Finite Group Databasebeta

← #16854973

Subgroups — #16854973

Order 3968.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
2C2×#11758486C212yesmaximal, complemented
3C2×#11758501C211yesmaximal, characteristic, complemented
4#14521213C214yesmaximal, complemented
5#14521207C212yesmaximal, complemented
6#14521202C212yesmaximal, complemented
7#14521051C212yesmaximal, complemented
8C31×#407C211yesmaximal, characteristic, complemented
9C23×C4×C31C2211yescharacteristic
10C22×C22⋊D31—21no—
11C2×#92495—21no—
12C2×#92495—21no—
13C2×#92495—28no—
14C2×#92495C2212yes—
15C2×#92495C2212yes—
16C22×D124C2211yescharacteristic
17C22×C4×D31—21no—
18C22×Dic62C2211yescharacteristic
19C22×Dic62C2211yescharacteristic, complemented
20C2×M16×C31C2212yescomplemented
21C2×C22⋊Dic31—21no—
22C2×C4⋊Dic31C2211yescharacteristic, complemented
23#11758501—21no—
24#11758501—21no—
25#11758501C2212yes—
26#11758501C2212yescomplemented
27C2×Dic31⋊C4—21no—
28#11758486—28no—
29C2×Dic124C2214yescomplemented
30C2×#92470C2214yescomplemented
31C31×C22⋊C8C2212yes—
32C31×C22⋊C8C2212yescomplemented
33#11758409C2214yescomplemented
34#11758404C2214yescomplemented
35C24×C31D411yescharacteristic
36C23×D31—21no—
37C22×C4×C31—24no—
38C22×C4×C31D411yescharacteristic
39C22×C4×C31D411yescharacteristic
40C22×C4×C31D411yescharacteristic
41C22×C4×C31C2311yescharacteristic
42C22×C4×C31C2311yescharacteristic
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—28no—
45C2×C22⋊D31—22no—
46C2×C22⋊D31—22no—
47C22×Dic31—21no—
48C22×Dic31—21no—
49C22×Dic31—21no—
50#92495—28no—
51#92495—28no—
52#92495—24no—
53#92495—24no—
54#92495—24no—
55#92495D418yes—
56C2×D124—21no—
57C2×D124—24no—
58C2×D124C2312yes—
59C2×C4×D31—28no—
60C2×C4×D31—22no—
61C2×C4×D31—22no—
62C2×Dic62—21no—
63C2×Dic62—24no—
64C2×Dic62—24no—
65C2×Dic62—21no—
66C2×Dic62C2312yes—
67C2×Dic62C2312yes—
68M16×C31—24no—
69C2×C8×C31C2314yes—
70C22⋊Dic31—22no—
71C22⋊Dic31—22no—
72C4⋊Dic31—21no—
73C4⋊Dic31C2312yes—
74Dic31⋊C4—22no—
75Dic31⋊C4—22no—
76Dic124—28no—
77#92470—28no—
78C23×C31—22no—
79C23×C31—22no—
80C23×C31—41no—
81C23×C31C2×D411yescharacteristic
82C23×C31C2×D411yescharacteristic
83C23×C31C2×D411yescharacteristic
84C22×D31—24no—
85C22×D31—22no—
86C22×D31—42no—
87C22×D31—42no—
88C2×C4×C31—22no—
89C2×C4×C31—22no—
90C2×C4×C31—22no—
91C2×C4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2×D411yescharacteristic
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2×D414yes—
97C2×C4×C31C2411yescharacteristic
98C22⋊D31—28no—
99C22⋊D31—28no—
100C2×Dic31—24no—
101C2×Dic31—22no—
102C2×Dic31—22no—
103C2×Dic31—22no—
104C2×Dic31—42no—
105C2×Dic31—42no—
106D124—24no—
107D124—22no—
108D124C2×D414yes—
109C4×D31—28no—
110C4×D31—28no—
111Dic62—24no—
112Dic62—22no—
113Dic62—24no—
114Dic62—42no—
115Dic62C2×D414yes—
116C8×C31—24no—
117#3007—311nomaximal, Sylow
118C22×C31—21no—
119C22×C31—21no—
120C22×C31—21no—
121C22×C31—21no—
122C22×C31—21no—
123C22×C31—21no—
124C22×C31—21no—
125C22×C31—44no—
126C22×C31#12012yes—
127C22×C31#13712yescomplemented
128C22×C31C22×D411yescharacteristic
129C2×D31—24no—
130C2×D31—48no—
131C2×D31—42no—
132C2×D31—42no—
133C4×C31—21no—
134C4×C31—21no—
135C4×C31#12012yes—
136C4×C31C22×D412yes—
137Dic31—24no—
138Dic31—42no—
139Dic31—42no—
140C22×#41—311no—
141C2×#137—312no—
142C2×C22⋊Q8—311no—
143#452—314no—
144#451—312no—
145#450—312no—
146#417—312no—
147#407D3111yescharacteristic, complemented
148C2×C31—22no—
149C2×C31—22no—
150C2×C31—41no—
151C2×C31C2×#12011yescharacteristic
152C2×C31C2×#13711yescharacteristic
153C2×C31#57511yescharacteristic
154D31—44no—
155C2×#41—312no—
156C2×#41—312no—
157C2×#41—628no—
158C2×#41—621no—
159C2×#41—621no—
160C22×Q8—311no—
161C22×Q8—311no—
162C22×D4—311no—
163C22×D4—621no—
164C23×C4—621no—
165C23×C4C2×D3111yescharacteristic
166#137—628no—
167C2×Q16—314no—
168C2×SD16—314no—
169C2×M16C2×D3112yescomplemented
170C22⋊Q8—312no—
171C22⋊Q8—312no—
172C22⋊Q8—621no—
173C22⋊Q8—621no—
174C2×C4⋊C4—311no—
175C2×C4⋊C4—621no—
176C2×C22⋊C4—621no—
177#103—314no—
178#102—314no—
179C22⋊C8C2×D3112yes—
180C22⋊C8C2×D3112yescomplemented
181C31#300711yescharacteristic, complemented, Sylow
182C24—621no—
183C24D12411yescharacteristic
184#41—318no—
185#41—624no—
186#41—624no—
187#41—628no—
188#41—628no—
189#41—624no—
190C2×Q8—312no—
191C2×Q8—312no—
192C2×Q8—624no—
193C2×Q8—621no—
194C2×Q8—621no—
195C2×Q8—624no—
196C2×D4—312no—
197C2×D4—624no—
198C2×D4—628no—
199C2×D4—622no—
200C2×D4—622no—
201C2×D4—621no—
202C22×C4—24no—
203C22×C4—628no—
204C22×C4—621no—
205C22×C4—622no—
206C22×C4—622no—
207C22×C4—621no—
208C22×C4—621no—
209C22×C4D12411yescharacteristic
210C22×C4D12411yescharacteristic
211C22×C4D12411yescharacteristic
212C22×C4C22×D3111yescharacteristic
213C22×C4C22×D3111yescharacteristic
214C22×C4C22×D3111yescharacteristic
215Q16—628no—
216SD16—628no—
217M16—24no—
218C2×C8C22×D3114yes—
219C4⋊C4—312no—
220C4⋊C4—622no—
221C4⋊C4—622no—
222C4⋊C4—621no—
223C22⋊C4—622no—
224C22⋊C4—622no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—624no—
229C23—622no—
230C23—1242no—
231C23—1242no—
232C23C2×D12411yescharacteristic
233C23C2×D12411yescharacteristic
234C23C2×D12411yescharacteristic
235Q8—314no—
236Q8—624no—
237Q8—622no—
238Q8—624no—
239Q8—1242no—
240D4—314no—
241D4—624no—
242D4—628no—
243D4—628no—
244D4—622no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—628no—
251C2×C4—624no—
252C2×C4—622no—
253C2×C4—628no—
254C2×C4—622no—
255C2×C4—622no—
256C2×C4—1242no—
257C2×C4—1242no—
258C2×C4C2×D12411yescharacteristic
259C2×C4C2×D12411yescharacteristic
260C2×C4C2×D12411yescharacteristic
261C2×C4D4×D3114yes—
262C2×C4C23×D3111yescharacteristic
263C8—24no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—624no—
273C22—1248no—
274C22—1242no—
275C22—1242no—
276C22#1175845912yes—
277C22#1175848612yescomplemented
278C22C22×D12411yescharacteristic
279C4—21no—
280C4—21no—
281C4—624no—
282C4—1242no—
283C4—1242no—
284C4#1175845912yes—
285C4C2×D4×D3112yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—1244no—
290C2C2×#1175845911yescharacteristic
291C2C2×#1175848611yescharacteristic
292C2#1452184911yescharacteristic
subgroupG/H
centerC22C22×D124
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#407C2
socleC22×C31C22×D4
prime radicalC22×#92495C2
hypercenter#407D31
hypocenterC31#3007
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
2. C2×#11758486C2semidirect
3. C2×#11758501C2semidirect
4. #14521213C2semidirect
5. #14521207C2semidirect
6. #14521202C2semidirect
7. #14521051C2semidirect
8. C31×#407C2semidirect
19. C22×Dic62C22semidirect
20. C2×M16×C31C22semidirect
22. C2×C4⋊Dic31C22semidirect
26. #11758501C22semidirect
29. C2×Dic124C22semidirect
30. C2×#92470C22semidirect
32. C31×C22⋊C8C22semidirect
33. #11758409C22semidirect
34. #11758404C22semidirect
127. C22×C31#137semidirect
147. #407D31semidirect
169. C2×M16C2×D31semidirect
180. C22⋊C8C2×D31semidirect
181. C31#3007semidirect
277. C22#11758486semidirect

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×#117585012
#145212132
#145212072
#145212022
#145210512
C31×#4072
#300731