complemented
2C2×#11758519C211yesmaximal, characteristic, complemented
3C2×#11758485C211yesmaximal, characteristic, complemented
4#14521561C211yesmaximal, characteristic, complemented
5#14521214C214yesmaximal, complemented
6#14521197C214yesmaximal, complemented
7#14521053C211yesmaximal, characteristic, complemented
8#14521012C211yesmaximal, characteristic, complemented
9C31×#419C211yesmaximal, characteristic, complemented
10C22×Q8×C31C2211yescharacteristic
11C2×#92499—24no—
12C2×#92499—24no—
13C2×#92499—22no—
14C2×#92499C2214yes—
15C2×#92495C2211yescharacteristic
16C22×D124—21no—
17C22×D124C2211yescharacteristic
18C22×C4×D31—21no—
19C22×C4×D31C2211yescharacteristic
20C2×M16×C31C2211yescharacteristic
21C31×#117C2211yescharacteristic
22C2×#92482C2212yescomplemented
23#11758519—24no—
24C2×#92481C2212yescomplemented
25C2×#92474C2211yescharacteristic
26#11758485—24no—
27C2×D248C2212yescomplemented
28C2×#92470C2212yescomplemented
29#11758474C2212yescomplemented
30#11758472C2212yescomplemented
31#11758463—21no—
32#11758459—21no—
33C4×D124—22no—
34C31×#103C2214yescomplemented
35(C2×D31)⋊C8C2214yescomplemented
36#11758397C2214yescomplemented
37C23×D31—21no—
38C23×D31D411yescharacteristic
39C2×Q8×C31—22no—
40C2×Q8×C31—21no—
41C2×Q8×C31D414yescomplemented
42C2×Q8×C31C2312yes—
43C22×C4×C31—21no—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—21no—
46C22×Dic31D411yescharacteristic
47#92499—216no—
48#92499—28no—
49#92499—28no—
50#92495—22no—
51#92495—22no—
52C2×D124—28no—
53C2×D124—24no—
54C2×D124—21no—
55C2×D124—22no—
56C2×D124—22no—
57C2×D124D414yescomplemented
58C2×D124C2311yescharacteristic
59C2×D124C2312yes—
60C2×C4×D31—28no—
61C2×C4×D31—22no—
62C2×C4×D31—22no—
63C2×C4×D31—24no—
64C2×C4×D31—21no—
65C2×C4×D31D412yes—
66C2×C4×D31C2312yes—
67C2×Dic62C2311yescharacteristic
68M16×C31—22no—
69C2×C8×C31C2312yes—
70C4⋊C4×C31C2312yes—
71C22⋊C4×C31—21no—
72C42×C31—21no—
73#92482—24no—
74#92481—24no—
75(C2×D31)⋊C4—22no—
76(C2×D31)⋊C4—22no—
77C4⋊Dic31—22no—
78#92474—22no—
79C2×C31⋊C8C2312yes—
80D248—24no—
81#92470—24no—
82C23×C31C2×D411yescharacteristic
83C22×D31—22no—
84C22×D31—24no—
85C22×D31—22no—
86C22×D31—21no—
87C22×D31—42no—
88C22×D31—44no—
89C22×D31C2×D412yes—
90Q8×C31—24no—
91Q8×C31—22no—
92Q8×C31C2×D414yes—
93C2×C4×C31—24no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31C2×D412yes—
97C2×C4×C31C2×D411yescharacteristic
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2411yescharacteristic
100C22⋊D31—42no—
101C2×Dic31—22no—
102C2×Dic31—21no—
103C2×Dic31C2×D412yes—
104D124—216no—
105D124—24no—
106D124—22no—
107D124—21no—
108D124—21no—
109D124—44no—
110D124C2×D414yes—
111C4×D31—216no—
112C4×D31—24no—
113C4×D31—22no—
114C4×D31—22no—
115C4×D31—42no—
116Dic62—21no—
117Dic62—21no—
118C8×C31—22no—
119C31⋊C8—22no—
120#3005—311nomaximal, Sylow
121C22×C31—22no—
122C22×C31#12012yes—
123C22×C31C22×D411yescharacteristic
124C2×D31—22no—
125C2×D31—24no—
126C2×D31—22no—
127C2×D31—22no—
128C2×D31—44no—
129C2×D31—48no—
130C2×D31—44no—
131C2×D31—41no—
132C2×D31—42no—
133C4×C31—24no—
134C4×C31—42no—
135C4×C31#12012yes—
136C4×C31C22×D411yescharacteristic
137C4×C31C22×D411yescharacteristic
138Dic31—22no—
139Dic31—41no—
140C22×#41—311no—
141C2×#137—311no—
142C2×AGL(1,ℤ/8)—311no—
143#525—311no—
144#452—314no—
145#449—314no—
146#419D3111yescharacteristic, complemented
147#418—311no—
148#407—311no—
149C2×C31—22no—
150C2×C31C2×#12011yescharacteristic
151C2×C31#57512yes—
152D31—42no—
153D31—44no—
154D31—81no—
155C2×#41—314no—
156C2×#41—311no—
157C2×#41—622no—
158C2×#41—624no—
159C2×#41—624no—
160C22×Q8C2×D3111yescharacteristic
161C22×D4—311no—
162C22×D4—621no—
163C23×C4—311no—
164C23×C4—621no—
165#137—624no—
166AGL(1,ℤ/8)—624no—
167C2×Q16—312no—
168C2×SD16—312no—
169C2×SD16—312no—
170C2×D8—312no—
171C2×M16—311no—
172C2×M16C2×D3111yescharacteristic
173#123—621no—
174C22⋊Q8—312no—
175#121—312no—
176#120—621no—
177C4×D4—622no—
178#117C2×D3111yescharacteristic
179#103C2×D3114yescomplemented
180#102—314no—
181C22⋊C8—314no—
182C31#300511yescharacteristic, complemented, Sylow
183C24—311no—
184C24—621no—
185#41—628no—
186#41—6216no—
187#41—628no—
188#41—622no—
189#41—622no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—311no—
193C2×Q8D12414yescomplemented
194C2×Q8C22×D3112yes—
195C2×D4—314no—
196C2×D4—312no—
197C2×D4—311no—
198C2×D4—624no—
199C2×D4—628no—
200C2×D4—622no—
201C2×D4—621no—
202C2×D4—622no—
203C2×D4—621no—
204C22×C4—21no—
205C22×C4—311no—
206C22×C4—312no—
207C22×C4—312no—
208C22×C4—624no—
209C22×C4—628no—
210C22×C4—622no—
211C22×C4—622no—
212C22×C4—621no—
213C22×C4C22×D3111yescharacteristic
214Q16—624no—
215SD16—624no—
216SD16—624no—
217D8—624no—
218M16—22no—
219M16—622no—
220C2×C8—312no—
221C2×C8C22×D3112yes—
222C4⋊C4—622no—
223C4⋊C4C22×D3112yes—
224C22⋊C4—21no—
225C22⋊C4—622no—
226C22⋊C4—622no—
227C42—21no—
228C23—312no—
229C23—624no—
230C23—622no—
231C23—622no—
232C23—621no—
233C23—1242no—
234C23—1244no—
235C23D4×D3111yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—621no—
239Q8—621no—
240Q8C2×D12414yes—
241D4—314no—
242D4—6216no—
243D4—624no—
244D4—622no—
245D4—621no—
246D4—621no—
247D4—1244no—
248D4—1242no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—312no—
253C2×C4—6216no—
254C2×C4—622no—
255C2×C4—624no—
256C2×C4—622no—
257C2×C4—622no—
258C2×C4—621no—
259C2×C4—1242no—
260C2×C4C2×D12412yes—
261C2×C4D4×D3111yescharacteristic
262C2×C4D4×D3112yes—
263C2×C4C23×D3111yescharacteristic
264C8—22no—
265C8—622no—
266C22—22no—
267C22—624no—
268C22—622no—
269C22—622no—
270C22—622no—
271C22—1244no—
272C22—1248no—
273C22—1244no—
274C22—1241no—
275C22—1242no—
276C22#1175845912yes—
277C22C2×D4×D3111yescharacteristic
278C4—24no—
279C4—42no—
280C4—622no—
281C4—1241no—
282C4#1175845912yes—
283C4C22×D12411yescharacteristic
284C4C2×D4×D3111yescharacteristic
285C2—22no—
286C2—1242no—
287C2—1244no—
288C2—2481no—
289C2C2×#1175845911yescharacteristic
290C2#1452184812yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#419C2
socleC22×C31C22×D4
hypercenter#419D31
hypocenterC31#3005
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#92499C2semidirect
2. C2×#11758519C2semidirect
3. C2×#11758485C2semidirect
4. #14521561C2semidirect
5. #14521214C2semidirect
6. #14521197C2semidirect
7. #14521053C2semidirect
8. #14521012C2semidirect
9. C31×#419C2semidirect
22. C2×#92482C22semidirect
24. C2×#92481C22semidirect
27. C2×D248C22semidirect
28. C2×#92470C22semidirect
29. #11758474C22semidirect
30. #11758472C22semidirect
34. C31×#103C22semidirect
35. (C2×D31)⋊C8C22semidirect
36. #11758397C22semidirect
41. C2×Q8×C31D4semidirect
57. C2×D124D4semidirect
146. #419D31semidirect
179. #103C2×D31semidirect
182. C31#3005semidirect
193. C2×Q8D124semidirect

Maximal subgroups

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

subgroupindex
C22×#924992
C2×#117585192
C2×#117584852
#145215612
#145212142
#145211972
#145210532
#145210122
C31×#4192
#300531